One-Step Estimation of Differentiable Hilbert-Valued Parameters

Abstract

We present estimators for smooth Hilbert-valued parameters, where smoothness is characterized by a pathwise differentiability condition. When the parameter space is a reproducing kernel Hilbert space, we provide a means to obtain efficient, root-n rate estimators and corresponding confidence sets. These estimators correspond to generalizations of cross-fitted one-step estimators based on Hilbert-valued efficient influence functions. We give theoretical guarantees even when arbitrary estimators of nuisance functions are used, including those based on machine learning techniques. We show that these results naturally extend to Hilbert spaces that lack a reproducing kernel, as long as the parameter has an efficient influence function. However, we also uncover the unfortunate fact that, when there is no reproducing kernel, many interesting parameters fail to have an efficient influence function, even though they are pathwise differentiable. To handle these cases, we propose a regularized one-step estimator and associated confidence sets. We also show that pathwise differentiability, which is a central requirement of our approach, holds in many cases. Specifically, we provide multiple examples of pathwise differentiable parameters and develop corresponding estimators and confidence sets. Among these examples, four are particularly relevant to ongoing research by the causal inference community: the counterfactual density function, dose-response function, conditional average treatment effect function, and counterfactual kernel mean embedding.

1. Introduction

There has been much recent work on combining tools from semiparametric efficiency and machine learning to estimate finite-dimensional parameters (Baiardi and Naghi, 2021; Kennedy, 2022; Hines et al., 2022). These works often focus on pathwise differentiable parameters, which are characterized by their smoothness along regular univariate submodels of the statistical model (Pfanzagl, 1990; van der Vaart, 1991; Bickel et al., 1993). When a finite-dimensional parameter is pathwise differentiable, it also has an efficient influence function (EIF), which corresponds to the Riesz representation of its pathwise derivative. Efficient influence functions are the critical ingredient used to define various estimation strategies, such as those based on one-step estimation (Pfanzagl, 1982; Newey and McFadden, 1994), estimating equations (van der Laan et al., 2003; Tsiatis, 2006), targeted learning (van der Laan and Rubin, 2006; van der Laan et al., 2011), and double machine learning (Chernozhukov et al., 2017, 2018). When paired with cross-fitting (Schick, 1986; Klaassen, 1987), these frameworks yield asymptotically efficient estimators provided the nuisance functions are estimated well enough as the sample size n grows to make a certain remainder term negligible. Often, this amounts to requiring an n^−1/4-rate condition, which will most plausibly hold if the nuisance functions are estimated flexibly.

Another line of research has focused on leveraging machine learning tools to estimate function-valued parameters, such as the causal dose-response function (Díaz and van der Laan, 2013), counterfactual density function (Kennedy et al., 2021), and conditional average treatment effect function (Nie and Wager, 2021). Possibly owing to the wealth of available methods for estimating real-valued functionals, many of these works have focused on the evaluation of these functions at a point. As has been noted in van der Laan et al. (2018) and Chernozhukov et al. (2018), the resulting point evaluations tend not to be pathwise differentiable except in trivial cases (e.g., when the data are discrete). To overcome this challenge, kernel-smoothed approximations of the function evaluation parameter have been considered in those two works and others (Colangelo and Lee, 2020; Luedtke and Wu, 2020; Chernozhukov et al., 2021; Jung et al., 2021), and local polynomial approximations have also been introduced for several parameters (Kennedy et al., 2017; Takatsu and Westling, 2022; Kennedy et al., 2022). These smoothed approximations tend to yield pathwise differentiable parameters, which enables the use of one-step estimators. Slower-than-n^1/2 convergence rates are typically attained because the fineness of the approximation must improve with sample size. The guarantees provided for these estimators tend to be pointwise in nature. Given that pointwise convergence does not generally imply norm convergence or uniform convergence without additional regularity conditions, these pointwise-based estimators usually only facilitate inference for the evaluation of the unknown function at one or finitely many points, rather than for the entire function.

Some works have focused on estimating unknown function-valued parameters in a norm sense. Many of these works incorporate objects from semiparametric efficiency theory. For example, in the context of conditional average treatment effect estimation, risk functions have been developed (van der Laan, 2006; Luedtke and van der Laan, 2016; Nie and Wager, 2021), and Kennedy (2020) develops rate-of-convergence guarantees for the corresponding empirical risk minimizers. These estimators incorporate (weighted) variants of the EIF of the marginal average treatment effect in their construction. As further examples, EIFs have been used to construct norm-convergent estimators of the counterfactual density function (Kennedy et al., 2021) and dose-response function (Takatsu and Westling, 2022). A drawback to these approaches to estimating function-valued parameters is that, to date, it has seemed that a new estimator must be derived and new regularity conditions established for each new parameter considered. Others have presented general approaches to learning unknown functions based on empirical risk minimization, where the population risk depends on unknown nuisance functions that can be orthogonalized by conducting statistical learning using an efficient estimator of the risk function as an objective function (van der Laan and Dudoit, 2003; Foster and Syrgkanis, 2019). When the population regret takes the form of a squared norm, these approaches provide a means to derive estimators with norm-convergence guarantees. However, unlike standard approaches such as one-step-estimation that are used for estimating finite-dimensional quantities, these methods do not appear to easily lend themselves to the construction of confidence sets for the unknown functions. Instead, the available approaches to construct confidence sets rely on approaches that are generally distinct from those used for estimating the function, such as building them using higher-order influence functions (Robins et al., 2008), a restricted score test (Hudson et al., 2021), or a maximum mean discrepancy (MMD) criterion (Luedtke et al., 2019).

In this work, we establish that the one-step estimation methodology can be extended to estimate and make inference about pathwise differentiable parameters that take values in a Hilbert space. This pathwise differentiability condition turns out to be quite reasonable for many function-valued parameters of current interest. Indeed, we show that all of the parameters mentioned earlier in this Introduction satisfy it under regularity conditions. This is true in spite of the fact that these Hilbert-valued parameters are not pathwise differentiable when composed with an evaluation map.

The notion of pathwise differentiability that we focus on in this work is that studied in some early literature on semiparametric efficiency theory, which defined pathwise differentiability and EIFs not just for finite-dimensional parameters, but for general Banach-valued parameters (van der Vaart and Wellner, 1989; van der Vaart, 1991; page 179 of Bickel et al., 1993). Since all Hilbert spaces are Banach spaces, their definitions apply in our case, as do some useful results that they present, such as a convolution theorem. Nevertheless, existing works did not provide any examples of how to evaluate the pathwise differentiability of infinite-dimensional Hilbert-valued parameters — for example, see van der Vaart (1991) and Chapter 5.3 of Bickel et al. (1993), whose infinite-dimensional examples all pertain to parameters taking values in a Banach space equipped with the uniform norm. Since they are not even pathwise differentiable at a point, none of the aforementioned function-valued parameters are pathwise differentiable in such a Banach space. Previous works also do not indicate whether or how the pathwise differentiability of an infinite-dimensional Hilbert-valued parameter can be used to facilitate estimation or inference, whether via the one-step estimation methodology or otherwise. While a brief, one-paragraph sketch was given on page 405 of Bickel et al. (1993) suggesting that providing a general, efficient n^−1/2-rate estimation framework may be difficult for infinite-dimensional spaces, this sketch only discusses a single example where n^−1/2-rate estimation may not even be possible. Moreover, neither that work, nor any subsequent ones, appear to evaluate whether leveraging the pathwise differentiability of a Hilbert-valued parameter would be useful for constructing a performant, but slower than n^−1/2-rate, estimator, or for constructing a confidence set.

The main contributions of this work are as follows:

1. We characterize the EIF of a pathwise differentiable Hilbert-valued parameter, when it exists, and provide a means to obtain a regularized version thereof, when it does not.

2. We construct one-step estimators using (possibly regularized) EIFs. Any method can be used to estimate the needed nuisance functions provided it converges at a suitable rate.

3. We provide root-n-rate weak convergence and efficiency guarantees for these estimators, when an EIF exists, and slower rate-of-convergence guarantees, when one does not.

4. We show how to construct asymptotically-valid confidence sets for the Hilbert-valued estimands. These confidence sets take different forms depending on whether an EIF exists.

5. We study our framework in examples of current interest to the causal inference community and establish the pathwise differentiability of several more traditional parameters.

When the estimand is a function, our confidence sets will contain it with a specified probability. Thus, if the aim is to infer about the whole function, our methodology is likely preferable to pointwise approaches. To accomplish the last point above, we derive a general lemma that facilitates the evaluation of the pathwise differentiability of Hilbert-valued parameters. Finally, we have conducted a simulation study to evaluate the proposed approach. All proofs can be found in the appendix.

2. Pathwise differentiability in Hilbert spaces and constructing estimators

2.1. Notation

We work on a Polish space ($𝒵$, $\mathbf{B}$) with a collection of distributions $𝒫$, which we refer to as the model. Let $Z_1$, $Z_2,\cdots ,Z_n\sim P_0$ be an independent and identically distributed (iid) sample from a distribution $P_0\in 𝒫$, and let $P_n$ denote the corresponding empirical distribution. Let ${\widehat{P}}_n\in 𝒫$ be an estimate of $P_0$. To ease notation, for now we consider a sample splitting approach wherein ${\widehat{P}}_n$ is fitted using an iid sample that is independent of $Z_1$, $Z_2,\cdots ,Z_n$; in Section 2.5, we describe the case where cross-fitting is used (Schick, 1986; Klaassen, 1987), which is our preferred approach. For $Q$ a signed measure on ($𝒵$, $\mathbf{B}$) and a measurable function $f:𝒵\to \mathbb{R}$, we use the shorthand $Qf≔\int fdQ$. For any object indexed by $P_0$, we will abbreviate the notation by replacing ‘$P_0$’ by ‘0’; for example, we will write $f_0$ rather than $f_{P_0}$. Similarly, we will replace ‘${\widehat{P}}_n$’ by ‘$n$’ and write $f_n$ rather than $f_{{\widehat{P}}_n}$.

All Hilbert spaces mentioned in this paper are real Hilbert spaces. For a measure $\mu$ on a measurable space ($𝒳$, $\Sigma$), we write $L^2(\mu )$ to denote the Hilbert space of $\mu$-a.s. equivalence classes of $𝒳\to \mathbb{R}$ functions equipped with inner product ${\langle f,g\rangle }_{L^2(\mu )}≔\int fgd\mu$. If $𝒳\subset {\mathbb{R}}^d$, $\mu$ is the Lebesgue measure, and $\Sigma$ is the Borel $\sigma$-algebra on ${\mathbb{R}}^d$, we will sometimes write $L^2(𝒳)$ instead of $L^2(\mu )$. In what follows $𝒱$ denotes a generic Hilbert space. We let ${\Vert \cdot \Vert }_{𝒱}$ and ${\langle \cdot ,\cdot \rangle }_{𝒱}$ denote the norm and inner product associated with $𝒱$. The space $L^2(P;𝒱)$ is the Hilbert space containing all Bochner measurable functions $f:𝒵\to 𝒱$ such that

$${\Vert f\Vert }_{L^2(P,𝒱)}≔{\left(\int {\Vert f(z)\Vert }_{𝒱}^{2}P(dz)\right)}^{1∕2}<\infty .$$

The operator norm of a linear functional $f:𝒱\to \mathbb{R}$ is defined as ${\Vert f\Vert }_{\text{op}}≔\text{inf}\{c\ge 0:\mid f(v)\mid \le c{\Vert v\Vert }_{𝒱}$ for all $v\in 𝒱$}. If $𝒲$ is a closed subspace of $𝒱$, then let ${\Pi }_{𝒱}[h\mid 𝒲]$ denote the orthogonal projection of $h$ to $𝒲$. We let ${\ell }^2$ denote the space of all square-summable sequences and ${\Vert b\Vert }_{{\ell }^2}≔{[{\sum }_{k=1}^{\infty }{b}_k^2]}^{1∕2}$. We also let ${[0,1]}^{\mathbb{N}}$ denote the space of all [0, 1]-valued sequences. To avoid having to use different notation to treat finite- and infinite-dimensional Hilbert spaces, throughout we use the convention that, if $𝒱$ is finite-dimensional, then we call ${(v_k)}_{k=1}^{\infty }$ an orthonormal basis of $𝒱$ if ${(v_k)}_{k=1}^{\text{dim}(𝒱)}$ is an orthonormal system that spans $𝒱$ and $v_k=0$ for all $k>\text{dim}(𝒱)$.

2.2. Pathwise differentiability in Hilbert spaces

We start with a brief review of important definitions that can be used to characterize the smoothness of a Hilbert-valued parameter. These definitions are adapted from those given in (Bickel et al., 1993) for more general Banach-valued parameter settings. The subsequent parts of this section will involve developing estimators for our more specialized, but understudied, setting, where we heavily leverage the availability of an inner product in our Hilbert parameter space.

Let $𝒫$ be a collection of distributions defined on a common Polish space ($𝒵$, $\mathbf{B}$), which we refer to as the model. Suppose that the model is dominated by a $\sigma$-finite measure $\lambda$. A submodel $\{P_{ϵ}:ϵ\in [0,\delta )\}\subset 𝒫$ is said to be quadratic mean differentiable at $P$ if and only if there exists a score function $s\in L^2(P)$ such that

$${\Vert p_{ϵ}^{1∕2}-p^{1∕2}-ϵs{p}^{1∕2}∕2\Vert }_{L^2(\lambda )}=o(ϵ),$$ (1)

where, for $ϵ\ge 0$, $p_{ϵ}^{1∕2}=\sqrt{\frac{d{P}_{ϵ}}{d\lambda }}$ and $p^{1∕2}=\sqrt{\frac{dP}{d\lambda }}$. Let $𝒫(P,𝒫,s)$ refer to the set of quadratic mean differentiable submodels at $P$ with score function $s$. The set $\{s\in L^2(P):𝒫(P,𝒫,s)\ne \varnothing \}$ is called the tangent set, and its closed linear span is called the tangent space of $𝒫$ at $P$, denoted by ${\stackrel{.}{𝒫}}_P$. For all $s\in {\stackrel{.}{𝒫}}_P$, $Ps=\int sdP=0$. We let $L_0^2(P)≔\{h\in L^2(P):Ph=0\}$, which is the largest possible tangent space at $P$. Any model with this tangent space at all distributions $P$ it contains is referred to as locally nonparametric.

Let $\mathscr{H}$ be a set known as the action space and $\nu :𝒫\to \mathscr{H}$ a parameter whose value is to be estimated. Throughout we assume that $\mathscr{H}$ is a real separable Hilbert space. The parameter $\nu$ is said to be pathwise differentiable at $P$ if and only if there exists a continuous linear operator ${\stackrel{.}{\nu }}_P:{\stackrel{.}{𝒫}}_P\to \mathscr{H}$ such that, for all $\{P_{ϵ}:ϵ\in [0,\delta )\}\in 𝒫(P,𝒫,s)$,

$${\Vert \nu (P_{ϵ})-\nu (P)-ϵ{\stackrel{.}{\nu }}_P(s)\Vert }_{\mathscr{H}}=o(ϵ).$$ (2)

The operator ${\stackrel{.}{\nu }}_P$ is called the local parameter of $\nu$ at $P$ and its Hermitian adjoint, denoted by ${\stackrel{.}{\nu }}_P^{\ast }:\mathscr{H}\to {\stackrel{.}{𝒫}}_P$, is referred to as the efficient influence operator. The image of the local parameter ${\stackrel{.}{\nu }}_P$, denoted by ${\stackrel{.}{\mathscr{H}}}_P$, is a closed subspace of $\mathscr{H}$ that is referred to as the local parameter space. Throughout we equip ${\stackrel{.}{\mathscr{H}}}_P$ with the inner product ${\langle \cdot ,\cdot ,\rangle }_{\mathscr{H}}$, so that ${\stackrel{.}{\mathscr{H}}}_P$ is itself a Hilbert space. The efficient influence operator can be shown to only depend on its argument through its projection onto the local parameter space, in the sense that ${\stackrel{.}{\nu }}_P^{\ast }(h)={\stackrel{.}{\nu }}_P^{\ast }({\Pi }_{\mathscr{H}}[h\mid {\stackrel{.}{\mathscr{H}}}_P])$ for all $h\in \mathscr{H}$. At times in this work, we will consider pointwise evaluations of the efficient influence operator of the form ${\stackrel{.}{\nu }}_P^{\ast }(h)(z)$. When doing so, we always assume that suitably ‘nice’ elements of the $P$-a.s. equivalence classes defined by the elements ${\stackrel{.}{\nu }}_P^{\ast }(h)$ of $L^2(P)$ are used to define these evaluations. In particular, we select these elements so that the efficient influence process, which we define as $\{{\stackrel{.}{\nu }}_P^{\ast }(h):h\in \mathscr{H}\}$, is a separable stochastic process, in the sense that there exists a countable dense subset ${\mathscr{H}}^{\prime }$ of $\mathscr{H}$ and a $P$-probability one subset ${𝒵}^{\prime }$ of $𝒵$ such that, for all $h\in \mathscr{H}$ and $z\in {𝒵}^{\prime }$, there exists an ${\mathscr{H}}^{\prime }$-valued sequence ${(h_j)}_{j=1}^{\infty }$ that converges to $h$ and satisfies ${\stackrel{.}{\nu }}_P^{\ast }(h_j)(z)\to {\stackrel{.}{\nu }}_P^{\ast }(h)(z)$ as $j\to \infty$.

Analogous to the case for Euclidean parameters, in some semiparametric models it may be natural to describe a Hilbert-valued parameter as the restriction of a parameter defined on a larger, possibly nonparametric, model. If the true parameter lies in a model ${𝒫}^{\prime }\subset 𝒫$ with tangent space ${\stackrel{.}{𝒫}}_P^{\prime }$ and $\nu :𝒫\to \mathscr{H}$ is a parameter defined on $𝒫$, then the restriction ${\nu \mid }_{{𝒫}^{\prime }}$ has local parameter ${\stackrel{.}{\nu }\mid }_{{\stackrel{.}{𝒫}}_P^{\prime }}$ and efficient influence operator $h\mapsto {\Pi }_{L^2(P)}[{\stackrel{.}{\nu }}_P^{\ast }(h)\mid {\stackrel{.}{𝒫}}_P^{\prime }]$. Armed with this fact, results can easily be transferred from a larger nonparametric model to a semiparametric model provided the form of the projection operator ${\Pi }_{L^2(P)}[\cdot \mid {\stackrel{.}{𝒫}}_P^{\prime }]$ is known. As a simple example, we may have that $\nu (P)(\cdot )=E_P[Y\mid X=\cdot ]$ for each $P$ in a nonparametric model $𝒫$, and the model ${𝒫}^{\prime }=\{P\in 𝒫:{\text{var}}_P(Y)=1\}$ may reflect knowledge that the variance of an outcome $Y$ is 1. The form of the local parameter and efficient influence operator of $\nu$ relative to $𝒫$ are given in Example 5 in the appendix when $\mathscr{H}$ is an $L^2$ space, and the form of the projection onto ${\stackrel{.}{𝒫}}_P^{\prime }$ is given in Example 3.2.3 of Bickel et al. (1993).

The parameter $\nu$ is said to have an EIF ${\varphi }_P:𝒵\to \mathscr{H}$ when there exists a $P$-probability-one set ${𝒵}^{\prime }$ such that

$${\stackrel{.}{\nu }}_P^{\ast }(h)(z)={\langle h,{\varphi }_P(z)\rangle }_{\mathscr{H}}\text{for all}(h,z)\in \mathscr{H}\times {𝒵}^{\prime }.$$ (3)

By the Riesz representation theorem, $\nu$ has an EIF if and only if ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z):\mathscr{H}\to \mathbb{R}$ is a bounded linear functional $P$-almost surely; in those cases, ${\varphi }_P(z)$ is $P$-a.s. equal to the Riesz representation of ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z)$. The fact that ${\stackrel{.}{\nu }}_P^{\ast }(h)={\stackrel{.}{\nu }}_P^{\ast }({\Pi }_{\mathscr{H}}[h\mid {\stackrel{.}{\mathscr{H}}}_P])$ implies that the image of ${\varphi }_P$ is necessarily contained in ${\stackrel{.}{\mathscr{H}}}_P$. When $\mathscr{H}={\mathbb{R}}^d$, the EIF of $\nu$ at $P$ takes the form ${\varphi }_P(z)={({\stackrel{.}{\nu }}_P^{\ast }(e_t)(z))}_{t=1}^d$, where ${\{e_t\}}_{t=1}^d$ is the standard basis. To our knowledge, the existence and form of this object have not previously been studied in infinite-dimensional Hilbert spaces. Given that knowing the form of the EIF readily facilitates the construction of estimators in finite-dimensional settings, this appears to constitute an important gap in the literature. We therefore focus the remainder of this section on studying the existence of EIFs in Hilbert spaces and providing ways to construct estimators based on EIFs, when we can show they exist, or imitations thereof, when we cannot.

The cases where EIFs do not exist become particularly salient in Section 3, where we demonstrate through examples that, for several interesting $L^2$-valued parameters, ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z):\mathscr{H}\to \mathbb{R}$ depends on a point evaluation functional, and so is not bounded $P$-almost surely. Nevertheless, even when $\nu$ does not have an EIF, we will show in Section 5.2 that it is always possible to define an injective transformation of $\nu$ that has one. Consequently, the procedure we shall present to construct confidence sets for parameters with EIFs can also be used to construct them for those without one: first construct a confidence set for the transformation of $\nu (P_0)$, and then invert it to obtain one for $\nu (P_0)$.

Before proceeding, we note that inefficient influence operators can be defined when the model is semiparametric at $P$, in that ${\stackrel{.}{𝒫}}_P$ is a strict subspace of $L_0^2(P)$. Under a condition akin to (3), inefficient influence functions can also be defined. To streamline presentation, we defer the presentation of these objects and their use for constructing estimators to Appendix H. There, we also show that inefficient influence functions can only exist if an EIF exists.

2.3. One-step estimation based on the efficient influence function

On the one hand, if $\mathscr{H}$ is finite-dimensional, then the EIF can be used to construct what is known as a one-step estimator, which is known to be efficient under conditions (Pfanzagl, 1982). This estimator takes the form $\nu ({\widehat{P}}_n)+P_n{\varphi }_n$, where ${\widehat{P}}_n\in 𝒫$ is an initial estimate of the data-generating distribution $P_0$ and we recall the convention that ${\varphi }_n≔{\varphi }_{{\widehat{P}}_n}$. On the other hand, if $\mathscr{H}$ is infinite-dimensional, then previously studied one-step estimators cannot be applied. In this section, we provide a natural means to extend the one-step estimation framework to infinite-dimensional settings. Similarly to the one-step estimator in finite-dimensional settings, this one-step estimator takes the form ${\widehat{\nu }}_n≔\nu ({\widehat{P}}_n)+P_n{\varphi }_n$ for an $\mathscr{H}$-valued EIF ${\varphi }_n$. This estimator is applicable whenever $\nu$ has an EIF at ${\widehat{P}}_n$ with $P_0$-probability one. We will see that, under conditions that include that $\nu$ also has an EIF ${\varphi }_0$ at $P_0$, ${\widehat{\nu }}_n$ is an asymptotically linear estimator of $\nu (P_0)$ with influence function ${\varphi }_0$, in the sense that

$${\widehat{\nu }}_n-\nu (P_0)=\frac{1}{n}{\sum }_{i=1}^n{\varphi }_0(Z_i)+o_p(n^{-1∕2}),$$ (4)

where throughout we let Hilbert-valued quantities of the form $o_p(n^{-a})$ denote terms whose Hilbert norm goes to zero in probability even after being multiplied by $n^a$. We will be especially interested in cases where $n^{1∕2}[{\widehat{\nu }}_n-\nu (P_0)]$ will converge weakly to a tight random element, since this can be used to facilitate the construction of confidence sets for $\nu (P_0)$ (see Section 4.2). To be able to apply a central limit theorem to establish the weak convergence of $n^{1∕2}[{\widehat{\nu }}_n-\nu (P_0)]$, it suffices that ${\varphi }_0$ is $P_0$-Bochner square integrable, in the sense that ${\Vert {\varphi }_0\Vert }_{L^2(P_0;\mathscr{H})}<\infty$ (Example 1.8.5 of van der Vaart and Wellner, 1996). Therefore, we will focus on settings where ${\varphi }_0\in L^2(P_0;\mathscr{H})$.

We begin by establishing the existence and form of the EIF at a generic $P\in 𝒫$ in an interesting class of problems. In particular, we consider cases where $\mathscr{H}$ is an RKHS over a space $𝒯$ or, more generally, the local parameter space ${\stackrel{.}{\mathscr{H}}}_P$ is an RKHS over $𝒯$. Denote the feature map of ${\stackrel{.}{\mathscr{H}}}_P$ by $t\mapsto K_t$. For $P\in 𝒫$, define ${\tilde{\varphi }}_P:𝒵\to \mathscr{H}$ as follows for each $t\in 𝒯$:

$${\tilde{\varphi }}_P(z)(t)={\stackrel{.}{\nu }}_P^{\ast }(K_t)(z)P-\text{a.s}.z.$$ (5)

The following result shows that ${\tilde{\varphi }}_P$ both provides the form of the EIF of $\nu$, when it exists, and also a sufficient condition that can be used to verify this existence.

Theorem 1 (Form of the efficient influence function in RKHS settings). Suppose $\nu$ is pathwise differentiable at $P$ and ${\stackrel{.}{\mathscr{H}}}_P$ is an RKHS. Both of the following implications hold:

1. If $\nu$ has an EIF ${\varphi }_P$ at $P$, then ${\varphi }_P={\tilde{\varphi }}_P$ $P$-almost surely.

2. If ${\Vert {\tilde{\varphi }}_P\Vert }_{L^2(P;\mathscr{H})}<\infty$, then $\nu$ has an EIF at $P$.

The form of the EIF in (5) naturally generalizes its form in finite-dimensional spaces, where the feature $K_t$ replaces the $t$-th standard basis element $e_t$. The proof of (i) is a straightforward extension of results about the Riesz representation of a bounded linear functional to our setting, where ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z)$ is only known to be bounded and linear $P$-almost surely (cf. Lemma 10 of Berlinet and Thomas-Agnan, 2011). The proof of (ii) is more subtle, and involves showing that, when ${\Vert {\tilde{\varphi }}_P\Vert }_{L^2(P;\mathscr{H})}<\infty$, any separable version of the efficient influence process must $P$-a.s. have sample paths ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z)$ that are both bounded and linear. In the remainder of this subsection, we suppose that $\nu$ has an EIF ${\varphi }_P$ at each $P\in 𝒫$.

In Section 4.1, we establish that, under conditions, a cross-fitted variant of the one-step estimator ${\widehat{\nu }}_n≔\nu ({\widehat{P}}_n)+P_n{\varphi }_n$ is efficient, in the sense that ${\Vert {\widehat{\nu }}_n-\nu (P_0)\Vert }_{\mathscr{H}}$ is as concentrated about zero as is possible for any estimator satisfying appropriate regularity conditions. Here, we provide two more heuristic arguments as to why the one-step correction should lead to improvements. The first, which applies specifically in cases where $\mathscr{H}$ is an RKHS, is based on the pointwise performance of the one-step estimator. In particular, the fact that norm convergence in an RKHS implies pointwise convergence can be used to show that the pathwise differentiability of $\nu :𝒫\to \mathscr{H}$ also implies the pathwise differentiability of $\nu (\cdot )(t):𝒫\to \mathbb{R}$ for each $t\in 𝒯$. Moreover, the EIF of $\nu (\cdot )(t)$ at $P\in 𝒫$ is equal to $z\mapsto {\stackrel{.}{\nu }}_P^{\ast }(K_t)(z)$, and so the one-step estimator for the real-valued parameter $\nu (P_0)(t)$ is equal to the evaluation of the $\mathscr{H}$-valued one-step estimator ${\widehat{\nu }}_n$ at the point $t$. While this pointwise justification of the one-step estimator ${\widehat{\nu }}_n$ is informative, it does not in itself explain why ${\widehat{\nu }}_n$ should be expected to perform well in a norm sense. Indeed, pointwise convergence in an RKHS does not necessarily imply norm convergence. For the same reason, pathwise differentiability of the real-valued parameters $\nu (\cdot )(t)$, $t\in 𝒯$, does not generally imply pathwise differentiability of the RKHS-valued parameter $\nu$.

The second heuristic justification that we provide here provides initial insights into why the one-step estimator ${\widehat{\nu }}_n$ should perform well in a norm sense. This justification applies regardless of whether ${\stackrel{.}{\mathscr{H}}}_P$ is an RKHS. Fix a submodel $\{P_{ϵ}:ϵ[0,\delta )\}\in 𝒫(P,𝒫,s)$. In the appendix, we establish that, under conditions on either the EIF (Lemma S5) or the submodel (Lemma S6), a first-order approximation to the local parameter ${\stackrel{.}{\nu }}_P(s)$ is given by ${ϵ}^{-1}(P_{ϵ}-P){\varphi }_P$, in the sense that the difference between these quantities converges to zero as $ϵ\to 0$. Combining this with (2) and the fact that $P{\varphi }_P=0$, this yields the approximation $\nu (P_{ϵ})\approx \nu (P)+P_{ϵ}{\varphi }_P$, which is valid up to an additive $o(ϵ)$ remainder term. Letting $P_0$ play the role of $P_{ϵ}$ and ${\widehat{P}}_n$ play the role of $P$, this suggests that the von Mises approximation

$$\nu (P_0)\approx \nu ({\widehat{P}}_n)+P_0{\varphi }_n$$ (6)

may also be valid up to a term that goes to zero in probability at a reasonable rate. Admittedly, caution is needed when making the leap from the approximation along the fixed quadratic mean differentiable submodel {$P_{ϵ}:ϵ\in [0,\delta )$} to an approximation that involves the random quantity ${\widehat{P}}_n$. For a given parameter $\nu$, the sense in which the above approximation holds can be made precise by directly studying the quantity $\nu (P_0)-\nu ({\widehat{P}}_n)-P_0{\varphi }_n$. In any case, the approximation above is appealing in that it only relies on $P_0$ through an expectation, which can naturally be approximated by an expectation under the empirical distribution. This, therefore, suggests the one-step estimator ${\widehat{\nu }}_n≔\nu ({\widehat{P}}_n)+P_n{\varphi }_n$.

As will follow from the upcoming Lemma 7, ${\stackrel{.}{\mathscr{H}}}_P$ need not be an RKHS for an EIF to exist. Consequently, when one does, it is natural to wonder whether there is a general expression for its form. The Riesz representation theorem provides an affirmative answer to this question, showing that, when an EIF exists, it is $P$-a.s. equal to the following convergent sum:

$${\varphi }_P(z)\sum _{k=1}^{\infty }{\stackrel{.}{\nu }}_P^{\ast }(h_k)(z)h_k,$$ (7)

where here and throughout we let ${(h_k)}_{k=1}^{\infty }$ denote an orthonormal basis of $\mathscr{H}$. If ${\stackrel{.}{\mathscr{H}}}_P$ is an RKHS, evaluating the expression for the EIF in (5) is typically easier than computing (or approximating) the infinite sum in (7). However, in non-RKHS settings, (7) is useful both as an explicit expression for the EIF, if it exists, and as a basis for generalizing the one-step estimator to settings where it does not.

2.4. Regularized one-step estimation when there is no efficient influence function

We now introduce a generalization of the one-step estimator that can be employed regardless of whether an EIF exists. This estimator is a type of series estimator (Schwartz, 1967; Chen, 2007) based on the Riesz representation of a regularized form of the efficient influence operator. This regularized form is motivated by the fact that, when $\nu$ has an EIF ${\varphi }_P$, it is $P$-a.s. true that ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z):h\mapsto {\sum }_{k=1}^{\infty }{\langle h,h_k\rangle }_{\mathscr{H}}{\stackrel{.}{\nu }}_P^{\ast }(h_k)(z)$. The regularized form is designed to ensure that the terms in this sum must decay as $k$ grows sufficiently large. For a square summable [0, 1]-valued sequence $\beta ≔{({\beta }_k)}_{k=1}^{\infty }$, the $\beta$-regularized efficient influence operator is given by $r_P^{\beta }(h)(z)≔{\sum }_{k=1}^{\infty }{\beta }_k{\langle h,h_k\rangle }_{\mathscr{H}}{\stackrel{.}{\nu }}_P^{\ast }(h_k)(z)$. We now show that $r_P^{\beta }(\cdot )(z)$ is always $P$-a.s. bounded and linear, and we also provide an explicit expression for its Riesz representation. In what follows we let ${\ell }_{\ast }^2≔{\ell }^2\cap {[0,1]}^{\mathbb{N}}$.

Lemma 1 ($\beta$-regularized EIF based on $\beta$-regularized efficient influence operator). If $\nu$ is pathwise differentiable at $P$ and $\beta \in {\ell }_{\ast }^2$, then $r_P^{\beta }(\cdot )(z):\mathscr{H}\to \mathbb{R}$ is a bounded linear functional on a P-probability one set ${𝒵}^{\beta }$ with Riesz representation

$${\varphi }_P^{\beta }(z)≔\sum _{k=1}^{\infty }{\beta }_k{\stackrel{.}{\nu }}_P^{\ast }(h_k)(z)h_k.$$

Moreover, ${\sigma }_P(\beta )≔{\Vert {\varphi }_P^{\beta }\Vert }_{L^2(P;\mathscr{H})}={[{\sum }_{k=1}^{\infty }{\beta }_k^{2}P{\stackrel{.}{\nu }}_P{(h_k)}^2]}^{1∕2}\le {\Vert {\stackrel{.}{\nu }}_P^{\ast }\Vert }_{\text{op}}{\Vert \beta \Vert }_{{\ell }^2}<\infty$.

When $\nu$ has an EIF, ${\varphi }_P^{\beta }$ is similar to the expression for it given in (7), but with the $k$-th term dampened by the multiplier ${\beta }_k\in [0,1]$. Given this similarity, we call ${\varphi }_P^{\beta }$ the $\beta$-regularized EIF of $\nu$ at $P$. The corresponding $\beta$-regularized one-step estimator is ${\widehat{\nu }}_n^{\beta }≔\nu ({\widehat{P}}_n)+P_n{\varphi }_n^{\beta }$.

We now provide a heuristic argument that is similar to one used in the previous subsection for justifying the (non-regularized) one-step estimator, but adapted to account for the regularization bias that arises from using a $\beta$-regularized EIF. Fix a submodel $\{P_{ϵ}:ϵ\}\in 𝒫(P,𝒫,s)$. In Lemma S7 in the appendix, we show that, under a regularity condition on the submodel,

$$\begin{gathered} {\Vert \nu (P)-\nu (P_{ϵ})+P_{ϵ}{\varphi }_P^{\beta }-{\sum }_{k=1}^{\infty }(1-{\beta }_k){\langle \nu (P)-\nu (P_{ϵ}),h_k\rangle }_{\mathscr{H}}{h}_k\Vert }_{\mathscr{H}} \\ =\left(1+{\Vert {\varphi }_P^{\beta }\Vert }_{L^2(P,\mathscr{H})}\right)\cdot o(ϵ), \end{gathered}$$ (8)

where the $o(ϵ)$ term does not depend on the choice of $\beta$. Similarly to how we did when deriving (6), we let $P_0$ play the role of $P_{ϵ}$ and ${\widehat{P}}_n$ play the role of $P$. Recalling that ${\widehat{\nu }}_n^{\beta }≔\nu ({\widehat{P}}_n)+P_n{\varphi }_n^{\beta }$ then yields that

$${\widehat{\nu }}_n^{\beta }-\nu (P_0)\approx (P_n-P_0){\varphi }_n^{\beta }+{ℬ}_{{\widehat{P}}_n}^{\beta },$$ (9)

where, for $P^{\prime }\in 𝒫$, we let ${ℬ}_{P^{\prime }}^{\beta }≔{\sum }_{k=1}^{\infty }(1-{\beta }_k){\langle \nu (P^{\prime })-\nu (P_0),h_k\rangle }_{\mathscr{H}}{h}_k$. Our formal study of the regularized one-step estimator in Section 5.1 builds on the above. Informally speaking, that study will show that the latter term above plays the role of a regularization bias term that decays as $\beta$ grows entrywise to (1, 1, 1, …) under conditions, and the leading term plays the role of a variance term whose magnitude typically grows with that of $\beta$. Hence, a bias-variance tradeoff must be considered when selecting a value for the tuning parameter $\beta$. In Section 5.3, we describe a cross-validation strategy for making this selection. There, we also discuss the selection of the basis ${(h_k)}_{k=1}^{\infty }$.

2.5. Cross-fitted (regularized) one-step estimation

So far, the estimators we have defined have assumed the availability of an iid sample that is independent of $Z_1,Z_2,\dots ,Z_n$ that can be used to obtain the estimate ${\widehat{P}}_n$ of $P_0$. We now describe how cross-fitting (Schick, 1986; Klaassen, 1987; Zheng and Laan, 2011; Chernozhukov et al., 2018) can be used to avoid the need for this independent sample. For simplicity, we focus on the case of 2-fold cross-fitting and suppose that the sample size is an even number. The generalizations to $k$-fold cross-fitting ($k\ge 2$) and to the case where $n$ is not divisible by $k$ are straightforward and so are omitted. Let ${\widehat{P}}_n^1\in 𝒫$ denote an estimate of $P_0$ based on ${\{Z_i\}}_{i=1}^{n∕2}$ and let $P_n^1$ denote the empirical distribution of the remainder of the sample ${\{Z_i\}}_{i=n∕2+1}^n$. Define ${\widehat{P}}_n^2$ and $P_n^2$ similarly, but with the roles of the two subsamples reversed. We note that, in a slight abuse of notation, $P_n^j$ denotes an empirical distribution derived from $n∕2$ observations rather than a $j$-fold product distribution derived from $j$ independent draws from the empirical distribution $P_n$ of the full sample ${\{Z_i\}}_{i=1}^n$. Cross-fitting enables the use of arbitrary estimation strategies when constructing ${\widehat{P}}_n^j,j\in \{1,2\}$, including those based on machine learning techniques.

We now present the form of our cross-fitted estimators. From a notational standpoint, these estimators will be denoted by replacing the hat accents used to denote the sample-splitting estimators in Sections 2.3 and 2.4 by bar accents — for example, the cross-fitted one-step estimator will be denoted by ${\overline{\nu }}_n$ rather than ${\widehat{\nu }}_n$. This cross-fitted one-step estimator takes the form ${\overline{\nu }}_n≔\frac{1}{2}{\sum }_{j=1}^2[\nu ({\widehat{P}}_n^j)+P_n^j{\varphi }_n^j]$, where ${\varphi }_n^j≔{\varphi }_{{\widehat{P}}_n^j}$. Let ${\varphi }_n^{j,\beta }≔{\varphi }_{{\widehat{P}}_n^j}^{\beta }$. The cross-fitted $\beta$-regularized one-step estimator takes the form ${\overline{\nu }}_n^{\beta }≔\frac{1}{2}{\sum }_{j=1}^2[\nu ({\widehat{P}}_n^j)+P_n^j{\varphi }_n^{j,\beta }]$.

3. Examples of pathwise differentiable parameters

In this section, we present examples of pathwise differentiable Hilbert-valued parameters and the forms of their efficient influence operators and, where applicable, EIFs. From these objects, cross-fitted (regularized) one-step estimators can be derived using the formulas at the end of the previous section. We study the performance of these estimators in Section 6.

In the main text, we focus on parameters that have recently become objects of interest to the causal inference community. Two of these examples (Examples 1b and 3) consider cases where the action space is an RKHS, and two (Examples 1a and 2) consider cases where the action space is an $L^2$ space, and is therefore not an RKHS. In Appendix A, we show that four more well-studied Hilbert-valued parameters are also pathwise differentiable. In particular, we show that regression functions, square-root density functions, and conditional average treatment effect functions are pathwise differentiable when viewed as elements of appropriate $L^2$ spaces, and we also show that a kernel mean embedding of a distribution (Gretton et al., 2012) is pathwise differentiable when viewed as an element of an RKHS. We are not aware of any reference establishing the pathwise differentiability of any of the eight Hilbert-valued parameters that we consider in this work.

All derivations for our examples are deferred to Appendix B. For most of these examples, our derivations make use of the following lemma, which we prove in Appendix C. In this lemma, $H(P,P^{\prime })≔{[\int {(\sqrt{dP}-\sqrt{d{P}^{\prime }})}^2]}^{1∕2}$ denotes the Hellinger distance.

Lemma 2 (Sufficient condition for pathwise differentiability). $\nu :𝒫\to \mathscr{H}$ is pathwise differentiable at $P$ with local parameter ${\stackrel{.}{\nu }}_P={\eta }_P$ if both of the following hold:

1. ${\eta }_P:{\stackrel{.}{𝒫}}_P\to \mathscr{H}$ is bounded and linear and there exists a set of scores $𝒮(P)$ whose $L^2(P)$-closure is equal to ${\stackrel{.}{𝒫}}_P$ such that, for all $s\in 𝒮(P)$, there is at least one submodel $\{P_{ϵ}:ϵ\}\in 𝒫(P,𝒫,s)$ for which ${\Vert \nu (P_{ϵ})-\nu (P)-ϵ{\eta }_P(s)\Vert }_{\mathscr{H}}=o(ϵ)$; and

2. $\nu$ is locally Lipschitz at $P$ in the sense that there exist $(c,\delta )\in {(0,\infty )}^2$ such that

$${\Vert \nu (P_1)-\nu (P_2)\Vert }_{\mathscr{H}}\le cH(P_1,P_2)\text{for all}{P}_1,P_2\in {ℬ}_{\delta }(P),$$ (10)

where $B_{\delta }(P)$ consists of all $P^{\prime }\in 𝒫$ for which $H(P,P^{\prime })\le \delta$.

This lemma is most useful when the set $𝒮(P)$ and corresponding submodels in $𝒫(P,𝒫,s)$ in (i) can be chosen to make establishing (2) for those submodels simple. For example, in a locally nonparametric model, we may take $𝒮(P)$ to be the set of bounded, $P$-mean zero functions, and we may take the chosen submodel in $𝒫(P,𝒫,s)$ to be such that, for all $ϵ\in [0,1∕{\text{ess sup}}_z\mid s(z)\mid )$, $\frac{d{P}_{ϵ}}{dP}=1+ϵs$, where the essential supremum is under $P$.

All of the examples presented in the main text are motivated by questions arising in causal inference. The data structure is common across them, with $Z=(X,A,Y)\sim P$, where $X$ is a vector of covariates with support on $𝒳$, $A$ is a treatment with support on either {0, 1} or $\mathbb{R}$, and $Y$ is an outcome with support on $𝒴$. In each example, we suppose that $𝒫$ is locally nonparametric. We further suppose, for simplicity, that all pairs of distributions in $𝒫$ are mutually absolutely continuous. For a given distribution $P$, we let $P_{Y\mid A,X}$ denote the conditional distribution of $Y$ given $(A,X)$ and $P_X$ denote the marginal distribution of $X$. We let $g_P(\cdot \mid x)$ denote the conditional probability mass function of $A$ given $X=x$ under $P$, when $A$ is binary, and the density of $A$ given $X=x$ under $P$, when $A$ is continuous. For $s\in {\stackrel{.}{𝒫}}_P$, we let $s_X(x)≔E_P[s(Z)\mid X=x]$ and $s_{Y\mid A,X}(y\mid a,x)≔s(z)-E_P[s(Z)\mid A=a,X=x]$.

Example 1a (Counterfactual density function). Suppose that the treatment $A$ is binary and the goal is to estimate the density function of the counterfactual outcome in a setting where everyone receives treatment $A=1$. This density can offer a more nuanced measure of causal effects than can a more commonly studied counterfactual mean outcome (Kennedy et al., 2021). Suppose that there is a $\sigma$-finite measure ${\lambda }_Y$ such that, for all $P\in 𝒫$, there is a regular conditional probability $P_{Y\mid A,X}$ such that $P_{Y\mid A,X}(\cdot \mid a,x)\ll {\lambda }_Y$ for $P$-almost all $(a,x)\in \{0,1\}\times 𝒳$. Define the propensity to receive treatment $a$ as $g_P(a\mid x)≔P(A=a\mid X=x)$ and let $p_{Y\mid A,X}(\cdot \mid 1,x)$ denote the conditional density of $Y$ given $(A,X)=(1,x)$. The parameter of interest $\nu :𝒫\to L^2({\lambda }_Y)$ takes the form

$$\nu (P)(y)=\int p_{Y\mid A,X}(y\mid 1,x)P_X(dx).$$ (11)

Under typical causal assumptions, $\nu$ corresponds to the density of the counterfactual outcome that would be seen under treatment $A=1$. We require that $𝒫$ satisfy the following conditions:

$$\underset{P\in 𝒫}{\text{inf}}\underset{x\in 𝒳}{\text{ess inf}}{g}_P(1\mid x)>0\text{and}\underset{P\in 𝒫}{\text{sup}}\underset{(x,y)\in 𝒳\times 𝒴}{\text{ess sup}}{p}_{Y\mid A,X}(y\mid 1,x)<\infty ,$$ (12)

where the essential infimum is under $P_X$ and the essential supremum is under $P_X\times {\lambda }_Y$. The first inequality, referred to as a strong positivity assumption, holds when the propensity to receive treatment 1 is not vanishingly small. The second holds when the conditional density of $Y$ given $(A,X)=(1,x)$ is uniformly bounded across all distributions in the model. While in principle the second condition could be weakened, this condition appears to be sufficiently general to capture many statistical models of interest.

The local parameter takes the form

$${\stackrel{.}{\nu }}_P(s)(y)=\int \{s_{Y\mid A,X}(y\mid 1,x)+s_X(x)\}{p}_{Y\mid A,X}(y\mid 1,x)P_X(dx),$$ (13)

and the efficient influence operator takes the form

$$\begin{gathered} {\stackrel{.}{\nu }}_P^{\ast }(h)(y,a,x)=\frac{1\{a=1\}}{g_P(a\mid x)}\{h(y)-E_P[h(Y)\mid A=a,X=x]\} \\ +\left(E_P[h(Y)\mid A=1,X=x]-\int E_P[h(Y)\mid A=1,X=x^{\prime }]P_X(d{x}^{\prime })\right). \end{gathered}$$ (14)

Unless ${\lambda }_Y$ is a discrete measure, ${\stackrel{.}{\nu }}_P^{\ast }$ will not generally be a bounded operator. This can be shown to follow from the facts that point evaluation is not continuous in $L^2$ spaces (page 8 of Berlinet and Thomas-Agnan, 2011) and ${\stackrel{.}{\nu }}_P^{\ast }(h)(y,a,x)$ depends on the evaluation of $h$ at $y$.

Example 1b (Bandlimited counterfactual density function). The setting is the same as in Example 1a, except that $Y$ must be real-valued and continuous, ${\lambda }_Y$ must be the Lebesgue measure, and, for fixed $b>0$, the target of inference is the following transformation of the counterfactual density $\nu (P)$ that was defined in Example 1a:

$$\underset{¯}{\nu }(P)(y)≔{\int }_{-\infty }^{\infty }{\underset{¯}{K}}_y(\tilde{y})\nu (P)(\tilde{y}){\lambda }_Y(d\tilde{y}),$$ (15)

where ${\underset{¯}{K}}_y(\tilde{y})≔\{\sin [b(\tilde{y}-y)]\}∕[\pi (\tilde{y}-y)]$. This estimand corresponds to a bandlimiting of the counterfactual density function. In particular, letting $ℱ$ and ${ℱ}^{-1}$ denote the Fourier transform and inverse Fourier transform and fixing $b>0$, the $b$-bandlimiting of a Lebesgue square integrable function $f:\mathbb{R}\to \mathbb{R}$ is the function $ℬ(f):\mathbb{R}\to \mathbb{R}$ given by

$$ℬ(f)≔{ℱ}^{-1}\left(1_{[-b,b]}\cdot ℱ(f)\right)(y)={\int }_{-\infty }^{\infty }{\underset{¯}{K}}_y(\tilde{y})f(\tilde{y}){\lambda }_Y(d\tilde{y}),$$ (16)

where $1_{[-b,b]}\cdot ℱ(f)$ represents the function $\xi \mapsto 1_{[-b,b]}(\xi )\cdot ℱ(f)(\xi )$ and the latter equality holds by the convolution theorem. The estimand is equal to $\underset{¯}{\nu }(P)≔ℬ(\nu (P))$. Lemma S1 in Appendix B.2 shows that $ℬ(\nu (P))$ corresponds to an $L^2({\lambda }_Y)$ projection of $\nu (P)$ onto

$$\underset{¯}{\mathscr{H}}≔\{h\in C(\mathbb{R})\mid \text{support}[ℱ(h)]\subseteq [-b,b]\},$$

where $C(\mathbb{R})$ denotes the set of continuous, Lebesgue square integrable functions. A little care is needed to make this result precise since $L^2({\lambda }_Y)$ is a space of equivalence classes of functions whereas $\underset{¯}{\mathscr{H}}$ is a space of functions (see the statement of Lemma S1 for details). The space $\underset{¯}{\mathscr{H}}$ is an RKHS when equipped with the $L^2({\lambda }_Y)$ inner product ${\langle h,\tilde{h}\rangle }_{\underset{¯}{\mathscr{H}}}=\int h(y)\tilde{h}(y){\lambda }_Y(dy)$ (Yao, 1967). The kernel function in this RKHS is given by $(y,\tilde{y})\mapsto {\underset{¯}{K}}_y(\tilde{y})$. This RKHS is a smoothness class consisting of all square integrable functions that have an analytic continuation to the complex plane that satisfies an exponential growth restriction (Theorem 19.3 and page 372 of Rudin, 1987). When a (non-counterfactual) density function belongs to $\underset{¯}{\mathscr{H}}$, a particular kernel density estimator has been shown to attain mean integrated squared error (MISE) that decays at an $n^{-1}$ rate (Ibragimov and Khas’minskii, 1983; Agarwal et al., 2015). Our setting differs from earlier ones in that (i) we focus on a counterfactual density, and (ii) we define our estimand as a nonparametric projection onto the space of $b$-bandlimited functions, rather than requiring that our estimand belong to this class. Naturally, when the counterfactual density $\nu (P)$ already belongs to $\underset{¯}{\mathscr{H}}$, $\underset{¯}{\nu }(P)$ will equal this density and so our approach will yield estimators of it.

In the appendix, we rely heavily on the calculations performed in Example 1a when showing that $\underset{¯}{\nu }:𝒫\to \underset{¯}{\mathscr{H}}$ is pathwise differentiable. We show that the local parameter ${\underset{¯}{\stackrel{.}{\nu }}}_P:{\stackrel{.}{𝒫}}_P\to \underset{¯}{\mathscr{H}}$ of $\underset{¯}{\nu }$ at $P$ is closely related to that of $\nu$. In particular, ${\underset{¯}{\stackrel{.}{\nu }}}_P(s)=ℬ({\stackrel{.}{\nu }}_P(s))$, where ${\stackrel{.}{\nu }}_P$ is as defined in Example 1a. Letting [$h$] denote the equivalence class of functions that are equal to some $h\in \underset{¯}{\mathscr{H}}$ Lebesgue-almost everywhere, the efficient influence operator ${\underset{¯}{\stackrel{.}{\nu }}}_P^{\ast }:\underset{¯}{\mathscr{H}}\to {\stackrel{.}{𝒫}}_P$ takes the form ${\underset{¯}{\stackrel{.}{\nu }}}_P^{\ast }(h)={\stackrel{.}{\nu }}_P^{\ast }([h])$ where ${\stackrel{.}{\nu }}_P^{\ast }$ is as defined in Example 1a. Since $\underset{¯}{\mathscr{H}}$ is an RKHS, we can also look to define the EIF ${\underset{¯}{\varphi }}_P$ of $\underset{¯}{\nu }$ at $P$. In particular, (12) can be used to show that the function ${\underset{¯}{\varphi }}_P(z):y\mapsto {\underset{¯}{\nu }}_P^{\ast }{\underset{¯}{K}}_y(z)$ belongs to $L^2(P;\underset{¯}{\mathscr{H}})$, and so ${\underset{¯}{\varphi }}_P$ is indeed the EIF of $\underset{¯}{\nu }$ at $P$. See (S5) in the appendix for a more explicit expression for the EIF ${\underset{¯}{\varphi }}_P$.

Figure 1 shows how a one-step estimator can improve an initial estimator in this example.

Figure 1:

Illustration of how a sample-splitting one-step estimator is constructed in Example 1b. The cross-fitted estimator averages two such estimators, one obtained as above and the other with the roles of the two halves of the data reversed. Details of the data-generating process, along with a Monte Carlo assessment of the performance of a cross-fitted one-step estimator, are given in Appendix I.1.

Example 2 (Counterfactual mean outcome under a continuous treatment). In the previous example, the treatment was considered to be binary. In this example, we take $A$ to be a continuous treatment taking values in $𝒜=[0,1]$, such as a dosage, duration, or frequency of intervention. Denote the marginal distribution of $A$ by $P_A$ and suppose that the Lebesgue measure ${\lambda }_A$ dominates the conditional distribution $P_{A\mid X}(\cdot \mid x)$ of $A\mid X=x$ under $P$ for $P$-almost all $x$. Let $g_P(\cdot \mid x)$ denote the conditional density of $A$ given that $X=x$. The target of estimation is $\nu :𝒫\to L^2({\lambda }_A)$, where $\nu (P)(a)=\int E_P[Y\mid A=a,X=x]P_X(dx)$. Under causal conditions, $\nu (P)(a)$ corresponds to the mean outcome under a continuous treatment (Díaz and van der Laan, 2013). We suppose the strong positivity assumption that ${\text{inf}}_{P\in 𝒫}{\text{ess inf}}_{(a,x)}{g}_P(a\mid x)>0$, where the essential infimum is under ${\lambda }_A\times P_X$. We further suppose that $Y$ has a bounded conditional second moment, in the sense that ${\text{sup}}_{P\in 𝒫}{\text{ess sup}}_{(a,x)}{E}_P[Y^2\mid A=a,X=x]<\infty$, where the essential supremum is under ${\lambda }_A\times P_X$.

The local parameter takes the form

$$\begin{gathered} {\stackrel{.}{\nu }}_P(s)(a)=\iint \{y-{\mu }_P(a,x)\}{s}_{Y\mid A,X}(y\mid a,x)P_{Y\mid A,X}(d_y\mid a,x)P_X(dx) \\ +\int [{\mu }_P(a,x)-\nu (P)(a)]s_X(x)P_X(dx), \end{gathered}$$ (17)

where ${\mu }_P(a,x)=E_P[Y\mid A=a,X=x]$. The efficient influence operator takes the form

$${\stackrel{.}{\nu }}_P^{\ast }(h)(y,a,x)=\frac{y-{\mu }_P(a,x)}{g_P(a\mid x)}h(a)+\int [{\mu }_P(a^{\prime },x)-\nu (P)(a^{\prime })]h(a^{\prime }){\lambda }_A(d{a}^{\prime }).$$ (18)

Similarly to Example 1a, ${\stackrel{.}{\nu }}_P^{\ast }$ is not generally a bounded operator.

Example 3 (Counterfactual kernel mean embedding). Let $A$ be a binary treatment and suppose the strong positivity assumption that ${\text{inf}}_{P\in 𝒫}{\text{ess inf}}_{x\in 𝒳}{g}_P(1\mid x)>0$, where the essential infimum is under $P_X$. Let $\kappa :𝒴\times 𝒴\to \mathbb{R}$ be a bounded, symmetric, positive define function, $\mathscr{H}$ be the RKHS associated with the kernel $\kappa$, and $K_y≔\kappa (y,\cdot )$ be the associated feature map. The counterfactual kernel mean embedding (Muandet et al., 2021; Fawkes et al., 2022) is a parameter $\nu :𝒫\to \mathscr{H}$ such that

$$\nu (P)=\int E_P[K_Y\mid A=1,X=x]P_X(dx).$$

Under standard causal conditions (Robins, 1986), $\nu (P)$ can be shown to be equal to the kernel mean embedding (Gretton et al., 2012) of the distribution of a counterfactual outcome in a world where treatment 1 was given to everyone. We suppose that the strong positivity assumption in (12) holds.

The local parameter takes the form

$${\stackrel{.}{\nu }}_P(s)=\iint K_y[s_{Y\mid A,X}(y\mid 1,x)+s_X(x)]P_{Y\mid A,X}(dy\mid 1,x)P_X(dx).$$ (19)

The efficient influence operator takes the form

$$\begin{gathered} {\stackrel{.}{\nu }}_P^{\ast }(h)(y,a,x)=\frac{a}{g_P(1\mid x)}\{h(y)-E_P[h(Y)\mid A=a,X=x]\} \\ +E_P[h(Y)\mid A=1,X=x]-\int E_P[h(Y)\mid A=1,X=x^{\prime }]P_X(d{x}^{\prime }), \end{gathered}$$

and the EIF is $P$-Bochner square integrable and takes the form

$${\varphi }_P(y,a,x)=\frac{a}{g_P(1\mid x)}\{K_y-{\mu }_P^K(x)\}+{\mu }_P^K(x)-\nu (P),$$ (20)

where ${\mu }_P^K:𝒳\to \mathscr{H}$ is defined as ${\mu }_P^K(x)≔E_P[K_Y\mid A=1,X=x]$.

4. Performance guarantees and inference when there is an EIF

4.1. Performance guarantees for one-step estimators

In this section, we provide conditions under which a cross-fitted one-step estimator ${\overline{\nu }}_n$ iss both asymptotically linear and efficient. These conditions concern the terms arising in the following decomposition:

$$\begin{gathered} {\Vert {\overline{\nu }}_n-\nu (P_0)-P_n{\varphi }_0\Vert }_{\mathscr{H}}=\frac{1}{2}{\Vert \sum _{j=1}^2\left[\nu ({\widehat{P}}_n^j)+P_0{\varphi }_n^j-\nu (P_0)\right]+\sum _{j=1}^2(P_n^j-P_0)({\varphi }_n^j-{\varphi }_0)\Vert }_{\mathscr{H}} \\ \le \underset{j}{\max }{\Vert {ℛ}_n^j\Vert }_{\mathscr{H}}+\underset{j}{\max }{\Vert {𝒟}_n^j\Vert }_{\mathscr{H}}, \end{gathered}$$ (21)

where ${ℛ}_n^j≔\nu ({\widehat{P}}_n^j)+P_0{\varphi }_n^j-\nu (P_0)$ and ${𝒟}_n^j≔(P_n^j-P_0)({\varphi }_n^j-{\varphi }_0)$. We call ${ℛ}_n^j$ the remainder terms and ${𝒟}_n^j$ the drift terms, $j\in \{1,2\}$. Asymptotic linearity, as defined in (4), holds whenever both of these quantities are negligible, in the sense that they are $o_p(n^{-1∕2})$. The remainder terms ${ℛ}_n^j$, $j\in \{1,2\}$, quantify the error in the approximation (6) across the two splits of the sample. As described heuristically in the text surrounding (6), it is reasonable to expect that this remainder term will be negligible under appropriate conditions. The following result provides a reasonable condition under which the drift terms will be negligible.

Lemma 3 (Sufficient condition for negligible drift terms). Suppose that $\nu$ is pathwise differentiable at $P_0$ with EIF ${\varphi }_0$. For each $j\in \{1,2\}$, ${\Vert {\varphi }_n^j-{\varphi }_0\Vert }_{L^2(P_0;\mathscr{H})}=o_p(1)$ implies that ${\Vert {𝒟}_n^j\Vert }_{\mathscr{H}}=o_p(n^{-1∕2})$.

In the appendix, this lemma is proved via a conditioning argument that makes use of Chebyshev’s inequality for Hilbert-valued random variables (Grenander, 1963) and the dominated convergence theorem. In the next section, we provide lower-level conditions under which the remainder and drift terms will be small in the contexts of Examples 1b and 3.

In the process of showing that ${\overline{\nu }}_n$ is asymptotically linear, we also show that it is regular. An estimator sequence ${({\tilde{\nu }}_n)}_{n=1}^{\infty }$ is said to be regular at $P_0\in 𝒫$ if and only if there is a tight $\mathscr{H}$-valued random variable $\tilde{\mathbb{H}}$ such that, for every score in the tangent set, quadratic mean differentiable submodel $\{P_{ϵ}:ϵ\}\in 𝒫(P_0,𝒫,s)$, and every sequence ${ϵ}_n=O(n^{-1∕2})$, $\sqrt{n}[{\tilde{\nu }}_n-\nu (P_{{ϵ}_n})]$ converges weakly to $\tilde{\mathbb{H}}$ under iid sampling of $n$ observations from $P_{{ϵ}_n}$. We say that an estimator ${\tilde{\nu }}_n$ is regular when the implied estimator sequence ${({\tilde{\nu }}_n)}_{n=1}^{\infty }$ is clear from context. In the upcoming theorem, we write ‘⇝’ to denote weak convergence in $\mathscr{H}$.

Theorem 2 (Asymptotic linearity and weak convergence of a one-step estimator). Suppose that $\nu$ is pathwise differentiable at $P_0$ with EIF ${\varphi }_0\in L^2(P_0;\mathscr{H})$ and, for $j\in \{1,2\}$, ${ℛ}_n^j=o_p(n^{-1∕2})$ and ${𝒟}_n^j=o_P(n^{-1∕2})$. Under these conditions, (4) holds, ${\overline{\nu }}_n$ is regular, and

$$n^{1∕2}[{\overline{\nu }}_n-\nu (P_0)]⇝\mathbb{H},$$ (22)

where $\mathbb{H}$ is a tight $\mathscr{H}$ -valued Gaussian random variable that is such that, for each $h\in \mathscr{H}$, the marginal distribution ${\langle \mathbb{H},h\rangle }_{\mathscr{H}}$ is $N(0,E_0[{\langle {\varphi }_0(Z),h\rangle }_{\mathscr{H}}^2])$.

The convolution theorem for Banach-valued estimators can be used to characterize one sense in which ${\overline{\nu }}_n$ is an efficient estimator of $\nu (P_0)$ (see Theorem 3.11.2 and Lemma 3.11.4 of van der Vaart and Wellner, 1996, for a convenient version). Rather than present the theorem in full generality, here we present a particularly interpretable consequence of it. Specifically, under the conditions of Theorem 2, that theorem shows that ${\overline{\nu }}_n$ is optimally concentrated about $\nu (P_0)$ in the sense that, for any regular estimator sequence ${({\tilde{\nu }}_n)}_{n=1}^{\infty }$ and $c\ge 0$,

$$\underset{n\to \infty }{\text{lim}}{P}_0^n\{n{\Vert {\overline{\nu }}_n-\nu (P_0)\Vert }_{\mathscr{H}}^2>c\}\le \underset{n\to \infty }{\text{lim}}{P}_0^n\{n{\Vert {\tilde{\nu }}_n-\nu (P_0)\Vert }_{\mathscr{H}}^2>c\}.$$ (23)

The above describes a sense in which ${\overline{\nu }}_n$ is optimal among all regular estimators of $\nu (P_0)$. Under the conditions of Theorem 2, ${\overline{\nu }}_n$ can also be shown to outperform all estimators, including non-regular ones, in a local asymptotic minimax sense — see Theorem 3.11.5 in van der Vaart and Wellner (1996) for details.

4.2. Construction of confidence sets based on one-step estimators

As we will now show, the weak convergence result in Theorem 2 can be used to facilitate the construction of $(1-\alpha )$-level confidence sets for the Hilbert-valued parameter $\nu (P_0)$, where $\alpha \in (0,1)$ is some fixed constant. Our proposed confidence set is built based upon a quadratic form $w(\cdot ;\Omega ):h\mapsto {\langle \Omega (h),h\rangle }_{\mathscr{H}}$ that is parameterized by a standardization operator $\Omega$ that belongs to the set $𝒪$ of continuous, self-adjoint, positive definite linear operators mapping from $\mathscr{H}$ to $\mathscr{H}$. In particular, letting $\zeta \ge 0$ be a specified threshold and ${\Omega }_n\in 𝒪$ an estimator of a some possibly-$P_0$-dependent operator ${\Omega }_0\in 𝒪$, our confidence set will take the form

$${𝒞}_n(\zeta )≔\{h\in \mathscr{H}:w({\overline{\nu }}_n-h;\Omega )\le \zeta ∕n\}.$$ (24)

We will see that, for an appropriate choice of $\zeta$ and provided ${\Vert {\Omega }_n-{\Omega }_0\Vert }_{\text{op}}=o_p(1)$, the continuous mapping theorem justifies the asymptotic validity of this confidence set.

Before presenting that result, we discuss two interesting choices of ${\Omega }_0$. The first is the identity function. This choice yields a spherical confidence set that consists of all $h$ belonging to an $\mathscr{H}$-ball centered at ${\overline{\nu }}_n$ of radius ${(\zeta ∕n)}^{1∕2}$. Since the form of ${\Omega }_0$ does not rely on $P_0$ in this case, ${\Omega }_n$ can be taken to be equal to ${\Omega }_0$. The second is proportional to the inverse of a regularized form of the covariance operator of $\mathbb{H}$, which will yield a Wald-type confidence set for $\nu (P_0)$ that has an elliptical shape. Compared to spherical confidence sets, Wald-type confidence sets have the benefit of being narrower in the direction of unit vectors $h$ where estimation is easier, in the sense that the variance of ${\langle \mathbb{H},h\rangle }_{\mathscr{H}}$ is smaller. Regularization is needed when defining a Wald-type confidence set because the covariance operator of $\mathbb{H}$ will not generally be invertible and, even if it is, this inverse will not be continuous when ${\Vert {\varphi }_0\Vert }_{L^2(P_0;\mathscr{H})}<\infty$ unless $\mathscr{H}$ is finite-dimensional. A simple example of a regularized covariance operator ${\Omega }_0$, and an operator-norm consistent estimator thereof, is given in Appendix D. Many other types of regularization are also possible (Tikhonov et al., 1995).

The following result establishes the asymptotic validity of the confidence set in (24) based on a threshold ${\widehat{\zeta }}_n$ that is measurable with respect to the $\sigma$-field generated by the iid sample $Z_1,\dots ,Z_n$ from $P_0$. This threshold is an estimate of the $(1-\alpha )$-quantile ${\zeta }_{1-\alpha }$ of $w(\mathbb{H};{\Omega }_0)$.

Theorem 3 (Asymptotically valid confidence set). Suppose the conditions of Theorem 2 hold. Further suppose that ${\Vert {\varphi }_0\Vert }_{L^2(P_0;\mathscr{H})}<\infty$, ${\Omega }_n\in 𝒪$, ${\Omega }_0\in 𝒪$, and ${\Vert {\Omega }_n-{\Omega }_n\Vert }_{\mathrm{op}}=o_p(1)$.

1. if ${\widehat{\zeta }}_n\to {\zeta }_{1-\alpha }$ in probability, then ${\lim }_{n\to \infty }{P}_0^n\{\nu (P_0)\in {𝒞}_n({\widehat{\zeta }}_n)\}=1-\alpha$.

2. if ${\widehat{\zeta }}_n$ is an asymptotically conservative estimator of ${\zeta }_{1-\alpha }$, in the sense that $P_0^n\{{\widehat{\zeta }}_n\ge {\zeta }_{1-\alpha }-\delta \}\stackrel{n\to \infty }{\to }1$ for all $\delta >0$, then $\lim {\inf }_{n\to \infty }{P}_0^n\{\nu (P_0)\in {𝒞}_n({\widehat{\zeta }}_n)\}\ge 1-\alpha$.

The proof of this result is similar in structure to those used to establish the validity of Wald-type confidence sets for finite-dimensional parameters. It consists in applying the continuous mapping theorem and Slutsky’s lemma to show that $n\cdot w[{\overline{\nu }}_n-\nu (P_0);{\Omega }_n]⇝w(\mathbb{H};{\Omega }_0)$, showing that $w(\mathbb{H};{\Omega }_0)$ is a continuous random variable, and finally using that convergence in distribution implies convergence of cumulative distribution functions at continuity points.

A consistent estimator of ${\zeta }_{1-\alpha }$ can be defined using the bootstrap (Efron, 1979). To define this estimator, we let $Z_1^{\#},\dots ,Z_{n∕2}^{\#}\stackrel{\mathrm{iid}}{\sim }{P}_n^2$ be sampled independently of $Z_{n∕2+1}^{\#},\dots ,Z_n^{\#}\stackrel{\mathrm{iid}}{\sim }{P}_n^1$. We then let $P_n^{2,\#}$ be the empirical distribution of $Z_1^{\#},\dots ,Z_{n∕2}^{\#}$ and $P_n^{1,\#}$ be the empirical distribution of $Z_{n∕2+1}^{\#},\dots ,Z_n^{\#}$. We let ${\mathbb{H}}_n^{\#}≔n^{1∕2}{\sum }_{j=1}^2(P_n^{j,\#}-P_n^j){\varphi }_n^j∕2$. The threshold ${\widehat{\zeta }}_n$ is taken to be equal to the ($1-\alpha$)-quantile of $w({\mathbb{H}}_n^{\#},{\Omega }_n)$, conditionally on the original sample $(Z_1,\dots ,Z_n)$ used to define $P_n^1$ and $P_n^2$. In practice this quantile can be well-approximated by selecting $m$ Monte Carlo draws of the sample $Z_1^{\#},\dots ,Z_n^{\#}$ and then returning the empirical ($1-\alpha$)-quantile of $w({\mathbb{H}}_n^{\#},{\Omega }_n)$ over these draws. A computational benefit of this bootstrap procedure is that it does not require refitting the initial estimators ${\widehat{P}}_n^j$ of $P_0$; indeed, the same EIFs ${\varphi }_n^j≔{\varphi }_{{\widehat{P}}_n^j}$ are used for each bootstrap replication.

Theorem 4 (Consistent estimation of ${\zeta }_{1-\alpha }$ via the bootstrap). Suppose the conditions of Theorem 2 hold. Further suppose that ${\Omega }_n\in 𝒪$, ${\Omega }_0\in 𝒪$, ${\Vert {\Omega }_n-{\Omega }_0\Vert }_{\mathrm{op}}=o_p(1)$,and ${\max }_{j\in \{1,2\}}{\Vert {\varphi }_n^j-{\varphi }_0\Vert }_{L^2(P_0;\mathscr{H})}=o_p(1)$. Under these conditions, ${\widehat{\zeta }}_n\to {\zeta }_{1-\alpha }$ in probability.

In brief, the proof of the above consists in showing that ${\mathbb{H}}_n^{\#}$ is asymptotically equivalent to ${\mathbb{H}}_{n,0}^{\#}≔n^{1∕2}\frac{1}{2}{\sum }_{j=1}^2(P_n^{j,\#}-P_n^j){\varphi }_0$ in probability, invoking a guarantee from Giné and Zinn (1990) regarding the weak convergence of the bootstrap for Hilbert-valued sample means and the continuous mapping theorem to show that $w({\mathbb{H}}_{n,0}^{\#};{\Omega }_0)⇝w(\mathbb{H};{\Omega }_0)$ conditionally on ${(Z_i)}_{i=1}^{\infty }$ with probability one, and finally applying a Slutsky-type argument to replace the $P_0$-dependent quantities ${\mathbb{H}}_{n,0}^{\#}$ and ${\Omega }_0$ in $w({\mathbb{H}}_{n,0}^{\#};{\Omega }_0)$ by ${\mathbb{H}}_n^{\#}$ and ${\Omega }_n$, respectively.

When ${\Omega }_0$ is the identity function, Theorem 1 in Székely and Bakirov (2003) provides a means to derive an alternative estimator of ${\zeta }_{1-\alpha }$. This estimator does not require the bootstrap, but is asymptotically conservative. See Appendix E for details.

In practice, it will typically be necessary to use numerical techniques to compute the quadratic form $w({\overline{\nu }}_n-h;{\Omega }_0)$ that is used to define our confidence set. We discuss some such approaches in Appendix F.

5. Performance guarantees and inference when there is no EIF

5.1. Performance guarantees for regularized one-step estimation

In this subsection, we provide performance guarantees for the cross-fitted ${\beta }_n$-regularized one-step estimator ${\overline{\nu }}_n^{{\beta }_n}$, where, for each $n$, ${\beta }_n$ is an ${\ell }_{\ast }^2$-valued regularization parameter. Before doing so, we acknowledge a minor abuse of notation. We will denote the $k^{\mathrm{th}}$ entry of a generic regularization parameter $\beta \in {\ell }_{\ast }^2$ by ${\beta }_k$, which should not be mistaken for the sample-size-n dependent regularization parameter ${\beta }_n$, whose $k^{\mathrm{th}}$ entry we will denote by ${\beta }_{n,k}$. This should not cause confusion, as we always denote sample size by n and a generic index of a vector in ${\ell }_{\ast }^2$ by k.

We will show that, under conditions, ${\overline{\nu }}_n^{{\beta }_n}$ satisfies the following biased and slower-than-n^−1/2-rate asymptotically linear expansion, which formalizes the approximation in (9) for a cross-fitted one-step estimator:

$${\overline{\nu }}_n^{{\beta }_n}-\nu (P_0)=\frac{1}{2}\sum _{j=1}^2{ℬ}_n^{j,{\beta }_n}+P_n{\varphi }_0^{{\beta }_n}+O_p\left({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2}\right).$$ (25)

Above ${\varphi }_0^{{\beta }_n}$ is as defined in Lemma 1 and ${ℬ}_n^{j,{\beta }_n}≔{ℬ}_{{\widehat{P}}_n^j}^{{\beta }_n}$ denotes the bias term defined below (9). When $\nu$ does not have an EIF, there is generally a tradeoff between the bias term, whose magnitude is smaller when the entries of ${\beta }_n$ are closer to 1, and the linear ‘variance’ term $P_n{\varphi }_0^{{\beta }_n}$, whose magnitude scales as $O_p[{\sigma }_0({\beta }_n)∕n^{1∕2}]$, where ${\sigma }_0({\beta }_n)=O({\Vert {\beta }_n\Vert }_{{\ell }^2})$ is as defined in Lemma 1. These two terms will typically be of the same order when ${\beta }_n$ is selected to minimize the mean-squared error $E_{P_0^n}[{\Vert {\overline{\nu }}_n^{{\beta }_n}-\nu (P_0)\Vert }_{\mathscr{H}}^2]$, which makes it so that ${\overline{\nu }}_n^{{\beta }_n}-\nu (P_0)$ converges to zero in probability slower than does n^−1/2. Owing to the bias term, and also to the fact that there is generally not a scaling of $P_n{\varphi }_0^{{\beta }_n}$ that will converge to a nondegenerate, tight random element when $\nu$ does not have an EIF (see Lemma S11 in the appendix), our focus in this subsection will be on deriving rates of convergence for the regularized estimator ${\overline{\nu }}_n^{{\beta }_n}$. We provide a means to construct confidence sets for $\nu (P_0)$ in Section 5.2.

To establish (25), we introduce regularized versions of the drift and remainder terms considered in Section 4.1. In particular, for $j\in \{1,2\}$ and $\beta ={({\beta }_k)}_{k=1}^{\infty }\in {\ell }^2$, define the $\mathscr{H}$-valued random elements ${𝒟}_n^{j,\beta }≔(P_n^j-P_0)({\varphi }_n^{j,\beta }-{\varphi }_0^{\beta })$ and ${ℛ}_n^{j,\beta }≔{ℛ}_{{\widehat{P}}_n^j}^{\beta }$, where, for $P\in 𝒫$,

$${ℛ}_P^{\beta }≔\nu (P)-\nu (P_0)+P_0{\varphi }_P^{\beta }-\sum _{k=1}^{\infty }(1-{\beta }_k){\langle \nu (P)-\nu (P_0),h_k\rangle }_{\mathscr{H}}{h}_k.$$ (26)

The main result of this subsection is as follows.

Theorem 5 (Rate of convergence of regularized one-step estimator). Suppose $\nu$ is pathwise differentiable at $P_0$, ${\beta }_n\in {\ell }_{\ast }^2$ for each $n\in \mathbb{N}$, and both ${ℛ}_n^{j,{\beta }_n}$ and ${𝒟}_n^{j,{\beta }_n}$ are $O_p[{\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2}]$ for $j\in \{1,2\}$. Under these conditions, (25) holds. Moreover, if ${ℬ}_n^{j,{\beta }_n}=O_p({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2})$ for $j\in \{1,2\}$, then

$${\overline{\nu }}_n^{{\beta }_n}-\nu (P_0)=O_p\left({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2}\right).$$ (27)

Eq. 27 suggests it is desirable to select ${\beta }_n$ as small as possible, while still ensuring that the drift, remainder, and bias terms are all $O_p({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2})$. Below we provide three general results that can aid in establishing these conditions. The first two provide ways to guarantee the drift and remainder terms are of no larger order than the variance term $P_n{\varphi }_0^{{\beta }_n}$ in (25), implying that the rate of convergence is determined by the variance and bias terms. The third makes precise our earlier statement that the bias term is smaller when the entries of ${\beta }_n$ are closer to 1, thereby enforcing a lower bound on how small ${\beta }_n$ can be to ensure that the bias term is of the same order as the variance term and, therefore, (27) holds.

Lemma 4 (Sufficient condition for negligible regularized drift terms). Suppose that $\nu$ is pathwise differentiable at $P_0$ and ${(r_n)}_{n=1}^{\infty }$ is a nonnegative sequence. Fix $j\in \{1,2\}$ and, for each $n\in \mathbb{N}$, let ${\beta }_n\in {\ell }_{\ast }^2$. If ${\Vert {\varphi }_n^{j,{\beta }_n}-{\varphi }_0^{{\beta }_n}\Vert }_{L^2(P_0;\mathscr{H})}=o_p(r_n)$, then ${\Vert {𝒟}_n^{j,{\beta }_n}\Vert }_{\mathscr{H}}=o_p(r_n∕n^{1∕2})$.

By taking $r_n={\Vert {\beta }_n\Vert }_{{\ell }^2}$, the above gives a condition for ${𝒟}_n^{j,{\beta }_n}$ to be $o_p({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2})$, and therefore $O_p({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2})$. For the regularized remainder term, the following can be useful.

Lemma 5 (Bound on regularized remainder term). Fix $j\in \{1,2\}$ and $\beta \in {\ell }_{\ast }^2$. If $\nu$ is pathwise differentiable at $P\in 𝒫$, $P_0\ll P$, and ${\varphi }_P^{\beta }\in L^2(P_0;\mathbb{H})$, then

$${\Vert {ℛ}_P^{\beta }\Vert }_{\mathscr{H}}^2=\sum _{k=1}^{\infty }{\beta }_k^2\cdot {({ℛ}_{P,k})}^2\le {\Vert \beta \Vert }_{{\ell }^2}^2\underset{k\in \mathbb{N}}{\text{sup}}{({ℛ}_{P,k})}^2,$$

where ${ℛ}_{P,k}≔{\langle \nu (P)-\nu (P_0),h_k\rangle }_{\mathscr{H}}+P_0{\stackrel{.}{\nu }}_P^{\ast }(h_k)$.

For a given $k\in \mathbb{N}$, ${ℛ}_{P,k}$ corresponds to the remainder term in a von Mises expansion of the real-valued parameter ${\psi }_k:P^{\prime }\mapsto {\langle \nu (P^{\prime }),h_k)\rangle }_{\mathscr{H}}$ (von Mises, 1947), where we note that the pathwise differentiability of $\nu$ at $P$ implies the pathwise differentiability of ${\psi }_k$ at $P$ with canonical gradient ${\stackrel{.}{\nu }}_P^{\ast }(h_k)$.

We now turn to the bias term. For $u\ge 0$, let ${\Vert \cdot \Vert }_u:\mathscr{H}\to [0,+\infty ]$ denote the norm defined by ${\Vert h\Vert }_u^2≔{\sum }_{k=1}^{\infty }{k}^{2u}{\langle h,h_k\rangle }_{\mathscr{H}}^2$, where the dependence of ${\Vert \cdot \Vert }_u$ on the basis ${(h_k)}_{k=1}^{\infty }$ used to construct the regularized one-step estimator is suppressed in the notation.

Lemma 6 (Bound on bias term). For any $u\ge 0$ and $\beta \in {\ell }_{\ast }^2$, ${\Vert {ℬ}_P^{\beta }\Vert }_{\mathscr{H}}\le {\Vert \nu (P)-\nu (P_0)\Vert }_u{\sup }_{k\in \mathbb{N}}(1-{\beta }_k)∕k^u$. If there exists $k\in \mathbb{N}$ such that ${\beta }_k=1$ for all $k\le K$ and ${\beta }_k=0$ for all $k>K$, then this implies that

$${\Vert {ℬ}_P^{\beta }\Vert }_{\mathscr{H}}\le {(K+1)}^{-u}{\Vert \nu (P)-\nu (P_0)\Vert }_u\le 2{(K+1)}^{-u}\underset{P^{\prime }\in 𝒫}{\text{sup}}{\Vert \nu (P^{\prime })\Vert }_u.$$ (28)

Naturally, the upper bounds are only informative if the evaluations of ${\Vert \cdot \Vert }_u$ upon which they rely are finite. Conditions for the finiteness of this norm have been evaluated in several settings. In particular, $\{h\in \mathscr{H}:{\Vert h\Vert }_u<\infty \}$ corresponds to a periodic Sobolev space when $\mathscr{H}=L^2([0,1])$ and ${(h_k)}_{k=1}^{\infty }$ is the trigonometric basis (Proposition 1.14 of Tsybakov, 2009), Sobolev-Laguerre space when $\mathscr{H}=L^2([0,\infty ))$ and ${(h_k)}_{k=1}^{\infty }$ consists of the Laguerre functions (Bongioanni and Torrea, 2008), and Sobolev-Hermite space when $\mathscr{H}=L^2(\mathbb{R})$ and ${(h_k)}_{k=1}^{\infty }$ consists of the Hermite functions (Bongioanni and Torrea, 2006).

In Section 6, we study the selection of ${\beta }_n$ in the context of our examples. We do this by leveraging the bounds from the preceding three lemmas and then deriving the choice of ${\beta }_n$ that balances the variance and bias terms.

5.2. Construction of confidence sets

In what follows we fix $\beta \in {\ell }_{\ast }^2$ and define ${\Gamma }_{\beta }:\mathscr{H}\to \mathscr{H}$ as ${\Gamma }_{\beta }={\sum }_{k=1}^{\infty }{\beta }_k{\langle h,h_k\rangle }_{\mathscr{H}}{h}_k$. The following is the key observation that we use to construct our confidence set for $\nu (P_0)$.

Lemma 7. If $\nu$ is pathwise differentiable at $P$, then its transformation ${\nu }^{\beta }≔{\Gamma }_{\beta }\circ \nu$ is pathwise differentiable at $P$ with local parameter ${\stackrel{.}{\nu }}_P^{\beta }≔{\Gamma }_{\beta }\circ {\stackrel{.}{\nu }}_P$ and $EIF{\varphi }_P^{\beta }\in L^2(P;\mathscr{H})$.

Since ${\nu }^{\beta }$ has an EIF, the methods from Section 4.2 can be used to construct a confidence set for ${\nu }^{\beta }(P_0)$ based on a one-step estimator. This one-step estimator takes the form ${\tilde{\nu }}_n^{\beta }≔\frac{1}{2}{\sum }_{j=1}^2[{\Gamma }_{\beta }\circ \nu ({\widehat{P}}_n^j)+P_n^j{\varphi }_n^{j,\beta }]$. By Theorem 3, the main condition for the asymptotic validity of the resulting confidence set is that ${\tilde{\nu }}_n^{\beta }$ is an asymptotically linear estimator of ${\nu }^{\beta }(P_0)$ with influence function ${\varphi }_0^{\beta }$. Since ${\tilde{\nu }}_n^{\beta }$ is a one-step estimator of ${\nu }^{\beta }(P_0)$, rather than $\nu (P_0)$, ${\tilde{\nu }}_n^{\beta }$ generally differs from the regularized one-step estimator ${\overline{\nu }}_n^{\beta }$ of $\nu (P_0)$ – indeed, ${\tilde{\nu }}_n^{\beta }-{\overline{\nu }}_n^{\beta }=\frac{1}{2}{\sum }_{j=1}^2[{\Gamma }_{\beta }\circ \nu ({\widehat{P}}_n^j)-\nu ({\widehat{P}}_n^j)]$. Nevertheless, conditions on the same regularized remainder and drift terms studied to establish rate guarantees for ${\overline{\nu }}_n^{\beta }$ can ensure the asymptotic linearity of ${\tilde{\nu }}_n^{\beta }$ — see Corollary S1 in the appendix.

Let ${𝒞}_n^{\beta }({\widehat{\zeta }}_n)$ denote an asymptotically valid (1 – α)-level confidence set for ${\nu }^{\beta }(P_0)$ constructed according to the methods in Section 4.2. Any standardization operator ${\Omega }_n$ satisfying the conditions of Theorem 3 may be used when doing this. For example, if ${\Omega }_n$ is taken to be the identity, then, for a cutoff ${\widehat{\zeta }}_n$ selected via the bootstrap, a spherical confidence set for ${\nu }^{\beta }(P_0)$ would take the form

$${𝒞}_n^{\beta }({\widehat{\zeta }}_n)≔\left\{h\in \mathscr{H}:{\Vert {\tilde{\nu }}_n^{\beta }-h\Vert }_{\mathscr{H}}^2\le {\widehat{\zeta }}_n∕n\right\}.$$ (29)

Since the methods in Section 4.2 require the parameter of interest to be fixed and not depend on sample size, when constructing ${𝒞}_n^{\beta }({\widehat{\zeta }}_n)$ we require the choice of $\beta$ to remain fixed as $n\to \infty$. Handling cases where $\beta$ changes with $n$ or is selected data-adaptively is an interesting area for future work. To transform the confidence set for ${\nu }^{\beta }(P_0)$ into one for $\nu (P_0)$, we take the preimage ${\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]≔\{h\in \mathscr{H}:{\Gamma }_{\beta }(h)\in {𝒞}_n^{\beta }({\widehat{\zeta }}_n)\}$. Since ${\nu }^{\beta }(P_0)≔{\Gamma }_{\beta }\circ \nu (P_0)$, this preimage is an asymptotically valid (1 – α)-level confidence set for $\nu (P_0)$ provided ${𝒞}_n^{\beta }({\widehat{\zeta }}_n)$ is an asymptotically valid (1 – α)-level confidence set for ${\nu }^{\beta }(P_0)$. The transformation ${\Gamma }_{\beta }$ has a left inverse if all entries of $\beta$ are nonzero, with ${\Gamma }_{\beta }^{-1}(h)={\sum }_{k=1}^{\infty }{\beta }_k^{-1}{\langle h,h_k\rangle }_{\mathscr{H}}{h}_k$ for $h$ in the image of ${\Gamma }_{\beta }$. Figure 2 illustrates how the map ${\Gamma }_{\beta }^{-1}$ stretches spherical and Wald-type confidence sets for the regularized parameter ${\nu }^{\beta }(P_0)$ into confidence sets for $\nu (P_0)$.

Figure 2:

Projections of the boundaries of confidence sets for the regularized parameter ${\nu }^{\beta }(P_0)$ (top) and original parameter $\nu (P_0)$ (bottom), plotted via pairs of generalized Fourier coefficients with respect to the basis ${(h_k)}_{k=1}^{\infty }$. The transformation applied to confidence sets for ${\nu }^{\beta }(P_0)$ to obtain those for $\nu (P_0)$ stretch the axes by the reciprocals of entries of the regularization parameter $\beta$. Since ${\beta }_k\to 0$ as $k\to \infty$, this stretch factor becomes arbitrarily large as $k\to \infty$.

To simplify the discussion, hereafter we focus on the special case where ${𝒞}_n^{\beta }({\widehat{\zeta }}_n)$ takes the spherical form in (29). In this case, ${\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]$ takes the elliptical form

$$\left\{h\in \mathscr{H}:{\sum }_{k=1}^{\infty }{\beta }_k^2{\left[\frac{1}{2}{\sum }_{j=1}^2\left\{{\langle \nu ({\widehat{P}}_n^j),h_k\rangle }_{\mathscr{H}}+P_n^j{\stackrel{.}{\nu }}_n^{j,\ast }(h_k)\right\}-{\langle h,h_k\rangle }_{\mathscr{H}}\right]}^2\le {\widehat{\zeta }}_n∕n\right\}.$$ (30)

Because ${\beta }_k$ must tend to zero as $k\to \infty$ in order for $\beta$ to belong to ${\ell }^2$, the ${\Vert \cdot \Vert }_{\mathscr{H}}$-diameter of this confidence set, namely ${\sup }_{h,h^{\prime }\in {\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]}{\Vert h-h^{\prime }\Vert }_{\mathscr{H}}$, will not converge to zero with sample size. In contrast, the ${\Vert \cdot \Vert }_{\beta }$-diameter of this confidence set will generally shrink to zero at an n^−1/2-rate, where $\Vert h\Vert \beta ≔{[{\sum }_{k=1}^{\infty }{\beta }_k^2{\langle h,h_k\rangle }_{\mathscr{H}}^2]}^{1∕2}$. Here we note that ${\Vert \cdot \Vert }_{\beta }$ is a norm on $\mathscr{H}$ if all of the entries of $\beta$ are nonzero and is otherwise a seminorm.

The confidence set in (30) also satisfies another desirable property, which can be most easily described by studying a corresponding hypothesis test. For fixed $h_0\in \mathscr{H}$, this test rejects the null hypothesis that $\nu (P_0)=h_0$ in favor of the complementary alternative precisely when $h_0\notin {\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]$. This test asymptotically controls the type I error at level $\alpha$ when ${\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]$ has asymptotically valid coverage, and, by the triangle inequality, is consistent against fixed alternatives when ${\tilde{\nu }}_n^{\beta }$ is a consistent estimator of ${\nu }^{\beta }(P_0)$, ${\mathrm{plim}}_{n\to \infty }{\widehat{\zeta }}_n<\infty$, and all entries of $\beta$ are nonzero. We now show that this test also achieves nontrivial power against a class of n^−1/2-rate local alternatives. In what follows we let ${\mathbb{H}}^{\beta }$ denote a tight $\mathscr{H}$-valued Gaussian random variable that is such that, for each $h\in \mathscr{H}$, the marginal distribution ${\langle {\mathbb{H}}^{\beta },h\rangle }_{\mathscr{H}}$ follows a $N(0,E_0[{\langle {\varphi }_0^{\beta }(Z),h)\rangle }_{\mathscr{H}}^2])$ distribution, where ${\varphi }_0^{\beta }$ is as defined in Lemma 7. Unlike in the rest of the paper, the following theorem requires all entries of $\beta$ to be nonzero, since its proof will rely on ${\Vert \cdot \Vert }_{\beta }$ being a norm.

Theorem 6 (Local power of regularized hypothesis test). Fix $\beta \in {\ell }^2\cap {(0,1]}^{\mathbb{N}}and{h}_0\in \mathscr{H}$. Suppose $\nu$ is pathwise differentiable at $P_0$, $\nu (P_0)=h_0$, ${\tilde{\nu }}_n^{\beta }$ is an asymptotically linear estimator of ${\nu }^{\beta }(P_0)$ with influence function ${\varphi }_0^{\beta }$, and ${\widehat{\zeta }}_n$ is a consistent estimator of the $(1-\alpha )$-quantile ${\zeta }_{1-\alpha }of{\Vert {\mathbb{H}}^{\beta }\Vert }_{\mathscr{H}}^2$. Fix $\{P_{ϵ}:ϵ\}\in 𝒫(P_0,𝒫,s)$ such that ${\Vert {\stackrel{.}{\nu }}_0(s)\Vert }_{\mathscr{H}}>0$. If ${\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]$ is as defined in (30), then

$$P_{ϵ=n^{-1∕2}}^n\left\{h_0\notin {\Gamma }_{\beta }^{-1}[{𝒞}_n^{\beta }({\widehat{\zeta }}_n)]\right\}\stackrel{n\to \infty }{\to }\text{Pr}\left\{{\Vert {\mathbb{H}}^{\beta }+{\stackrel{.}{\nu }}_0^{\beta }(s)\Vert }_{\mathscr{H}}^2>{\zeta }_{1-\alpha }\right\}>\alpha .$$

Also, $h_n≔\nu (P_{ϵ=n^{-1∕2}})$ is an n^−1/2-rate local alternative in that ${\Vert h_n-h_0\Vert }_{\mathscr{H}}=O(n^{-1∕2})$.

The above focuses on local alternatives that are defined via smooth parametric submodels of $𝒫$. It is worth noting, however, that by selecting such a submodel, the first-order direction of the local alternative, defined by the value of the local parameter ${\stackrel{.}{\nu }}_0(s)$, is fixed as the sample size grows. Since the ${\Vert \cdot \Vert }_{\mathscr{H}}$-diameter of our confidence set does not decay with sample size, it does not appear that our test will generally have nontrivial asymptotic power against local alternatives whose direction is not fixed and whose ${\Vert \cdot \Vert }_{\mathscr{H}}$-magnitude decays at an $n^{-1∕2}$ rate.

5.3. Tuning parameter selection

We begin by discussing tuning parameter selection for the regularized one-step estimator of $\nu (P_0)$, and then we subsequently discuss confidence set construction. Evaluating the regularized one-step estimator requires selecting three key components: the initial estimator ${\widehat{P}}_n^j$, orthonormal basis ${(h_k)}_{k=1}^{\infty }$, and regularization parameter ${\beta }_n$. Similarly to finite-dimensional problems, the suitability of an initial estimator ${\widehat{P}}_n^j$ will depend on the parameter of interest and the form of its corresponding remainder term as defined in (26). In the next section, we will study these remainder terms in our illustrative examples. In what follows we discuss the choice of basis ${(h_k)}_{k=1}^{\infty }$ and regularization parameter ${\beta }_n$.

Following the literature on series estimators (Chen, 2007) and motivated by the bound in (28), we suggest choosing the basis ${(h_k)}_{k=1}^{\infty }$ so that the span of finitely many initial basis elements yields an accurate approximation of $\nu (P)$, $P\in 𝒫$, provided these Hilbert random elements are smooth enough. Here, smoothness is characterized by the rate of decay of the generalized Fourier coefficients ${\langle \nu (P),h_k)\rangle }_{\mathscr{H}}$ as $k\to \infty$. If $\mathscr{H}$ is an $L^2(\lambda )$ space with $\lambda$ the Lebesgue measure on the real line or a bounded subset thereof, then common choices of bases include Legendre polynomials, Laguerre functions, Hermite functions, trigonometric polynomials, and wavelets, among others. If $\mathscr{H}$ is instead an $L^2(Q)$ space with $Q$ an absolutely continuous probability measure on ${\mathbb{R}}^d$, then an orthonormal basis for $\mathscr{H}$ can be obtained in several ways. One is to multiply an orthornormal basis ${(g_k)}_{k=1}^{\infty }$ for $L^2({\mathbb{R}}^d)$ by the root-density q^1/2 of $Q$; in particular, ${(q^{1∕2}{g}_k)}_{k=1}^{\infty }$ is an orthonormal basis for $L^2(Q)$. The suitability of these bases for characterizing the smoothness of $\nu (P_0)$ can be assessed on a case-by-case basis. If d = 1, then another approach involves transforming an orthornormal basis ${(g_k)}_{k=1}^{\infty }$ for $L^2([0,1])$ via the cumulative distribution function $F_Q$ of $Q$; in particular, ${(g_k\circ F_Q)}_{k=1}^{\infty }$ is an orthonormal basis of $L^2(Q)$. Other orthonormal bases of $L^2(Q)$-spaces are also readily available for certain choices of $Q$, such as if $Q$ is a Gaussian measure (Chapter 9 of Da Prato, 2006). Orthonormal bases for some non-$L^2$ spaces, such as Sobolev Hilbert spaces, are also well studied (Marcellán and Xu, 2015).

We propose using cross-validation to choose the regularization parameter ${\beta }_n$. If there is uncertainty about which orthonormal basis ${(h_k)}_{k=1}^{\infty }$ should be used, this could also be selected via cross-validation, though the discussion that follows focuses on selecting ${\beta }_n$. Our proposal is based on the following loss for $\nu (P_0)$, which relies on an estimate $P$ of $P_0$:

$${ℒ}_P(z;h)≔\frac{1}{2}{\Vert h-\nu (P)\Vert }_{\mathscr{H}}^2-{\stackrel{.}{\nu }}_P^{\ast }[h-\nu (P)](z).$$ (31)

The algorithm to implement the proposed cross-validation scheme can be found in Appendix G. There, we also explain why ${ℒ}_P$ is a reasonable loss function to use for estimating $\nu (P_0)$. The cross-validation algorithm will be easiest to implement when the search for a regularization parameter is reduced to a search over a finite subset $B_n$ of ${\ell }_{\ast }^2$. A particularly simple choice of $B_n$ consists of the $K_n+1$ elements of ${\ell }^2$ that take the value 1 in their first $k\in \{0,1,\dots ,K_n\}$ entries and zero in all remaining entries. Selecting over a finite set of possible values is also desirable since there are oracle inequalities for cross-validation selectors over finite sets provided the loss function satisfies appropriate regularity conditions (van der Laan and Dudoit, 2003; van der Vaart et al., 2006). Exploring the applicability of these conditions in our setting is an interesting area for future study.

We now turn to tuning parameter selection for confidence set construction. The considerations for selecting the basis ${(h_k)}_{k=1}^{\infty }$ are similar to those discussed above for estimation, and so we focus on selecting the regularization parameter $\beta$. As our coverage guarantees rely on the regularization parameter $\beta$ being fixed and not depending on sample size, cross-validation should not be used to select this quantity. Instead, we recommend choosing a fixed, square-summable sequence $\beta$. One natural family of choices is given by setting $\beta ={({\beta }_k)}_{k=1}^{\infty }$ with ${\beta }_k=1∕[1+{(k∕c)}^{1∕2+d}]$ for $c$, $d>0$. The parameters c and d control the stretch and polynomial rate of decay of the function $k\mapsto {\beta }_k$, respectively. Finally, we note that, to ensure computational feasibility, the infinite sum used to define the confidence set in (30) can be truncated at a large, finite number of terms $K_n^{\star }$ that grows with n, without adversely affecting coverage. This follows from the fact that the set on the right-hand side of (30) can only be made larger by replacing the sum from k = 1 to ∞ with one from k = 1 to $K_n^{\ast }$.

6. Study of (regularized) one-step estimators in our examples

We now revisit Examples 1a and 1b from Section 3. For each, we evaluate the plausibility of the regularity conditions that guarantee our theoretical results hold. We revisit the other two examples from Section 3 in Appendices B.3.3 and B.4.4. In what follows, C denotes a generic finite constant whose value may differ from display to display.

Example 1a (Counterfactual density function, continued). Since there is no EIF in this example, we study a regularized one-step estimator ${\overline{\nu }}_n^{{\beta }_n}$. This estimator is defined based on an orthonormal basis ${(h_k)}_{k=1}^{\infty }$ of $\mathscr{H}$ and a regularization parameter ${\beta }_n\in {\ell }_{\ast }^2$. Guidance on how to choose these quantities is given in Section 5.3.

Theorem 5 relies on the negligibility of regularized remainder and drift terms ${ℛ}_n^{j,{\beta }_n}$ and ${𝒟}_n^{j,{\beta }_n}$ and bias terms ${ℬ}_n^{j,{\beta }_n}$. In Appendix B.1.3, we use Lemma 5 and the strong positivity assumption to show there exists a constant $C$ that does not depend on $\beta$ such that, for all $P\in 𝒫$,

$${\Vert {ℛ}_P^{\beta }\Vert }_{\mathscr{H}}\le C{\Vert \beta \Vert }_{{\ell }^2}{\Vert g_P(1\mid \cdot )-g_0(1\mid \cdot )\Vert }_{L^2(P_{0,X})}{\Vert p_{Y\mid A=1,X}-p_{0,Y\mid A=1,X}\Vert }_{L^2({\tau }_0)},$$ (32)

where ${\tau }_0$ denotes the product measure ${\lambda }_Y\times P_{0,X}$ and, within the $L^2({\tau }_0)$ norm, $p_{Y\mid A=1,X}-p_{0,Y\mid A=1,X}$ denotes the function $(y,x)\mapsto p_{Y\mid A,X}(y\mid 1,x)-p_{0,Y\mid A,X}(y\mid 1,x)$. The upper bound in (32) depends on three quantities: the ${\ell }^2$-magnitude of $\beta$, the $L^2(P_{0,X})$-distance between the propensities $g_P(1\mid \cdot )$ and $g_0(1\mid \cdot )$, and a root-MISE of the conditional distribution of $Y\mid A=1$, $X$ under $P$ relative to that under $P_0$, where the mean is taken across values of $X\sim P_0$. Applying the above inequality to study the remainder term ${ℛ}_n^{j,{\beta }_n}$ that Theorem 5 requires to be $O_p[{\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2}]$, we see that ${ℛ}_n^{j,{\beta }_n}$ will satisfy this condition provided typical n^−1/4-rate conditions are satisfied by the estimators of two nuisance functions, namely the propensity to receive treatment and the conditional density of the outcome given treatment and covariates. Such conditions have been discussed extensively in the literature across a variety of problems (e.g,. van der Laan and Rubin, 2006; Chernozhukov et al., 2018), and tend to hold when the needed nuisance functions are sufficiently smooth or parsimonious relative to the dimension of $X$ and an appropriate estimation strategy is used. For example, suppose that $X$ is continuous and ${\mathbb{R}}^d$-valued, $g_0(1\mid \cdot )$ and $(x,y)\mapsto p_{0,Y\mid A,X}(y\mid 1,x)$ are Hölder smooth with Hölder exponents b and c, respectively (Robins et al., 2008). If $g_0(1\mid \cdot )$ is estimated via a kernel regression and $(x,y)\mapsto p_{0,Y\mid A,X}(y\mid 1,x)$ is estimated via conditional kernel density estimation, each using kernels of sufficiently high orders, then the above can be used to show that ${ℛ}_n^{j,{\beta }_n}=O_p[{\Vert {\beta }_n\Vert }_{{\ell }^2}{n}^{-\frac{b}{2b+d}}{n}^{-\frac{c}{2c+d+1}}]$, and so ${ℛ}_n^{j,{\beta }_n}$ achieves the desired $O_p[{\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2}]$ rate provided $bc\ge d(d+1)∕4$. Alternative estimation strategies that often perform well in practice even when these smoothness assumptions fail, such as those based on random forests (Ho, 1995) or gradient boosting (Friedman, 2001), could also be used. For the regularized drift terms, Lemma 4 shows that ${𝒟}_n^{j,{\beta }_n}$ is $o_p({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2})$ whenever ${\Vert {\varphi }_n^{j,\beta }-{\varphi }_0^{\beta }\Vert }_{L^2(P_0;\mathscr{H})}=o_p({\Vert {\beta }_n\Vert }_{{\ell }^2})$. To provide conditions under which this is true, we use that there exists a constant $C>0$ that does not depend on $\beta$ such that, for all $P\in 𝒫$, ${\Vert {\varphi }_P^{\beta }-{\varphi }_0^{\beta }\Vert }_{L^2(P_0;\mathscr{H})}$ is upper bounded by

$$𝒞{\Vert \beta \Vert }_{{\ell }^2}\left({\Vert g_P(1\mid \cdot )-g_0(1\mid \cdot )\Vert }_{L^2(P_{0,X})}+{\Vert p_{Y\mid A=1,X}-p_{0,Y\mid A=1,X}\Vert }_{L^2({\tau }_0}\right).$$

Hence, ${\Vert {\varphi }_n^{j,{\beta }_n}-{\varphi }_0^{{\beta }_n}\Vert }_{L^2(P_0;\mathscr{H})}=o_p({\Vert {\beta }_n\Vert }_{{\ell }^2})$ whenever the propensity and conditional density of $Y\mid A=1,X$ under ${\widehat{P}}_n^j$ are consistent according to the norms above. Consistency is a weaker requirement than the rate conditions imposed to ensure the negligibility of ${ℛ}_n^{j,{\beta }_n}$, so it is reasonable to expect that ${𝒟}_n^{j,{\beta }_n}$ will be negligible when ${ℛ}_n^{j,{\beta }_n}$ is negligible.

From Lemma 6, an upper bound on the rate at which the bias terms ${ℬ}_n^{j,{\beta }_n}$ will decay to zero can be derived by bounding either ${\Vert \nu ({\widehat{P}}_n^j)-\nu (P_0)\Vert }_u$ or ${\sup }_{P\in 𝒫}{\Vert \nu (P)\Vert }_u$ for some u ≥ 0. The latter of these quantities is no more than c < ∞ if the parameter space {$\nu (P):P\in 𝒫$} is a subset of the Sobolev ellipsoid {$h\in \mathscr{H}:{\Vert h\Vert }_u\le c$}. In this case, when the first K_n entries of ${\beta }_n$ are one and all others are zero, Lemma 6 shows that ${\Vert {ℬ}_n^{j,{\beta }_n}\Vert }_{\mathscr{H}}\le c∕{(K_n+1)}^u$; if the earlier-discussed regularity conditions hold so that the regularized remainder and drift terms are $O_p({\Vert {\beta }_n\Vert }_{{\ell }^2}∕n^{1∕2})$, then this yields that, when K_n is of the order $n^{1∕(2u+1)}$,

$${\overline{\nu }}_n^{{\beta }_n}-\nu (P_0)=O_p(n^{-u∕(2u+1)}).$$ (33)

This analysis bears similarity to the study of projection estimators (Theorem 1.9 of Tsybakov, 2009), but with the added requirement that drift and remainder terms must be considered.

The rate of convergence in (33) was derived based on the looser of the two bounds in (28). While the former bound would give tighter bounds on the bias term when ${\Vert \nu ({\widehat{P}}_n^j)-\nu (P_0)\Vert }_u$ converges to zero in probability at some rate, it is unclear whether there are initial estimators ${\widehat{P}}_n^j$ of P₀ that would achieve this. Indeed, since ${\Vert \cdot \Vert }_u$ is a stronger norm than ${\Vert \cdot \Vert }_{\mathscr{H}}$, probabilistic convergence relative to ${\Vert \cdot \Vert }_{\mathscr{H}}$ is insufficient to guarantee convergence relative to ${\Vert \cdot \Vert }_u$. Looking to identify or develop initial estimators of $P_0$ for which ${\Vert \nu ({\widehat{P}}_n^j)-\nu (P_0)\Vert }_u\stackrel{p}{\to }0$ is an interesting area for future study, since, when such an initial estimator is used, faster rates of convergence for ${\overline{\nu }}_n^{{\beta }_n}$ than that given in (33) may be established.

While the discussion above focused on the regularized one-step estimator, similar arguments can be used to analyze the confidence sets introduced in Section 5.2 for fixed $\beta \in {\ell }_{\ast }^2$. Indeed, Corollary S1 in the appendix shows that the key quantities to bound to establish the validity of these confidence sets are ${ℛ}_n^{j,{\beta }_n}$ and ${𝒟}_n^{j,{\beta }_n}$ – in particular, both of these quantities should be $o_p(n^{-1∕2})$. We have already bounded these quantities above when studying the regularized one-step estimator of $\nu (P_0)$. In particular, these conditions will hold if each of $p_{0,Y\mid A,X}$ and $g_0(1\mid \cdot )$ is estimated at a faster-than-n^−1/4 rate according to the norms in (32).

In the special case where $\beta$ is a vector whose first K entries are 1 and whose remaining entries are 0, the estimator ${\tilde{\nu }}_n^{\beta }$ used to construct our confidence sets coincides with the $L^2({\lambda }_Y)$ projection estimator studied in Corollary 2 of Kennedy et al. (2021). In our notation, that estimator can be viewed as estimating the parameter ${\Gamma }_{\beta }\circ \nu (P_0)$, though if K grows with n, as would typically occur under the model selection strategy described by Kennedy et al., then it can be viewed as estimating $\nu (P_0)$ as well. This estimator differs from the regularized one-step estimator ${\overline{\nu }}_n^{\beta }$ that we have recommended using for estimation of $\nu (P_0)$, with the estimators differing by the $L^2({\lambda }_Y)$ projection of $\frac{1}{2}{\sum }_{j=1}^2\nu ({\widehat{P}}_n^j)$ onto the orthogonal complement of the linear span of the first K elements of the chosen basis for $L^2({\lambda }_Y)$. It is not immediately clear whether one of these two estimators should be preferred over the other in general, though our upcoming simulation study supports using ${\overline{\nu }}_n^{\beta }$, especially when n is small. The decision between using these estimators of $\nu (P_0)$ can be summarized as follows: ${\overline{\nu }}_n^{\beta }$ should be used to estimate $\nu (P_0)$ if $\frac{1}{2}{\sum }_{j=1}^2{\Gamma }_{\beta }\circ \nu ({\widehat{P}}_n^j)$ attains a lower MISE for estimating ${\Gamma }_{\beta }\circ \nu (P_0)$ than does the zero function, and ${\tilde{\nu }}_n^{\beta }$ should be preferred otherwise.

Kennedy et al. (2021) also proposes an approach for making inference about the difference between two counterfactual densities using any of several distance metrics. For the $L^2({\lambda }_Y)$ metric, this inference is based upon first-order asymptotics the parameter $\psi (P_0)≔{\Vert {\nu }_1(P_0)-{\nu }_0(P_0)\Vert }_{L^2({\lambda }_Y)}^2$, where ${\nu }_1$ is equal to the counterfactual density parameter $\nu$ defined in (11) and ${\nu }_0$ takes the same form but with $p_{Y\mid A,X}(y\mid 1,x)$ replaced by $p_{Y\mid A,X}(y\mid 0,x)$. There, they note an oft-confronted difficulty (Luedtke et al., 2019; Williamson et al., 2021) wherein their estimator of $\psi (P_0)$ converges to zero at a faster-than-n^−1/2 rate under the null hypothesis that ${\psi }_1(P_0)={\psi }_0(P_0)$, leading them to propose a conservative threshold to test this null based on the maximum of the estimated standard error of their estimator and n^−1/2. Since the pathwise differentiability of ${\nu }_1$ and ${\nu }_0$ implies the pathwise differentiability of ${\nu }_1-{\nu }_0$ — with efficient influence operator equal to the difference of the efficient influence operators of ${\nu }_1$ and ${\nu }_0$ — our regularized one-step estimation framework provides an alternative, non-conservative means to test this hypothesis by constructing a confidence set for this parameter for fixed $\beta \in {\ell }^2\cap {(0,1]}^{\mathbb{N}}$ and checking whether it contains zero.

We now compare our inferential procedure in this example to that of Kennedy et al. (2021). We start by comparing the size of the dual confidence sets. The method from Kennedy et al. can be used to construct a confidence set by inverting tests of whether ${\psi }_h(P_0)≔{\Vert {\nu }_1(P_0)-{\nu }_0(P_0)\Vert }_{L^2({\lambda }_Y)}^2$ is equal to zero across values of $h\in L^2({\lambda }_Y)$; the threshold for each h-dependent test is determined using the same conservative threshold methodology as when h = 0. The $L^2({\lambda }_Y)$ and ${\Vert \cdot \Vert }_{\beta }$ diameters of this confidence set both decay at rates no faster than n^−1/4. In contrast, the ${\Vert \cdot \Vert }_{\beta }$ diameter of our confidence set decays at a quadratically-faster rate of n^−1/2, while the $L^2({\lambda }_Y)$ diameter does not decay at all. Nevertheless, since ${\Vert h\Vert }_{\beta }=0$ if and only if ${\Vert h\Vert }_{L^2({\lambda }_Y)}=0$, our confidence set will exclude any particular $h\ne {\nu }_1(P_0)-{\nu }_0(P_0)$ with probability tending to one. As a practical matter, our confidence set will exclude functions h that differ smoothly from ${\nu }_1(P_0)-{\nu }_0(P_0)$ relative to the basis ${(h_k)}_{k=1}^{\infty }$ at smaller sample sizes than will the n^−1/4-rate confidence set, and will otherwise require larger sample sizes; here, smoothness is characterized by the decay rate of ${\langle {\nu }_1(P_0)-{\nu }_0(P_0)-h,h_k\rangle }_{\mathscr{H}}$ as $k\to \infty$. Our dual hypothesis test of whether ${\nu }_1(P_0)-{\nu }_0(P_0)=0$ will also satisfy the local power guarantee from Theorem 6. We investigate the properties of this test and compare it to the test proposed in Kennedy et al. (2021) in our upcoming simulation study.

Example 1b (Bandlimited counterfactual density function, continued). Since there is an EIF in this example, we study a (non-regularized) one-step estimator. Theorem 2 relies on the negligibility of the remainder and drift terms, namely that they are $o_p(n^{-1∕2})$. Let ${\underset{¯}{ℛ}}_P≔\underset{¯}{\nu }(P)+P_0{\underset{¯}{\varphi }}_P-\underset{¯}{\nu }(P_0)$ denote the remainder term for a generic $P\in 𝒫$. In Appendix B.2.5 we show that there exists a C < ∞ that does not depend on $P\in 𝒫$ such that

$${\Vert {\underset{¯}{ℛ}}_P\Vert }_{\underset{¯}{\mathscr{H}}}\le C{\Vert g_P(1\mid \cdot )-g_0(1\mid \cdot )\Vert }_{L^2(P_{0,X})}{\Vert p_{Y\mid A=1,X}-p_{0,Y\mid A=1,X}\Vert }_{L^2({\tau }_0)}.$$ (34)

Hence, for ${\underset{¯}{ℛ}}_n^j:{\underset{¯}{ℛ}}_{{\widehat{P}}_n^j}$ to be $o_p(n^{-1∕2})$, the products of the rate of convergence of $g_{{\widehat{P}}_n^j}(1\mid \cdot )$ to $g_0(1\mid \cdot )$ and $p_{Y\mid A=1,X}$ to $p_{0,Y\mid A=1,X}$ according to the norms above must be faster than n^−1/2. This results in the same n^−1/4-type requirement that we discussed below (32) for Example 1a, except, because we only focus on rates of convergence for regularized one-step estimators (rather than weak convergence), there we only required this product to be at least as fast as n^−1/2 rather than faster, as we require here. Also similarly to Example 5.3, for each $j\in \{1,2\}$, ${\Vert {\underset{¯}{\varphi }}_n^j-{\underset{¯}{\varphi }}_0\Vert }_{L^2(P_0\underset{¯}{\mathscr{H}})}$ can be shown to be $o_p(1)$ provided the propensity and conditional density of $Y\mid A=1$, $X$ under ${\widehat{P}}_n^j$ converge to in probability $g_0(1\mid \cdot )$ and $p_{0,Y\mid A=1,X}$ according to the norms in (34). Hence, Lemma 3 ensures that the drift term ${\underset{¯}{𝒟}}_n^j$ is $o_p(n^{-1∕2})$ under this condition, and so the conditions of Theorem 2 hold under reasonable conditions. If the operator ${\Omega }_0$ used to construct a confidence set for $\underset{¯}{\nu }(P_0)$ is fixed, then the conditions of Theorem 4 are also satisfied, justifying the use of the bootstrap in confidence set construction. If instead the regularized covariance operator described in Appendix D is used, then the bootstrap will still yield an asymptotically valid confidence set for $\underset{¯}{\nu }(P_0)$ provided the estimator of ${\Omega }_0$ described in that appendix is used (see Lemma S12 for details).

7. Simulation study

7.1. Overview

We conduct a simulation study to evaluate the finite-sample properties of our one-step estimation framework, both in settings where an EIF exists and in ones where it does not. All of these settings involve drawing inferences about the distributions or densities of counterfactual outcomes (Examples 1 and 3). Our implemented methods are available in the HilbertOneStep R package (Luedtke, 2023).

We consider multiple data-generating processes, each indexed by real-valued probability distributions $Q(0)$ and $Q(1)$. Sampling from a generic such process involves drawing n iid samples from P₀, where n takes values in {250, 500, 1000, 2000, 4000}. An observation Z = (X, A, Y) from P₀ is sampled as follows:

$$\begin{gathered} (Y(0),Y(1))\sim Q(0)\times Q(1),V\mid Y(0),Y(1)\sim N(0_5,{\text{Id}}_5), \\ X=\frac{V}{2}+(V+1)1\{Y(1)>0\};A\mid Y(0),Y(1),V,X\sim \text{Bern}[\frac{1}{20}+\frac{9}{10}\cdot \text{expit}(X_1)], \end{gathered}$$

and then letting $Y=AY(1)+(1-A)Y(0)$. For a ∈ {0, 1}, Y (a) is the counterfactual outcome if treatment A = a were assigned. Since $Y(a)⫫A\mid X$ and the positivity assumption is satisfied, the density of Y (a) takes the form in (11) when a = 1 and otherwise is the same but with $p_{Y\mid A,X}(y\mid 1,x)$ replaced by $p_{Y\mid A,X}(y\mid 0,x)$. Unless otherwise specified, estimates of performance are based on 1000 Monte Carlo repetitions.

We estimate all needed nuisance functions using the same approaches as Kennedy et al. (2021). In particular, we estimate the marginal of X with the empirical distribution, the conditional distribution of A given X using the ranger package (Wright and Ziegler, 2017), and the conditional density of Y given (A, X) using ranger and a Gaussian kernel weighted outcome with bandwidth selected by Silverman’s rule. When implementing the quadratic forms used to define our confidence sets, we use grids of 500 points on the support of Y — see Appendix F for details.

7.2. Performance of the regularized one-step estimator in Example 1a

We evaluate the performance of the regularized one-step estimator of the counterfactual density of Y (1) from Example 1a when $\mathscr{H}=L^2([0,1])$. We consider three choices of Q(1), which are displayed in Figure S2 in the appendix. The behavior of Y (1) on the boundaries of its support, namely 0 and 1, differs across the three settings; to emphasize this, we label them ‘zero on both sides’, ‘nonzero on both sides’, and ‘spike on left side’. The cross-validation strategy outlined in Section 5.3 is used to select the regularization parameter $\beta$ over the elements of ${\ell }^2$ that take the value 1 in their first K ≤ 16 entries and 0 in all others. To evaluate sensitivity to the choice of basis, we evaluate our estimator based on the cosine basis, with $h_k(y)=2^{1∕2}\cos [\pi (k-1)y]$, and a rescaled Legendre basis, with h_k(y) proportional to the k-th Legendre function applied to 2y – 1. We also evaluate the use of cross-validation to select between these bases.

Performance is compared to that of a plugin estimator and also the estimation strategy implemented in the npcausal package (Kennedy et al., 2021), which is a series estimator of the L²([0, 1]) projection of the counterfactual density onto the first K terms of the cosine basis. We use the cross-validation scheme implemented in that package to select a value of K ≤ 16. Though npcausal does not return the estimated density function, we tweaked its open-source code to extract this information.

Figure 3 displays the estimators’ MISEs. In all settings, the regularized one-step estimators are outperformed by the plug-in estimator at small sample sizes, but their relative performances improve as n grows and eventually exceed or trend towards exceeding that of the plugin. Compared to the regularized one-step estimator with the cosine basis, npcausal has MISE that is twice as large at small sample sizes in two of the three scenarios. In one of these scenarios, npcausal’s performance improves with n, but is still worse than all the other estimators. In the other, its performance dramatically improves between n = 2000 to 4000 from the worst of all the estimators to slightly better than the others. In the remaining scenario, npcausal and the regularized one-step estimator with the cosine basis perform similarly. Among the regularized one-step estimators, using the Legendre basis outperforms using the cosine basis in one scenario, while the two perform similarly otherwise. Selecting the basis via cross-validation yields an estimator that is about as good as the one based on the Legendre basis in all scenarios.

Figure 3:

Mean integrated squared error (MISE) versus sample size (n) in Example 1a for five estimators across the three data generating processes considered. Both axes are log-transformed.

7.3. Properties of hypothesis tests from Examples 1a and 3

We evaluate 5% level tests of the null hypothesis that Q(1) = Q(0) against the complementary alternative. The first class of tests uses the results from Example 1a to check if zero is included in an L²([0, 1]) confidence set for the difference $\nu (P_0)={\nu }_1(P_0)-{\nu }_0(P_0)$ of the densities of Q(1) and Q(0). We obtain spherical and Wald-type $L^2([0,1])$ confidence sets for the regularized parameter ${\nu }^{\beta }(P_0)$ using cosine and Legendre bases. Both are then transformed into elliptical confidence sets for $\nu (P_0)$ using the approach from Section 5.2. The Wald-type confidence sets are defined with the correlation-based standardization operator ${\Omega }_n$ from Appendix F with $\lambda =0.5$. For the regularization parameter $\beta$, we let ${\beta }_k=1∕[1+{(k∕c)}^2]$ and consider values of c ∈ 2.5, 5, 10. Results for c = 5 are reported in the main text, while others appear in the appendix. Additionally, we examine a test based on the Gaussian kernel MMD between $Q(1)$ and $Q(0)$, which depends on a bandwidth choice of 0.5, 1, and 2 times median{$\mid Y_i-Y_j\mid :1\le i<j\le n$} (Garreau et al., 2017). We report results for the middle value in the main text and others in the appendix. We compare performance to the asymptotically conservative test from Kennedy et al. (2021), implemented using npcausal.

We set Q(1) to its value from the ‘nonzero on both sides’ simulation setting and consider different values of Q(0). We explore the null hypothesis with Q(0) = Q(1), and, for k ∈ {1, 2, … , 7}, the alternative hypothesis with ${\nu }_0(P_0)(y)={\nu }_1(P_0)(y)+\cos (k^2\pi y)∕3$, denoted as ‘Alt k²’. By examining these alternatives, we assess the power decay of our tests for ${\nu }_1(P_0)-{\nu }_0(P_0)=0$ as the direction of the alternative corresponds to that of a higher-frequency function in the cosine basis.

Figure 4 presents the type I error and power of the tests. In terms of type I error, the five tests based on our one-step estimation framework achieve or nearly achieve the nominal 5% level. Surprisingly, the npcausal test has a type I error over three times the nominal level at smaller sample sizes, despite having an asymptotic rejection probability of zero. However, as n increases, the type I error converges towards its expected conservative asymptotic behavior. As for power, our tests that regularize using the cosine basis display the anticipated power decay for rejecting Alt k² as k increases. Tests using the Legendre basis do not exhibit the same monotonic dependence on k. The Wald-type test with the cosine basis demonstrates noticeably higher power for larger k alternatives than the spherical test, while this trend is less apparent for the Legendre basis. The MMD test has high power for detecting the smoothest alternative, Alt 1, but low or no power against all others. It is important to note that the alternatives we have considered become quite nonsmooth as k increases, making MMD’s poor finite-sample performance for detecting them potentially acceptable. Figure S3 in the appendix illustrates test performance with various tuning parameter choices. Overall, the results align with expectations: for tests based on Example 1a, enhanced power against rougher alternatives (k larger) comes at the cost of reduced power against smoother alternatives when later entries of the regularization parameter $\beta$ are increased, and vice versa. For MMD, modifying the bandwidth results in the same tradeoff.

Figure 4:

Type I error and power of tests from Examples 1a and 3.

Appendix I.1 presents simulation results evaluating our confidence sets for a bandlimited counterfactual density. Nominal coverage is achieved for all sample sizes considered. There, we also highlight the disadvantage of using our L² confidence sets when a point evaluation of the counterfactual density, rather than the function itself, is the true quantity of interest.

8. Discussion

The lack of existence of an EIF that we have confronted in parts of this work bears resemblance to the lack of existence of higher-order influence functions for many real-valued parameters (Robins et al., 2008; van der Vaart, 2014; Robins et al., 2017). There, the nonexistence of these objects owes to the lack of a suitable Riesz-representation-type theorem for multilinear forms. In our case, it owes to the fact that the efficient influence operator ${\stackrel{.}{\nu }}_P^{\ast }(\cdot )(z):\mathscr{H}\to \mathbb{R}$ typically fails to be (P-a.s.) bounded and linear, even though ${\stackrel{.}{\nu }}_P^{\ast }:\mathscr{H}\to {\stackrel{.}{𝒫}}_P$ is. Though the technical details of the two problems differ, similar solutions work for both: replace the operator that does not satisfy a Riesz-type representation by an approximation that does. In our case, this involved studying the $\beta$-regularized efficient influence operator $r_P^{\beta }$. In future work, it would be interesting to investigate the possibility of defining higher-order influence functions for Hilbert-valued parameters. We have shown that, under mild conditions, a first-order EIF exists when the Hilbert space is an RKHS, making this case a natural starting point for exploration.

Another interesting area for future work is to develop a systematic means to select the tuning parameter $\beta$ needed to define our confidence sets when an EIF does not exist. Although our asymptotic guarantees hold for any fixed choice of $\beta \in {\ell }_{\ast }^2$ and our numerical studies shed light on how different choices of $\beta$ improve power against different alternatives, it would be desirable to have an automated means of selecting this parameter. One possible approach would involve drawing inspiration from the choice of kernel for two-sample tests based on the MMD (Gretton et al., 2012). Like our approach, these tests rely on selecting a fixed tuning parameter — in their case, a kernel — as n grows. In that context, an appealing heuristic choice of the bandwidth parameter indexing the kernel has been developed, provided a radial basis function kernel is used (Garreau et al., 2017). It would be of interest to develop a similar heuristic in our setting.

Another area for future study involves extending our results to cases where the Hilbert space depends on the data-generating distribution. As a simple example, in the regression setting from Example 5 in the appendix, we may want to evaluate performance relative to $L^2({\lambda }_X)$ with ${\lambda }_X$ equal to the marginal distribution $P_{0,X}$ of X under P₀, rather than some fixed known measure, such as the Lebesgue measure. The definition of pathwise differentiability at P₀ goes through unchanged in that case. However, because the efficient influence operator at an initial estimate ${\widehat{P}}_n^j$ of P₀ depends on P₀ when $\mathscr{H}=L^2(P_{0,X})$, the regularized one-step estimator we have presented in this work cannot be evaluated. A natural workaround would be to modify the definition of this estimator to use the efficient influence operator of $\nu$ at ${\widehat{P}}_n^j$ relative to a Hilbert space that is indexed by ${\widehat{P}}_n^j$, rather than P₀, and replace ${(h_k)}_{k=1}^{\infty }$ by a basis of this space. We leave the study of this estimator to future work.

After establishing that many Hilbert-valued parameters of interest are pathwise differentiable, we focused on developing and studying a particular estimation framework that leverages this property, namely one-step estimation. This framework has the benefit that there is a closed-form expression for the resulting estimators, which simplifies the study of their convergence properties and construction of corresponding confidence sets. While this approach has advantages, it would be worth considering alternative frameworks in future work. As one example, an M-estimator based on the loss that we introduced in (31) could also be considered. Foster and Syrgkanis (2019) offers a general method for determining the convergence rates of these estimators. Additional research is needed to investigate their weak convergence properties and the possibility of using them to construct confidence sets.

Supplementary Material

Figure S1

Figure S1: Actual versus nominal coverage of confidence sets for the bandlimited density function based on 5000 Monte Carlo repetitions. The inset displays nominal coverage values that are of particular interest in practice.

Figure S2

Figure S2: Densities Q(1) of Y(1) used in the three settings considered in the evaluation of estimators in Example 1a. Each of these densities is a uniform mixture $\frac{1}{3}{\sum }_{k=1}^3\mathrm{Beta}(c_k,d_k)$, where the parameters indexing the beta distributions in this mixture differ across the three settings plotted in the figure. In particular, {(c₁, d₁), (c₂, d₂), (c₃, d₃)} is equal to {(2, 2), (3, 3), (4, 4)} for ‘Zero on Both Sides’, {(1, 1), (8, 4), (4, 8)} for ‘Nonzero on Both Sides’, and {(1, 5), (5, 2), (4, 8)} for ‘Spike on Left Side’. When evaluating the mean integrated squared error performance of estimators of the density of Y(1), Q(0) is set equal to Q(1).

Figure S3

Figure S3: Same as Figure 4, but at different choices of the tuning parameters indexing the tests. The tests based on Example 1a set the regularization parameter so that ${\beta }_k=1∕[1+{(k∕c)}^2]$ for a constant c and the MMD test selects a bandwidth equal to some constant s times median{$Y_1,\dots ,Y_n$}. The same npcausal panel from Figure 4 is shown as a benchmark in both subfigures.

Table S1

Table S1: Performance of the plug-in and one-step estimators at different sample sizes n, where performance is measured in terms of n times the mean integrated squared error. As would be predicted by theory, this criterion appears to stabilize with n for the one-step estimator. In contrast, it grows with n for the plug-in estimator.