NONPARAMETRIC GOODNESS-OF-FIT TESTS FOR UNIFORM STOCHASTIC ORDERING

Abstract

We propose L^p distance-based goodness-of-fit (GOF) tests for uniform stochastic ordering with two continuous distributions F and G, both of which are unknown. Our tests are motivated by the fact that when F and G are uniformly stochastically ordered, the ordinal dominance curve R = FG⁻¹ is star-shaped. We derive asymptotic distributions and prove that our testing procedure has a unique least favorable configuration of F and G for p ∈ [1,∞]. We use simulation to assess finite-sample performance and demonstrate that a modified, one-sample version of our procedure (e.g., with G known) is more powerful than the one-sample GOF test suggested by Arcones and Samaniego (2000, Annals of Statistics). We also discuss sample size determination. We illustrate our methods using data from a pharmacology study evaluating the effects of administering caffeine to prematurely born infants.

1. Introduction

Suppose X and Y are continuous random variables with distribution functions F and G, respectively. In many applications, it is of interest to compare F and G. The ordinal dominance curve (ODC), which plots (G(t), F(t)) for −∞ ≤ t≤∞, is a useful graphical tool that facilitates such a comparison (Bamber, 1975; Hsieh and Turnbull, 1996; Carolan and Tebbs, 2005; Davidov and Herman, 2012). The ODC can also be defined as R = FG⁻¹, where G⁻¹(u) = inf{t: G(t) ≥ u} is the quantile function of G. When F = G, the ODC follows the main diagonal of the unit square, the so-called equal distribution line.

We consider order-restricted comparisons of F and G. Define F̄ = 1− F and Ḡ = 1 − G. These are the survivor functions if X and Y are lifetime random variables, although herein we do not require X and Y to be nonnegative. Denote the corresponding densities by f and g, respectively. If F̄ ≤ Ḡ, then X and Y are stochastically ordered; this is written as F ≤_S G and means informally that X “tends to be smaller” than Y. Two stronger orders are the uniform stochastic order and the likelihood ratio order. When F̄/Ḡ is nonincreasing, X and Y satisfy a uniform stochastic order, written F ≤_US G. When f/g is nonincreasing, X and Y satisfy a likelihood ratio order, written F ≤_LR G. It is easy to show these orderings follow the nested structure: F ≤_LR G ⇒ F ≤_US G ⇒ F ≤_S G. A comprehensive account of these and other orderings is given in Shaked and Shanthikumar (2007).

Different stochastic orderings give rise to different functional forms of the ODC. The weakest ordering F ≤_S G holds if and only if R is at least as large as the equal distribution line; i.e., R(u) ≥ u, for 0 ≤ u ≤ 1. The strongest ordering F ≤_LR G holds if and only if R is concave. The intermediate ordering F ≤_US G holds if and only if R is star-shaped (Lehmann and Rojo, 1992). One way to characterize a star-shaped ODC is that the slope of the secant line from the point (1, 1) to (u,R(u)); i.e., r(u) = {1−R(u)}/(1−u), is nonincreasing in u. Figure 1 gives examples of ODCs that correspond to stochastic, uniform stochastic, and likelihood ratio orderings. This figure demonstrates the utility of the ODC in characterizing how two distributions are ordered and how the structure F ≤_LR G ⇒ F ≤_US G ⇒ F ≤_S G manifests itself graphically in the ODC.

Fig 1.

Ordinal dominance curves. Left: F ≤_S G. Middle: F ≤_US G. Right: F ≤_LR G. In each subfigure, the equal distribution line is shown dotted.

This article is motivated by a pharmacology study evaluating the effects of administering caffeine to prematurely born infants in Columbia, South Carolina; see Section 5. Among 404 infants in the study, m = 127 were administered caffeine and n = 277 were not. Each infant was then followed until he or she was discharged from the hospital. All infants were eventually discharged and were alive at the time of discharge; i.e., no discharge times were censored. One of the goals of the study was to understand how the distributions of discharge times F (caffeine) and G (no caffeine) compared for the two groups. In Figure 2 (left), we display the sample ODC for the data, which is defined as $R_{mn}(u)=F_m\{G_n^{-1}(u)\}$, for 0 ≤ u ≤ 1, where F_m and G_n are the empirical distribution functions and $G_n^{-1}(u)=inf\{t:G_n(t)\ge u\}$ is the empirical quantile function. The sample ODC and its large-sample properties were described in Hsieh and Turnbull (1996).

Fig 2.

Premature infant data. Left: The sample ODC $R_{mn}(u)=F_m\{G_n^{-1}(u)\}$ for the time to discharge (F = caffeine; G = no caffeine). Right: The least star-shaped majorant ℳR_mn is shown in blue. In each subfigure, the equal distribution line is shown dotted.

On the basis of Figure 2, which stochastic ordering, if any, characterizes the true relationship between the discharge time distributions? There is a substantive literature on nonparametric tests for stochastic orderings with two or more distributions; see Davidov and Herman (2012), El Barmi and McKeague (2016), and the references therein. In the two-sample case, most of this literature describes tests where the equal distribution assumption F = G is treated as the null hypothesis and the ordering (i.e., F ≤_S G, F ≤_US G, or F ≤_LR G) is placed in the alternative. A potential drawback with this type of test is that it is constructed assuming a specific order-restricted class of alternatives; if the assumed class is incorrect, the test may lead to misleading or vacuous conclusions. For example, applying tests of this type to the premature infant data, we obtain the following results:

• testing F = G versus F ≤_S G: p-value < 0.00002 (Davidov and Herman, 2012)

• testing F = G versus F ≤_US G: p-value < 0.00001 (Arcones and Samaniego, 2000)

• testing F = G versus F ≤_LR G: p-value < 0.00001 (Carolan and Tebbs, 2005).

Each test clearly dictates that the infant data are not consistent with F = G. However, we are no closer to identifying which specific ordering (if any) holds in this setting.

In this light, we consider goodness-of-fit (GOF) testing procedures instead. By “goodness-of-fit,” we mean the procedure places the ordering in the null hypothesis and attempts to detect departures from the ordering. By comparison, the literature on nonparametric GOF tests with two distributions is more sparse, perhaps because this type of testing problem is more difficult. The primary reason for the added difficulty is that the ordering can hold under different configurations of F and G. Therefore, one must determine the least favorable configuration of the two distributions before the test can be performed; i.e., so that the probability of type I error can be controlled. Carolan and Tebbs (2005) proposed nonparametric GOF tests for likelihood ratio ordering with two continuous distributions by using the least concave majorant of the sample ODC. This work was generalized and improved upon by Beare and Moon (2015) in the econometrics literature, who considered likelihood ratio ordering and its applications in finance.

GOF tests for uniform stochastic ordering have been proposed but only in limited settings. Dardanoni and Forcina (1998) considered likelihood-based tests against uniform stochastic ordering in a two-way contingency table. Park et al. (1998) used a nonparametric maximum likelihood approach to formulate GOF tests with two or more continuous distributions, but only after data from these distributions have been assigned to disjoint intervals in the form of counts. This essentially discretizes the problem and results in testing against uniform stochastic ordering among several multinomial distributions. Furthermore, this formulation gives rise to non-unique least favorable configurations that depend on how the intervals are selected, the number of distributions, and even the significance level used. Finally, in the two-population setting, Arcones and Samaniego (2000) suggested a GOF test for uniform stochastic ordering based on the family of order-restricted estimators in Mukerjee (1996). However, these authors assume that one of the population distributions is known (e.g., G is known) and do not determine the least favorable configuration for their procedure. Instead, the authors use critical values from an upper bound asymptotic distribution which leads to a conservative test.

In this article, we propose a family of GOF tests for uniform stochastic ordering with two continuous distributions F and G; that is, we are interested in testing H₀: F ≤_US G versus H₁: F ≰_US G, where both distributions are unknown. Motivated by the ODC approaches taken in Carolan and Tebbs (2005) and Beare and Moon (2015), we construct test statistics for H₀ versus H₁ based on the L^p difference between the sample ODC and its least star-shaped majorant (defined in Section 2). We then derive asymptotic distributions and prove that our testing procedure has a unique least favorable configuration for p ∈ [1,∞]. Interestingly, this theoretical result is different from the finding in Beare and Moon (2015), who showed that when using L^p distance-based GOF tests for likelihood ratio ordering, the least favorable configuration exists only when p ∈ [1,2]. Furthermore, unlike Park et al. (1998), our approach does not require one to discretize the support of the distributions which can only lead to a loss in power. Finally, we show that the one-sample version of our test (e.g., with G known) is not as conservative as the test proposed by Arcones and Samaniego (2000) and is generally better equipped to detect departures from H₀.

Formulating L^p distance-based GOF tests for uniform stochastic ordering in the two-sample problem is technically challenging. It is not possible to simply modify the proofs in Carolan and Tebbs (2005) and Beare and Moon (2015) under likelihood ratio ordering; see Section 3. At the same time, establishing that such an ordering exists has great practical implications. For example, if X and Y are lifetime random variables (and are absolutely continuous), then F ≤_US G is equivalent to the corresponding hazard rates being ordered. This is an important characterization in reliability and survival analysis applications. Our interest in uniform stochastic ordering is motivated by our collaboration with researchers in the premature infant study discussed earlier. Letting X and Y denote the times to discharge for the caffeine and no-caffeine groups, respectively, uniform stochastic ordering holds if and only if pr(X > t|X > t₀ ) ≤ pr(Y > t|Y > t₀ ), for all t, t₀ satisfying t > t₀ ≥ 0. In other words, no matter how much time t₀ ≥ 0 has subsequently passed, administering caffeine is consistent with shorter discharge times. Note that, in this context, stochastic ordering requires that the relationship above hold only initially (i.e., when t₀ = 0). Uniform stochastic ordering guarantees this type of dominance will hold for all t₀ ≥ 0.

2. Testing procedure

Suppose that X₁,X₂, …, X_m are independent and identically distributed (iid) from F and that Y₁, Y₂, …, Y_n are iid from G. We assume the two samples are independent and that both F and G are unknown. Let R = FG⁻¹ denote the corresponding ODC. For our asymptotic results in Section 3 to hold, as in Hsieh and Turnbull (1996), we assume F and G have continuous densities f and g and that the first derivative of R is bounded over [0, 1]. Throughout this article, we denote the parameter space of R by Θ, the collection of nondecreasing, continuously differentiable functions from [0, 1] to [0, 1]. Under our assumptions, the hypotheses H₀: F ≤_US G and H₁: F ≰_US G can be expressed equivalently as

$$H_0:R\in {\mathrm{\Theta }}_0=\{\theta \in \mathrm{\Theta }:\theta \text{is}\text{star}-\text{shaped}\}\text{and}{H}_1:R\in {\mathrm{\Theta }}_1=\mathrm{\Theta }\backslash {\mathrm{\Theta }}_0.$$

Recall that θ ∈ Θ is star-shaped if and only if {1 − θ(u)}/(1 − u) is nonincreasing in u.

Let $R_{mn}=R_{mn}(u)=F_m\{G_n^{-1}(u)\}$ denote the sample ODC, defined in Section 1. Informally, our testing procedure is based on measuring the distance between R_mn and an estimate of R subject to the constraint that F ≤_US G. Towards defining this restricted estimator, let l([0, 1]) denote the collection of bounded functions on [0, 1]. For any h ∈ l([0, 1]), its least star-shaped majorant is defined as

$$ℳh=inf\{h^{\ast }\in l([0,1]):h\le h^{\ast }\text{and}{h}^{\ast }\text{is}\text{star}-\text{shaped}\};$$

i.e., ℳh is the smallest star-shaped function in l([0,1]) that is at least as large as h. Throughout our work, we call ℳ: l([0, 1]) ↦ l([0, 1]) the least star-shaped majorant operator. Just as R_mn is an estimator of R under no restriction (Hsieh and Turnbull, 1996), the least star-shaped majorant ℳR_mn is an estimator of R under H₀: F ≤_US G. Using Lemma 1 in the supplementary article (Tang et al., 2016), we show that this restricted estimator can be calculated as

$$ℳR_{mn}(u)=1-\underset{\begin{array}{c}v\in \mathcal{V}\cup \{0\}\\ v\le u\end{array}}{min}\left\{\frac{1-R_{mn}(v)}{1-v}\right\}(1-u),$$

for 0 ≤ u < 1, where 𝒱 is the set of discontinuous (jump) points of R_mn and ℳR_mn(1) = 1. Figure 2 (right) shows the least star-shaped majorant of the sample ODC for the premature infant data described in Section 1.

Our testing procedure utilizes the sample ODC R_mn and its least star-shaped majorantℳR_mn. Specifically, we propose the family of test statistics

$$M_{mn}^p=c_{mn}{\Vert ℳR_{mn}-R_{mn}\Vert }_p,$$

where c_mn = {mn/(m+n)}^1/2 is a normalizing constant and || · ||_p is the L^p norm with respect to Lebesgue measure. We allow for p ∈ [1,∞]; i.e., ||h||_p = (∫_[0,1] |h(u)|^pdu)^1/p when p < ∞ and ||h||_∞ = sup_u∈[0,1] |h(u)|. For example, when p = 1, ||ℳR_mn − R_mn||₁ equals the area between the two estimators; when p = ∞, ||ℳR_mn−R_mn||_∞ equals the largest vertical distance between the estimators. For any p ∈ [1,∞], clearly large values of $M_{mn}^p$ are evidence against H₀.

3. Theoretical results

In this section, we first describe the asymptotic distribution of $M_{mn}^p$ for any star-shaped ODC; i.e., for any R ∈ Θ₀. We then demonstrate that, for any p ∈ [1,∞], all null distributions are dominated stochastically by the asymptotic distribution of $M_{mn}^p$ under R(u) = u, that is, when F = G. From this least favorable distribution, we can find the critical value c_α,p that satisfies ${lim}_{m,n\to \infty }\text{pr}(M_{mn}^p\ge c_{\alpha ,p})=\alpha$ when F = G and ${lim}_{m,n\to \infty }\text{pr}(M_{mn}^p\ge c_{\alpha ,p})\le \alpha$ when H₀: F ≤_US G is true. In other words, rejecting H₀ when $M_{mn}^p\ge c_{\alpha ,p}$ is an asymptotic size α decision rule. Finally, we examine relevant asymptotic distributions when R ∈ Θ₁ and then characterize large-sample power properties. We also discuss sample size calculations to detect departures from H₀. All theorems are proved in Section 7. Additional technical details are provided in the supplementary article (Tang et al., 2016).

3.1. Asymptotic results under H₀

Let ℐ denote the identity operator on l([0, 1]) and define 𝒟 = ℳ−ℐ. When H₀ is true; i.e., when R ∈ Θ₀, note that ℳR = R and

$$M_{mn}^p=c_{mn}{\Vert ℳR_{mn}-R_{mn}\Vert }_p=c_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_p.$$

At first glance, establishing the limiting distribution of $M_{mn}^p$ under H₀ might seem to be straightforward, that is, one could simply start with the asymptotic distribution of c_mn(R_mn − R) described in Hsieh and Turnbull (1996) and apply the functional delta method (see, e.g., Section 3.9 in van der Vaart and Wellner, 1996) and continuous mapping theorem. This was the approach taken by Beare and Moon (2015) with their L^p distance-based GOF test statistics under likelihood ratio ordering. In our setting, this direct approach is not possible because whereas the least concave majorant operator in Beare and Moon (2015) is Hadamard directionally differentiable (Shapiro, 1990, 1991), the least star-shaped majorant operator ℳ (and hence 𝒟) is not always so; see Lemma 5 in the supplementary article (Tang et al., 2016). Fortunately, this does not create insurmountable problems because weak convergence of c_mn(𝒟R_mn − 𝒟R) is not a necessary prerequisite to derive the asymptotic distribution of c_mn||𝒟R_mn −𝒟R||_p.

Before we state the asymptotic distribution of $M_{mn}^p$ for any R ∈ Θ₀, we need to describe R precisely because these distributions depend completely on the shape of R. Recall that when R ∈ Θ₀, the slope function r(u) = {1 − R(u)}/(1 − u) is nonincreasing in u. When r(u) is strictly decreasing over [0, 1], we say that R is strictly star-shaped. When R ∈ Θ₀ is not strictly star-shaped, then, analogously to Beare and Moon (2015), there exists a unique collection (finite or countable) of closed, pairwise disjoint intervals of the form [a_k, b_k], 0 ≤ a_k < b_k ≤ 1, where

• the slope r(u) is constant over each interval (i.e., R is affine over each interval)

• no two intervals possess the same value of r(u).

In this case, we say that R ∈ Θ₀ is non-strictly star-shaped. The reason we bifurcate Θ₀ using “strictly” and “non-strictly” descriptors is that the nondegenerate part of the asymptotic distribution of $M_{mn}^p$ depends only on those regions where R is non-strictly star-shaped. If R is strictly star-shaped over [0, 1], the distribution of $M_{mn}^p$ collapses to zero in the limit.

To make our description of the asymptotic distributions precise, we therefore introduce the following notation. For 0 ≤ a < b ≤ 1, define

$${ℳ}_{[a,b]}^{(1,0)}h=inf\{h^{\ast }\in l([0,1]):h\le h^{\ast }\text{and}{h}^{\ast }\text{is}\text{star}-\text{shaped}\text{over}[a,b]\text{with}\text{kernel}(1,0)\}.$$

A general definition of what it means for a function h^* to be star-shaped with kernel (c, d) is given directly before Lemma 1 in the supplementary article (Tang et al., 2016). For any h ∈ l([0, 1]), the function ${ℳ}_{[a,b]}^{(1,0)}h$ has two defining characteristics. First, ${ℳ}_{[a,b]}^{(1,0)}h(u)=h(u)$ whenever u/∈ [a, b]. Second, over [a, b], ${ℳ}_{[a,b]}^{(1,0)}h$ is the smallest function (at least as large as h) that is star-shaped with kernel (1, 0); i.e., the slope function $-{ℳ}_{[a,b]}^{(1,0)}h(u)/(1-u)$ over [a, b] is nonincreasing in u. The importance of the functional operator ${ℳ}_{[a,b]}^{(1,0)}:l([0,1])\mapsto l([0,1])$ becomes clear as we state our first main result.

Theorem 1

Suppose R ∈ Θ₀ and let ℬ denote a standard Brownian bridge. The asymptotic results below hold when min{m, n} → ∞ and n/(m + n) → λ ∈ (0, 1).

1. If R is strictly star-shaped over [0, 1], then $M_{mn}^p\stackrel{d}{\to }0$ 0 for all p ∈ [1,∞].

2. If R is non-strictly star-shaped, then for p ∈ [1,∞),

   $$M_{mn}^p\stackrel{d}{\to }{\left\{\sum _k{[\lambda R^{\prime }(a_k)+(1-\lambda ){\{R^{\prime }(a_k)\}}^2]}^{p/2}{\int }_{a_k}^{b_k}{\left\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\right\}}^{p}du\right\}}^{1/p};$$
   when p = ∞,

   $$M_{mn}^p\stackrel{d}{\to }\underset{k}{sup}\left\{{[\lambda R^{\prime }(a_k)+(1-\lambda ){\{R^{\prime }(a_k)\}}^2]}^{1/2}\underset{u\in [a_k,b_k]}{sup}\left\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\right\}\right\}.$$

   In both asymptotic distributions, R′ is the derivative of R and ${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}={ℳ}_{[a_k,b_k]}^{(1,0)}-ℐ$.

From Theorem 1, one can see that when F ≤_US G, the only randomness in the asymptotic distribution of $M_{mn}^p$ arises from the non-strictly star-shaped regions [a_k, b_k] and is described probabilistically by the ${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ$ processes. Furthermore, when F = G, the asymptotic distribution of $M_{mn}^p$ simplifies to ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$ for all p ∈ [1,∞]. When p = 1, for example, this quantity describes the distribution of the area between the least star-shaped majorant of a standard Brownian bridge ℬ and ℬ itself. When p = ∞, ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_{\infty }$ describes the distribution of the sup-norm distance between these two processes. Readers familiar with the GOF tests for likelihood ratio ordering in Carolan and Tebbs (2005) and Beare and Moon (2015) will no doubt recognize the homology between our Theorem 1 and the corresponding results in these articles. However, as noted earlier, GOF tests for uniform stochastic ordering present their own set of mathematical challenges and different conclusions are reached about the existence of a least favorable configuration.

Theorem 2

Suppose R ∈ Θ₀. For any p ∈ [1,∞], the asymptotic distribution of $M_{mn}^p$ is ordinary stochastically smaller than ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$; i.e.,

$$\underset{\begin{array}{c}m,n\to \infty \\ n/(m+n)\to \lambda \end{array}}{lim}{\text{pr}}_{R\in {\mathrm{\Theta }}_0}(M_{mn}^p\ge t)\le \text{pr}({\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p\ge t),$$

for all t ∈ ℝ, where λ is defined in Theorem 1.

Theorem 2 establishes that when using $M_{mn}^p$ to test H₀: F ≤_US G versus H₁: F ≰_US G, the equal distribution line R(u) = u represents the least favorable configuration of F and G for all p ∈ [1,∞]. Proving this result involves showing that each of the ${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ$ processes in Theorem 1 are mutually independent, a somewhat startling discovery because each process shares the same Brownian bridge ℬ and each operator ${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}$ shares the same kernel point (1, 0). The practical utility of Theorem 2 is that, for any p ∈ [1,∞], we can determine the critical value that maximizes the probability of type I error over all configurations of F and G in Θ₀. This result is different than the conclusion reached in Beare and Moon (2015), who showed that when testing against likelihood ratio ordering using L^p distance-based statistics involving the least concave majorant of R_mn, R(u) = u is the least favorable configuration when p ∈ [1, 2] and for p > 2 the least favorable configuration does not exist. Careful inspection of Theorem 1 and some intuition sheds insight on why this is true. When R is star-shaped, but not strictly star-shaped, each of the derivatives R′(a_k) in Theorem 1 satisfies R′(a_k) ≤ 1. However, when F ≤_LR G, there is no guarantee these derivatives are uniformly bounded for all concave R and hence anomalous limiting behavior can result when p is too large.

For given values of the significance level α and p ∈ [1,∞], denote the 1−α quantile of ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$ by c_α,p; i.e., c_α,p solves $\alpha =\text{pr}({\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p\ge c_{\alpha ,p})$. To approximate the distribution of ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$, we generated 100,000 Brownian bridge paths on a grid of 100,000 equally spaced points in [0, 1], and, for each p ∈ {1, 2, 3, 5, ∞}, we calculated ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$ for each path. For each p, these 100,000 values were used to approximate the density function of ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$ and quantiles c_α,p, for α = 0.01, 0.05, and 0.10. These functions and the selected quantiles c_α,p are provided in the supplementary article (Tang et al., 2016).

3.2. Asymptotic results under H₁

The difference between the asymptotic distribution of $M_{mn}^p$ under H₀: R ∈ Θ₀ and that under H₁: R ∈ Θ₁ arises from the non-star-shaped regions of R. To characterize a non-star-shaped ODC R ∈ Θ₁, start with ℳR, which is star-shaped, and note that (as in Section 3.1) one can partition the unit interval [0, 1] as [0, 1] = S∪(∪_kS_k), where ℳR is strictly star-shaped over S and non-strictly star-shaped over pairwise disjoint intervals of the form S_k = [a_k, b_k], 0 ≤ a_k < b_k ≤ 1, for k = 1, 2, …. One can further partition each S_k as S_k = S_k1 ∪ S_k2, where S_k1 = {u ∈ S_k: ℳR(u) = R(u)} and S_k2 = {u ∈ S_k: ℳR(u) > R(u)}. Each S_k1 must contain a_k so it is never empty, and the non-star-shaped regions of R can be written as ∪_kS_k2. In other words, R ∈ Θ₀ when ∪_kS_k2 is empty and R ∈ Θ₁ otherwise.

In general, these types of regions contribute differently to the limiting distribution of $M_{mn}^p$. Over the strictly star-shaped region S, ℳR(u) = R(u) for all u and the L^p norm of c_mn{𝒟R_mn(u) − 𝒟R(u)} converges in distribution to 0, as in Section 3.1. To clearly describe the contribution over the S_k regions, we introduce new notation. For any h ∈ l([0, 1]), define the functional operator ℒ_Sk: l([0, 1]) ↦ l([0, 1]) according to

$${ℒ}_{S_k}h(u)=-\underset{\begin{array}{c}v\in S_{k1}\\ v\le u\end{array}}{inf}\left\{\frac{-h(v)}{1-v}\right\}(1-u)I_{S_k}(u)+h(u)I_{S_k^c}(u),\text{for}u\in [0,1),$$

where I_A(·) is the indicator function over the set A and A^c denotes the complement of A. When u = 1, ℒ_Skh(u) = max{h(1), 0} or h(1) depending on whether the singleton {1} ∈ S_k1 or not; see Appendix C in the supplementary article (Tang et al., 2016). Using this new operator, we now characterize asymptotic distributions for any ODC R ∈ Θ with those in Θ₁ = Θ\Θ₀ of particular interest. A discussion on the large-sample power properties of our testing procedure follows.

Theorem 3

Suppose R ∈ Θ. Using the notation described in this subsection,

$$c_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_p\stackrel{d}{\to }{\left\{\sum _k{\int }_{u\in S_k}{\left|{ℒ}_{S_k}{T}_R^{\lambda }(u)-T_R^{\lambda }(u)\right|}^{p}du\right\}}^{1/p}$$

for p ∈ [1,∞); when p = ∞,

$$c_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_p\stackrel{d}{\to }\underset{k}{sup}\underset{u\in S_k}{sup}\left|{ℒ}_{S_k}{T}_R^{\lambda }(u)-T_R^{\lambda }(u)\right|.$$

Both results hold as min{m, n} → ∞ and n/(m + n) → λ ∈ (0, 1). In both cases, $T_R^{\lambda }(u)={\lambda }^{1/2}{ℬ}_1(R(u))+{(1-\lambda )}^{1/2}{R}^{\prime }(u){ℬ}_2(u)$, 0 ≤ u ≤ 1, where ℬ₁ and ℬ₂ denote two independent standard Brownian bridges.

Four remarks are in order. First, the process $T_R^{\lambda }=\{T_R^{\lambda }(u),0\le u\le 1\}$ in Theorem 3 is well known; as noted earlier, it represents the asymptotic distribution of c_mn(R_mn − R) for any R ∈ Θ; see, e.g., Theorem 2.2 in Hsieh and Turnbull (1996). Second, the asymptotic distributions identified in Theorem 3 apply for any R ∈ Θ, but we show in the supplementary article (Tang et al., 2016) that they quickly reduce to those in Theorem 1 when R ∈ Θ₀. Third, our L^p tests are consistent for p ∈ [1,∞]. To see why, consider the sup-norm (p = ∞) case in Theorem 3 and note that, by the triangle inequality,

$${\text{pr}}_{R\in {\mathrm{\Theta }}_1}(M_{mn}^{\infty }\ge c_{\alpha ,\infty })={\text{pr}}_{R\in {\mathrm{\Theta }}_1}\left(c_{mn}{\Vert \mathcal{D}{R}_{mn}\Vert }_{\infty }\ge c_{\alpha ,\infty }\right)\ge {\text{pr}}_{R\in {\mathrm{\Theta }}_1}\left(c_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_{\infty }\le c_{mn}{\Vert \mathcal{D}R\Vert }_{\infty }-c_{\alpha ,\infty }\right)$$

which can be approximated by

$${\text{pr}}_{R\in {\mathrm{\Theta }}_1}\left(\underset{k}{sup}\underset{u\in S_k}{sup}\left|{ℒ}_{S_k}{T}_R^{\lambda }(u)-T_R^{\lambda }(u)\right|\le c_{mn}{\Vert \mathcal{D}R\Vert }_{\infty }-c_{\alpha ,\infty }\right).$$

It is easy to show that ${sup}_k{sup}_{u\in S_k}\mid {ℒ}_{S_k}{T}_R^{\lambda }(u)-T_R^{\lambda }(u)\mid$ is bounded and that, for any R ∈ Θ₁, c_mn||𝒟R||_∞ → ∞, as min{m, n} → ∞, which establishes our claim. The finite p argument is analogous. Fourth, approximate lower bounds on the power, like the one above in the sup-norm case, can be used for sample size calculations. For an ODC R ∈ Θ₁ deemed to be clinically relevant, one can determine numerically the smallest m and n that solve ${\text{pr}}_{R\in {\mathrm{\Theta }}_1}({sup}_k{sup}_{u\in S_k}\mid {ℒ}_{S_k}{T}_R^{\lambda }(u)-T_R^{\lambda }(u)\mid \le c_{mn}{\Vert \mathcal{D}R\Vert }_{\infty }-c_{\alpha ,\infty })=1-\beta$, where β ∈ (0, 1). The resulting solution will be inexorably conservative but still potentially useful for planning purposes. We illustrate this approach with examples in Section 4.

We conclude this section with a brief discussion on local power. This discussion is ultimately not dissimilar from the local power discussion in Beare and Moon (2015) under likelihood ratio ordering. However, our interest in local power arises because we want to compare the one-sample version of our testing procedure to the GOF test suggested by Arcones and Samaniego (2000). This one-sample comparison is given in Section 4.3. The two-sample discussion is given now. Let {R^(r), r = 1, 2, …, } denote a sequence of ODCs in Θ₁. For each r ≥ 1, denote the corresponding distributions by F^(r) and G^(r) from which we have independent random samples $X_1^{(r)},X_2^{(r)},\dots ,X_m^{(r)}$ and $Y_1^{(r)},Y_2^{(r)},\dots ,Y_n^{(r)}$, respectively. We examine local power properties by letting R^(r) approach Θ₀ in the sense that ||𝒟R^(r)||_p = ||ℳR^(r)−R^(r) ||_p → 0 as r →∞ at different rates. Using the notation in this paragraph, our last theorem summarizes the salient results.

Theorem 4

Suppose the first derivative of R^(r) ∈ Θ₁ is uniformly bounded over [0, 1] for all r. Suppose p ∈ [1,∞]. All limits stated below assume that max{m, n} = O(r) and n/(m + n) → λ ∈ (0, 1), as r→∞.

1. If lim c_mn||𝒟R^(r)||_p = ∞, then $lim{\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(M_{mn}^p>c_{\alpha ,p})=1$.

2. For any β ∈ (0, 1), there exists η_p(β) > 0 such that

   $$liminf{\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(M_{mn}^p>c_{\alpha ,p})\ge 1-\beta$$

   whenever lim inf c_mn||𝒟R^(r)||_p ≥ η_p(β).

Part (a) of Theorem 4 indicates that when ||𝒟R^(r)||_p converges to 0 at a rate slower than $c_{mn}^{-1}$, c_mn||𝒟R^(r)||_p diverges and the power of our test converges to 1. Part (b) guarantees that when c_mn||𝒟R^(r)||_p remains bounded away from zero, the power of our test is still nontrivial; i.e., it does not converge to 0. This occurs when the “amount of information” c_mn increases and the “departure” ||𝒟R^(r)||_p decreases, and both do so at the same rate.

4. Simulation evidence

We use simulation to assess the finite-sample performance of our tests. In Section 4.1, we consider fixed ODCs under both H₀ : F ≤_US G and H₁ : F ≰ _US G to estimate type I error probability and power, respectively, and we illustrate the sample size calculations described in Section 3.2. In Section 4.2, we modify our testing procedure to allow for one of the population distributions to be known and compare this modified test to the one-sample GOF test in Arcones and Samaniego (2000). Local power results are provided in Section 4.3.

4.1. Fixed ODC comparisons

We consider four ODCs satisfying R ∈ Θ₀ (R₁, R₂, R₃, and R₄) and four ODCs satisfying R ∈ Θ₁ (R₅, R₆, R₇, and R₈). The H₀ ODCs (Figure 3, left) are each members of a family of star-shaped ODCs that we describe in the supplementary article (Tang et al., 2016). The H₁ ODCs (Figure 3, right) are not star-shaped and are also described in Tang et al. (2016). We also consider R₀ = R₀(u) = u, for u ∈ [0, 1], to examine finite-sample performance under the least favorable configuration F = G. All of our results are based on 10, 000 Monte Carlo data sets using independent samples from F and G with sample sizes m and n, respectively. To generate the samples, we let F(u) = R_i(u) and G(u) = u, for u ∈ [0, 1]. We then sample X₁, X₂, …, X_m from F using the inverse cumulative distribution function technique and independently sample Y₁, Y₂, …, Y_n from a uniform(0, 1) distribution. This provides independent samples for each ODC R under consideration.

Fig. 3.

Left: Star-shaped ODCs; i.e., R_i ∈ Θ₀. Right: Non-star-shaped ODCs; i.e., R_i ∈ Θ₁. A description of each curve is given in the supplementary article (Tang et al., 2016).

Table S.2 in the supplementary article (Tang et al., 2016) gives Monte Carlo estimates of the probability of rejecting H₀ : F ≤_US G for different sample sizes, values of p ∈ {1, 2,∞}, and α = 0.05. We experimented with other values of p (i.e., p = 3 and p = 5) but obtained results similar to those when p = 2. Of initial interest is the finite-sample performance when F = G. With 10,000 simulated data sets, the margin of error associated with the size estimates under F = G, assuming a 99 percent confidence level, is approximately 0.006. Therefore, one notes that our tests with p = 1 and p = 2 are slightly anticonservative with small samples and otherwise operate closely to the nominal level. Furthermore, examining the rejection rates for the other star-shaped ODCs (R₁, R₂, R₃, and R₄) supports Theorem 2 which, for p ∈ [1,∞], guarantees the probability of type I error will be at its maximum under F = G. Likewise, powers for the non-star-shaped ODCs (R₅, R₆, R₇, and R₈) all approach unity as m and n become large. This reinforces our consistency claim.

We also use the non-star-shaped ODCs in Figure 3 to illustrate sample size determination. For p ∈ [1,∞] and for a given R ∈ Θ₁, denote by d_R,β,p the 1 − β quantile of the asymptotic distributions in Theorem 3. Using our lower bound on the asymptotic power from Section 3.2 and taking m = n (for simplicity), we obtain a closed-form expression for the minimum sample size necessary to detect the departure ||𝒟R||_p = ||ℳR−R||_p with probability 1 − β when using an asymptotic size α test; i.e.,

$$m=2{\left(\frac{d_{R,\beta ,p}+c_{\alpha ,p}}{{\Vert \mathcal{D}R\Vert }_p}\right)}^2,\text{for}p\in [1,\infty ].$$

With α = 0.05 and 1−β = 0.8, the supplementary article (Tang et al., 2016) tables these solutions for each non-star-shaped ODC in Figure 3 and for each p ∈ {1, 2,∞}. For example, for the R₅ ODC, which corresponds to F and G being stochastically ordered (but not uniformly stochastically ordered), the minimum sample size solutions for p ∈ {1, 2,∞}, respectively, are m = 634, m = 461, and m = 582. Such sample sizes might seem dispiritingly large; however, it is not surprising these solutions are conservative. We describe in Section 6 alternative approaches that should reduce this conservatism.

4.2. Comparison with Arcones and Samaniego (2000)

We now turn our attention to the special case of testing H₀ : F ≤_US G versus H₁ : F ≰ _US G where G is known. Arcones and Samaniego (2000), who focused largely on optimal estimation of F (with F ≤_US G and G known), also suggested a conservative large-sample procedure to test against H₀. Their proposed test statistic, which we denote by D_m, can be expressed as a function of the one-sample ODC R_m = F_mG⁻¹; specifically,

$$D_m=m^{1/2}\underset{0\le v\le u\le 1}{sup}[(1-v)\{1-R_m(u)\}-(1-u)\{1-R_m(v)\}].$$

However, instead of deriving a least favorable (asymptotic) distribution for inference, the authors proved that the asymptotic distribution of D_m is bounded above by 2 sup_u∈[0,1] |ℬ (u)|, where ℬ is a standard Brownian bridge, and selected their critical value $c_{\alpha /2}^{\text{AS}}$ to satisfy $\alpha =\text{pr}({sup}_{u\in [0,1]}\mid ℬ(u)\mid \ge c_{\alpha /2}^{\text{AS}})$. On the other hand, one-sample versions of our GOF procedure are available and use the test statistics

$$M_m^p=m^{1/2}{\left[{\int }_{[0,1]}{\{\mathcal{D}{R}_m(u)\}}^{p}du\right]}^{1/p}\text{and}{M}_m^{\infty }=m^{1/2}\underset{u\in [0,1]}{sup}\{\mathcal{D}{R}_m(u)\},$$

where 𝒟 is the operator defined in Section 3.1 and R_m(u) = F_m{G⁻¹(u)}. The limiting distributions in Theorem 1 also apply here as m → ∞; in addition, it is straightforward to modify the proof of Theorem 2 to conclude that F = G admits the least favorable configuration for p ∈ [1,∞] in the known G case.

For different sample sizes m (now corresponding to F only), Table S.3 in the supplementary article (Tang et al., 2016) gives small-sample rejection rates of our one-sample tests and the test from Arcones and Samaniego (2000), both performed using α = 0.05. We used techniques similar to those described in Section 3.1 to approximate the critical value $c_{\alpha /2}^{\text{AS}}=c_{0.025}^{\text{AS}}=1.359$ and performed all simulations in the same way as before except G is now known. Clearly, there is a price to be paid for using the test based on the D_m statistic when F = G; type I error probability estimates remain significantly below the nominal level for all m ≤ 200. On the other hand, our p = 1 and p = 2 tests are only minimally conservative when m ≤ 75, and our sup-norm (p = ∞) test performs nominally even when m = 20. In addition, the sup-norm test can be markedly more powerful at detecting non-star-shaped alternatives with small to moderately sized samples.

4.3. Local power analysis

A consequence of Theorem 3 is that, for any fixed R ∈ Θ₁, our L^p GOF tests are consistent for all p ∈ [1,∞]. To glean additional insight on which values of p might be preferred in practice, we investigate the power associated with local alternatives. Starting in the lower left corner, Figure 4 depicts a sequence of ODCs in Θ₁ that approach Θ₀ (moving from lower left to upper right). Each ODC shown in Figure 4 belongs to a family of ODCs described in the supplementary article (Tang et al., 2016); the defining feature of this family is that it is indexed by a single parameter δ ∈ [0, 0.5]. The δ = 0 member, say R₍₀₎, is the initial ODC in the lower left corner of Figure 4; the δ = 0.5 member R_(0.5), shown in the upper right, is the limiting ODC in Θ₀. ODCs R_(δ) with intermediate values of δ ∈ (0, 0.5) are also identified in Figure 4.

Fig. 4.

Local power family of ODCs indexed by δ ∈ [0, 0.5]. The δ = 0 member R₍₀₎ is the initial ODC in Θ₁; the δ = 0.5 member R_(0.5) is the limiting ODC in Θ₀. This family is described in the supplementary article (Tang et al., 2016).

In our testing problem, a local power analysis involves examining a sequence of ODCs {R^(r), r = 1, 2, …, } in Θ₁ that converges to Θ₀ at different rates. We do so here by using the family of ODCs just described. Specifically, we consider the rates ζ_r ∈ {log r, r^2/5, r^1/2}. For each ζ_r, we first choose a sequence of constants δ^(r) such that lim_r→∞ ζ_r|δ^(r) − 0.5| = c_ζr > 0 and then select members from our ODC family identified by R^(r) = R_(δ^(r)), for r = 1, 2, …. The resulting sequence R^(r) satisfies ||𝒟R^(r)||_p = ||ℳR^(r) − R^(r)||_p → 0 and ${\zeta }_r{\Vert \mathcal{D}{R}^{(r)}\Vert }_p\to c_{{\zeta }_r,p}^{\ast }>0$, both as r→∞. This investigation allows us to learn more about the practical aspects of Theorem 4 (i.e., with both F and G unknown). We also use these ODC sequences, one for each rate ζ_r, to compare the one-sample versions of our tests with the test in Arcones and Samaniego (2000).

For each r ∈ {50, 100, 500, 1000, 5000, 10000}, we simulated 10,000 independent random samples, $X_1^{(r)},X_2^{(r)},\dots ,X_m^{(r)}$ from F^(r) and $Y_1^{(r)},Y_2^{(r)},\dots ,Y_n^{(r)}$ from G^(r), where F^(r)(u) = R^(r)(u) and G^(r)(u) = u, 0 ≤ u ≤ 1, and m = n = r. Figure 5 (top row) shows the estimated powers of our α = 0.05 tests associated with each rate: ζ_r = logr (left), ζ_r = r^2/5 (middle), and ζ_r = r^1/2 (right). Note that with m = n = r, considering the slower rates ζ_r = logr and ζ_r = r^2/5 allows us to assess part (a) of Theorem 4, while the fastest rate ζ_r = r^1/2 allows us to assess part (b). Both parts are supported by our empirical results in Figure 5. For the slower rates, the powers approach unity as expected; however, we find that there is no decisively preferred value of p among p ∈ {1, 2,∞}. On the other hand, when ζ_r = r^1/2, the p = 1 powers hover only slightly above 0.3 for all r, while the p = 2 and p = ∞ powers still approach unity.

Fig. 5.

Local power results with α = 0.05. Left: ζ_r = logr. Middle: ζ_r = r^2/5. Right: ζ_r = r^1/2. Top: Two-sample case. Bottom: One-sample case. Our L^p results are shown dotted for p = 1, dashed for p = 2, and dot-dashed for p = ∞. Arcones and Samaniego (2000) results (one-sample case only) are shown using a solid line.

Switching to the one-sample problem, we find quite different results. For each rate ζ_r, Figure 5 (bottom row) displays the estimated powers of our one-sample α = 0.05 tests which use $M_m^1,M_m^2$, and $M_m^{\infty }$. Powers were estimated in the same way as for the two-sample case except now we treat G^(r)(u) = u as known and take m = r. In this setting, the sup-norm test consistently provides the largest power, followed by the p = 2 test and the p = 1 test. In addition, all three distance-based tests outperform the corresponding α = 0.05 Arcones and Samaniego (2000) test in terms of local power, especially at the fastest rate ζ_r = r^1/2 where ${\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(D_m>c_{0.025}^{\text{AS}})$ appears to decrease towards zero.

5. Premature infant data

Caffeine is commonly used to treat newborn infants for apnea of prematurity (Schmidt et al., 2006) and to prevent the onset of respiratory distress syndrome, bronchopulmonary dysplasia, and extubation failure (Cox et al., 2015). Known as “the silver bullet” in the treatment of prematurely born infants at risk for these and other acute conditions (Aranda et al., 2010), caffeine is widely regarded within the neonatal care community to be safe and cost effective. It has also been approved by the United States Food and Drug Administration for use with preterm infants due to its history of providing beneficial outcomes with no long-term adverse side effects (Dobson and Hunt, 2013).

We now analyze the data from the study described in Section 1; for complete details, see Cox et al. (2015). Because assessing the use of caffeine with premature infants was a central focus of this study, we consider only those infants who were classified as “premature;” i.e., newborns whose gestational age was at or below 37 weeks. With F and G denoting the discharge time distributions for the caffeine and no-caffeine groups, respectively, recall that Figure 2 displays the sample ODC R_mn and its least star-shaped majorant ℳR_mn, calculated from samples of size m = 127 from F and n = 277 from G. As noted in Section 1, we performed the test in Davidov and Herman (2012) with these data and concluded that F ≤_S G was strongly supported over F = G. We also performed the GOF tests in Beare and Moon (2015) and concluded that F ≤_LR G would be rejected at α = 0.05; the L¹ and L² statistics based on the least concave majorant of R_mn are 0.717 and 0.999, respectively, which are larger than the corresponding 0.95 quantiles 0.664 and 0.753 identified by their least favorable distributions.

We therefore assess whether or not the data in Figure 2 are consistent with uniform stochastic ordering. Testing H₀ : F ≤_US G versus H₁ : F ≰ _US G based on the least star-shaped majorant of R_mn, our GOF test statistics are $M_{mn}^1=0.170,M_{mn}^2=0.263$, and $M_{mn}^{\infty }=0.949$, each of which is well below the α = 0.10 critical values identified in the supplementary article (0.496, 0.586, and 1.219, respectively), that is, H₀ cannot be discounted at any reasonable level of significance. Therefore, not only does caffeine therapy provide point-of-care health benefits and improved long-term outcomes for prematurely born infants, our analysis suggests that treating these infants with caffeine may also lead to hospital discharge times that are uniformly stochastically smaller than those for infants not treated with caffeine.

6. Concluding remarks

When two distributions F and G satisfy uniform stochastic ordering, F and G when conditioned on the interval [t₀,∞), for any t₀ ∈ ℝ, also satisfy uniform stochastic ordering. This desirable property could be exploited to increase the power of our tests under H₁ and simultaneously reduce the sample sizes necessary to detect departures from H₀. To see how, suppose that uniform stochastic ordering is suspected to be violated when t > t₀, either from historical information or from observing data in related applications. In this situation, one could apply our tests after conditioning to determine if R is non-star-shaped over the smaller region [G⁻¹(t₀), 1] and calculate sample sizes to detect departures over it instead of over [0, 1]. A similar approach was suggested by Carolan and Tebbs (2005) for detecting departures from likelihood ratio ordering. In the same spirit, Beare and Moon (2015) suggest that bootstrapping samples over departure regions could help to increase the power of GOF tests for likelihood ratio ordering. This strategy may also be fruitful in our setting, allowing one to reduce the conservatism arising from relying on the least favorable distribution over the entire unit interval.

We believe that our GOF tests could be generalized to allow for different types of censored data, but the theory underpinning these extensions would not be trivial. For example, with random right-censored data, there would be nothing to prevent one from simply replacing the empirical survival functions F̄_m and Ḡ_n with Kaplan-Meier estimators of F and G and then calculating R_mn and ℳR_mn using these estimates. However, asymptotic distributions of the corresponding test statistics may depend heavily on the latent censoring distributions, and there is no guarantee that the least favorable configuration of F and G will exist. Future work could investigate censored-data extensions of majorant-based inference–not only with uniform stochastic ordering, but with other orderings as well.

Finally, estimating distributions under a uniform stochastic ordering assumption has received considerable attention for two populations; see, e.g., Rojo and Samaniego (1993), Mukerjee (1996), and Arcones and Samaniego (2000). We view the one- and two-sample tests proposed herein as helpful inference procedures to determine if the uniform stochastic ordering assumption is plausible and hence restricted estimation methods for F and G are warranted. An anonymous referee has suggested that developing pointwise confidence intervals for R(u) under a uniform stochastic ordering constraint may be a worthwhile next step. We agree and comment on this further after Lemma 4 in the supplementary article (Tang et al., 2016). Another interesting avenue for future research would be to generalize our majorant-based tests to more than two populations. Estimation techniques in this setting are available in Dykstra et al. (1991) and El Barmi and Mukerjee (2016).

7. Proofs

In this section, we provide the proofs of Theorems 1–4. Lemmas cited in this section are stated and proved in the supplementary article (Tang et al., 2016), henceforth referred to as “the supplementary article.”

Proof of Theorem 1

We start with the asymptotic distribution of R_mn, suitably centered and scaled. Applying Theorem 2.2 in Hsieh and Turnbull (1996), it follows that c_mn(R_mn–R) converges weakly to $T_R^{\lambda }$ as min{m, n} → ∞ and n/(m + n) → λ ∈ (0, 1), where $T_R^{\lambda }$ satisfies $T_R^{\lambda }(u)={\lambda }^{1/2}{ℬ}_1(R(u))+{(1-\lambda )}^{1/2}{R}^{\prime }(u){ℬ}_2(u)$, for 0 ≤ u ≤ 1, and ℬ₁ and ℬ₂ are independent standard Brownian bridges. When R ∈ Θ₀, 𝒟R = 0 and $M_{mn}^p=c_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_p$. Define the functional operator d𝒟_R : l([0, 1]) ↦ l([0, 1]) by

$$d{\mathcal{D}}_{R}h(u)=\{\begin{array}{ll}\hfill max\{h(1),0\}-h(1),\hfill & \text{if}u=1\hfill \\ \hfill {ℳ}_{[a_k,b_k]}^{(1,0)}h(u)-h(u),\hfill & \text{if}\exists k\text{such}\text{that}{a}_k\le u\le b_k\hfill \\ \hfill 0,\hfill & \text{otherwise},\hfill \end{array}$$

for h ∈ l([0, 1]). Denote by C([0, 1]) the collection of all real continuous functions with domain [0, 1]. If 𝒟 is Hadamard directionally differentiable tangentially to C([0, 1]) at R, then d𝒟_R is the Hadamard directional derivative of 𝒟. Applying the functional delta method and continuous mapping theorem yields $M_{mn}^p\stackrel{d}{\to }{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$ for p ∈ [1,∞]. Those situations in which 𝒟 is Hadamard directionally differentiable are described in Lemma 5 in the supplementary article.

When 𝒟 is not Hadamard directionally differentiable, the functional delta method and continuous mapping theorem cannot be applied. However, by using Lemma 6 in the supplementary article, we are able to prove that $M_{mn}^p\stackrel{d}{\to }{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$ anyway. For convenience, let Z_mn = c_mn(R_mn – R) and $Z=T_R^{\lambda }$. From Theorem 12.2 in Billingsley (1999) and Skorohod’s representation theorem (see, e.g., Theorem 6.7 in Billingsley, 1999), there exist random elements $Z_{mn}^{\prime }$ and Z′ defined on a common probability space with $Z_{mn}^{\prime }\stackrel{L}{=}{Z}_{mn}$ and $Z^{\prime }\stackrel{L}{=}Z$ such that ${\Vert Z_{mn}^{\prime }-Z^{\prime }\Vert }_{\infty }\to 0$ almost surely. The notation “ $\stackrel{L}{=}$” denotes that two processes are equivalent in distribution. Define $R_{mn}^{\prime }=c_{mn}^{-1}{Z}_{mn}^{\prime }+R$. From Lemma 6 in the supplementary article, because $c_{mn}^{-1}$ decreases to 0 and ${\Vert Z_{mn}^{\prime }-Z^{\prime }\Vert }_{\infty }\to 0$ almost surely, then for all p ∈ [1,∞] we have

$$\underset{\begin{array}{c}m,n\to \infty \\ n/(m+n)\to \lambda \end{array}}{lim}{c}_{mn}{\Vert \mathcal{D}{R}_{mn}^{\prime }-\mathcal{D}R\Vert }_p={\Vert d{\mathcal{D}}_R{Z}^{\prime }\Vert }_p$$

almost surely. Because $c_{mn}{\Vert \mathcal{D}{R}_{mn}^{\prime }-\mathcal{D}R\Vert }_p\stackrel{d}{=}{c}_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_p$ and also ${\Vert d{\mathcal{D}}_R{Z}^{\prime }\Vert }_p\stackrel{d}{=}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$, where the notation “ $\stackrel{d}{=}$” means equal in distribution, we have

$$\underset{\begin{array}{c}m,n\to \infty \\ n/(m+n)/\lambda \end{array}}{lim}{c}_{mn}{\Vert \mathcal{D}{R}_{mn}-\mathcal{D}R\Vert }_p\stackrel{d}{=}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p.$$

This shows that $M_{mn}^p\stackrel{d}{\to }{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$ for all p ∈ [1,∞].

When R is strictly star-shaped over [0, 1], it is easy to see that ${\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p=0$ which quickly establishes part (a). The remainder of the proof focuses on establishing part (b). When R is non-strictly star-shaped,

$${\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p={\left[\sum _k{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)\}}^{p}du\right]}^{1/p}$$

for p ∈ [1,∞) and ${\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p={sup}_k\{{sup}_{u\in [a_k,b_k]}{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)\}$ for p = ∞. Using Lemma 1 in the supplementary article, we write ${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)={sup}_{v\in [a_k,u]}{Q}_k(u,v)$ for u ∈ [a_k, b_k], where

$$Q_k(u,v)=\left(\frac{1-u}{1-v}\right)T_R^{\lambda }(v)-T_R^{\lambda }(u),\text{for}v\in [a_k,u].$$

In Lemma 8 in the supplementary article, we show that the processes

$$\{Q_k(u,v),a_k\le v\le u<b_k\}$$

are mutually independent across k. Therefore, { ${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)$, u ∈ [a_k, b_k]} are also mutually independent. To prove further results, we note that over each non-strictly star-shaped region [a_k, b_k], we can write R(u) as a linear function; i.e., R(u) = 1–R′(a_k)(1–u). Thus, from Lemma 2 in the supplementary article, we have

$${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)={\mathcal{D}}_{[a_k,b_k]}^{(1,0)}\{W_R^{\lambda }(u)-l_{R,k}^{\lambda }(1)\},$$

for all k, where $W_R^{\lambda }(u)={\lambda }^{1/2}{\mathcal{W}}_1(R(u))+{(1-\lambda )}^{1/2}{R}^{\prime }(u){\mathcal{W}}_2(u)$,

$$l_{R,k}^{\lambda }(u)={\lambda }^{1/2}\{1-R^{\prime }(a_k)(1-u){\mathcal{W}}_1(1)+{(1-\lambda )}^{1/2}{R}^{\prime }(a_k)u{\mathcal{W}}_2(1)\},$$

and 𝒲₁ and 𝒲₂ are independent standard Wiener processes; i.e., 𝒲_i, for i = 1, 2, satisfies ℬ_i(u) = 𝒲_i(u) – u𝒲_i(1), 0 ≤ u ≤ 1, for i = 1, 2. Based on the properties of a standard Wiener process, it follows that for u ∈ [a_k, b_k],

$$\begin{array}{l}{\mathcal{W}}_i(R(u))-{\mathcal{W}}_i(1)={\mathcal{W}}_i(1-R^{\prime }(a_k)(1-u))-{\mathcal{W}}_i(1)\\ \stackrel{L}{=}{R}^{\prime }{(a_k)}^{1/2}\{{\mathcal{W}}_i(u)-{\mathcal{W}}_1(1)\},\end{array}$$

for i = 1, 2. Furthermore, for u ∈ [a_k, b_k], we have R′(u) = R′ (a_k) and

$$\begin{array}{l}{W}_R^{\lambda }(u)-l_{R,k}^{\lambda }(1)\stackrel{L}{=}{\lambda }^{1/2}{R}^{\prime }{(a_k)}^{1/2}\{{\mathcal{W}}_1(u)-{\mathcal{W}}_1(1)\}+{(1-\lambda )}^{1/2}{R}^{\prime }(a_k)\{{\mathcal{W}}_2(u)-{\mathcal{W}}_2(1)\}\\ \stackrel{L}{=}{\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{1/2}\{\mathcal{W}(u)-\mathcal{W}(1)\},\end{array}$$

where 𝒲 is a standard Wiener process. The last equivalence (in distribution) follows because both right-hand side processes above are Gaussian, they have the same mean $E\{W_R^{\lambda }(u)-l_{R,k}^{\lambda }(1)\}=0$, for u ∈ [a_k, b_k], and they have the same covariance $cov\{W_R^{\lambda }(u_1)-l_{R,k}^{\lambda }(1),W_R^{\lambda }(u_2)-l_{R,k}^{\lambda }(1)\}=\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}min\{1-u_1,1-u_2\}$, for u₁, u₂ ∈ [a_k, b_k]. Using Lemma 2 in the supplementary article again, we have

$${\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{1/2}\{\mathcal{W}(u)-\mathcal{W}(1)\}={\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{1/2}{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u),$$

where ℬ is a standard Brownian bridge formed by 𝒲; i.e., ℬ(u) = 𝒲(u) – u 𝒲(1), for u ∈ [0, 1]. We can therefore write

$${\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)\}}^{p}du\stackrel{d}{=}{\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{p/2}{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\}}^{p}du,$$

for p ∈ [1,∞), and

$$\underset{u\in [a_k,b_k]}{sup}{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)\stackrel{d}{=}{\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{1/2}\underset{u\in [a_k,b_k]}{sup}{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u),$$

for p = ∞. For p ∈ [1,∞), we have shown that ${\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)\}}^{p}du$ are mutually independent. One can show that ${\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\}}^{p}du$ are also mutually independent by replacing $T_R^{\lambda }(·)$ with ℬ(·) in the definition of Q_k(u, v) and repeating the same argument. Therefore, we have

$$\sum _k{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}{T}_R^{\lambda }(u)\}}^{p}du\stackrel{d}{=}\sum _k{\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{p/2}{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\}}^{p}du,$$

which completes the proof for p ∈ [1,∞). Completing the proof for the p = ∞ case is analogous.

Proof of Theorem 2

When F = G, the ODC is R₀ = R₀(u) = u, 0 ≤ u ≤ 1, and $T_{R_0}^{\lambda }\stackrel{d}{=}ℬ$. Because R₀ is non-strictly star-shaped over [0, 1], Theorem 1 yields $M_{mn}^p\stackrel{d}{\to }{\Vert d{\mathcal{D}}_{R_0}{T}_{R_0}^{\lambda }\Vert }_p\stackrel{d}{=}{\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p$ when F = G for p ∈ [1,∞]. It therefore suffices to show ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p{\ge }_{\mathrm{S}}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$ for p ∈ [1,∞] and for any other R ∈ Θ₀. If R ∈ Θ₀ is strictly star-shaped, then from Theorem 1, ${\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p=0$ for p ∈ [1,∞] and hence ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p{\ge }_{\mathrm{S}}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$. If R ∈ Θ₀ is non-strictly star-shaped, then for p ∈ [1,∞),

$$\begin{array}{r}{\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p={\left[{\int }_0^1{\{{\mathcal{D}}_{[0,1]}^{(1,0)}ℬ(u)\}}^{p}du\right]}^{1/p}\ge {\left[\sum _k{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[0,1]}^{(1,0)}ℬ(u)\}}^{p}du\right]}^{1/p}\\ \ge {\left[\sum _k{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\}}^{p}du\right]}^{1/p}.\end{array}$$ (7.1)

The first and second inequalities above hold because ${\mathcal{D}}_{[0,1]}^{(1,0)}ℬ(u)\ge 0$ and also ${\mathcal{D}}_{[0,1]}^{(1,0)}ℬ(u)\ge {\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\ge 0$, for all u ∈ [0, 1]. Because λ ∈ (0, 1) and R′(a_k) ≤ 1 for all k, λ R′(a_k) + (1–λ)R′(a_k)² ≤ 1 and the rightmost side of (7.1) is greater than or equal to

$${\left[\sum _k{\{\lambda R^{\prime }(a_k)+(1-\lambda )R^{\prime }{(a_k)}^2\}}^{p/2}{\int }_{a_k}^{b_k}{\{{\mathcal{D}}_{[a_k,b_k]}^{(1,0)}ℬ(u)\}}^{p}du\right]}^{1/p}\stackrel{d}{=}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p.$$

Therefore, for R ∈ Θ₀ non-strictly star-shaped, we have ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_p{\ge }_{\mathrm{S}}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_p$ for p ∈ [1,∞). Showing ${\Vert {\mathcal{D}}_{[0,1]}^{(1,0)}ℬ\Vert }_{\infty }{\ge }_{\mathrm{S}}{\Vert d{\mathcal{D}}_R{T}_R^{\lambda }\Vert }_{\infty }$ for R ∈ Θ₀ non-strictly star-shaped is analogous.

Proof of Theorem 3

When R ∈ Θ₁, we redefine the functional operator d𝒟_R : l([0, 1]) ↦ l([0, 1]) in Theorem 1 by

$$d{\mathcal{D}}_{R}h(u)=\{\begin{array}{ll}\hfill -h(1),\hfill & \text{if}u=1,R(u)<1\hfill \\ \hfill max\{h(1),0\}-h(1),\hfill & \text{if}u=1,R(u)=1\hfill \\ \hfill {ℒ}_{S_k}h(u)-h(u),\hfill & \text{if}\exists k\text{such}\text{that}u\in S_k\backslash \{1\}\hfill \\ \hfill 0,\hfill & \text{otherwise}.\hfill \end{array}$$

The proof proceeds in the same manner as in Theorem 1. If 𝒟 is not Hadamard directionally differentiable, one can use Skorohod’s representation theorem and part (b) of Lemma 7 in the supplementary article to obtain the result.

Proof of Theorem 4

For convenience, all limits stated in this proof assume that max{m, n} = O(r) and n/(m + n) → λ ∈ (0, 1), as r → ∞. We have independent random samples $X_1^{(r)},X_2^{(r)},\dots ,X_m^{(r)}$ and $Y_1^{(r)},Y_2^{(r)},\dots ,Y_n^{(r)}$ from F^(r) and G^(r), respectively. The sample ODC is $R_{mn}^{(r)}=F_m^{(r)}{(G_n^{(r)})}^{-1}$, where $F_m^{(r)}$ and ${(G_n^{(r)})}^{-1}$ are the empirical distribution and empirical quantile functions, respectively. Our test statistic is $M_{mn}^p=c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}\Vert }_p$. By the triangle inequality,

$${\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(M_{mn}^p\ge c_{\alpha ,p})\ge {\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p<c_{mn}{\Vert \mathcal{D}{R}^{(r)}\Vert }_p-c_{\alpha ,p})$$

for all p ∈ [1,∞]. Therefore, to prove part (a), it suffices to show that $c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p=O_P(1)$.

From Lemma 3 in the supplementary article, it follows that ${\Vert ℳR_{mn}^{(r)}-ℳR^{(r)}\Vert }_{\infty }\le {\Vert R_{mn}^{(r)}-R^{(r)}\Vert }_{\infty }$, which implies ${\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_{\infty }\le 2{\Vert R_{mn}^{(r)}-R^{(r)}\Vert }_{\infty }$. Because L^p norms are dominated by the sup-norm, it therefore suffices to show $c_{mn}{\Vert R_{mn}^{(r)}-R^{(r)}\Vert }_{\infty }$ is O_P (1). To accomplish this, we decompose $c_{mn}(R_{mn}^{(r)}-R^{(r)})$ into two parts:

$$c_{mn}(R_{mn}^{(r)}-R^{(r)})=c_{mn}\{F_m^{(r)}{(G_n^{(r)})}^{-1}-F^{(r)}{(G_n^{(r)})}^{-1}\}+c_{mn}\{F^{(r)}{(G_n^{(r)})}^{-1}-F^{(r)}{(G^{(r)})}^{-1}\}.$$ (7.2)

Define the two independent empirical processes

$$U_m(u)=\frac{1}{m}\sum _{i=1}^{m}I\{F^{(r)}(X_i^{(r)})\le u\}$$

and $V_n(u)=n^{-1}{\sum }_{i=1}^{n}I\{G^{(r)}(Y_i^{(r)})\le u\}$, for 0 ≤ u ≤ 1. This allows us to rewrite $F_m^{(r)}$ as U_mF^(r) and $F^{(r)}{(G_n^{(r)})}^{-1}$ as $R^{(r)}{V}_n^{(r)}$. Consequently, the two terms on the right-hand side of Equation (7.2) can be written as

$$c_{mn}\{F_m^{(r)}{(G_n^{(r)})}^{-1}-F^{(r)}{(G_n^{(r)})}^{-1}\}=c_{mn}[U_m\{F^{(r)}{(G_n^{(r)})}^{-1}\}-U\{F^{(r)}{(G_n^{(r)})}^{-1}\}]$$ (7.3)

and

$$c_{mn}\{F^{(r)}{(G_n^{(r)})}^{-1}-F^{(r)}{(G^{(r)})}^{-1}\}=c_{mn}(R^{(r)}{V}_n-R^{(r)}V),$$ (7.4)

where U(·) and V (·) both represent the cumulative distribution function of a uniform distribution on [0, 1]. These expressions allow us to unify all random samples (from different distributions) to be uniformly distributed.

We are now ready to show that the sup-norms of the right-hand sides of Equations (7.3) and (7.4) are uniformly bounded in probability. We begin with the uniform processes. From Theorem 3 in Komlós et al. (1975), there exist versions of independent standard Brownian bridges ${ℬ}_1^{(m)}$ and ${ℬ}_2^{(n)}$ such that, almost surely,

$${\Vert \sqrt{m}(U_m-U)-{ℬ}_1^{(m)}\Vert }_{\infty }=o(m^{-1/2}{(logm)}^2)$$ (7.5)

$${\Vert \sqrt{n}(V_n-V)-{ℬ}_2^{(n)}\Vert }_{\infty }=o(n^{-1/2}{(logn)}^2).$$ (7.6)

Because $lim{c}_{mn}/({\lambda }^{1/2}\sqrt{m})=1$, we have ${\Vert c_{mn}(U_m-U)-{\lambda }^{1/2}{ℬ}_1^{(m)}\Vert }_{\infty }=o(m^{-1/2}{(logm)}^2)$ from Equation (7.5). Consequently, the sup-norm of the right-hand side of Equation (7.3) is less than or equal to

$${\Vert c_{mn}[U_m\{F^{(r)}{(G_n^{(r)})}^{-1}\}-U\{F^{(r)}{(G_n^{(r)})}^{-1}\}]-{\lambda }^{1/2}{ℬ}_1^{(m)}\{F^{(r)}{(G_n^{(r)})}^{-1}\}\Vert }_{\infty }+{\Vert {\lambda }^{1/2}{ℬ}_1^{(m)}\{F^{(r)}{(G_n^{(r)})}^{-1}\}\Vert }_{\infty }$$

which is less than or equal to

$${\Vert c_{mn}(U_m-U)-{\lambda }^{1/2}{ℬ}_1^{(m)}\Vert }_{\infty }+{\Vert {\lambda }^{1/2}{ℬ}_1^{(m)}\Vert }_{\infty }=o(m^{-1/2}{(logm)}^2)+O_P(1).$$

The O_P (1) term arises because ${ℬ}_1^{(m)}$ is bounded with probability 1. Likewise, the o(m^−1/2(logm)²) term comes from Equation (7.5). Therefore, we have shown that the sup-norm of the right-hand side of Equation (7.3), that is, ${\Vert c_{mn}\{F_m^{(r)}{(G_n^{(r)})}^{-1}-F^{(r)}{(G_n^r)}^{-1}\}\Vert }_{\infty }=O_P(1)$.

For the right-hand side of Equation (7.4), we use the mean value theorem to write

$$R^{(r)}{V}_n(u)-R^{(r)}V(u)={\stackrel{.}{R}}^{(r)}({\tau }_u)\{V_n(u)-V(u)\},$$

where Ṙ^(r) denotes the derivative of R^(r) and where τ_u is between V_n(u) and V (u). Therefore,

$$\underset{u\in [0,1]}{sup}\left|\sqrt{n}\{R^{(r)}{V}_n(u)-R^{(r)}V(u)\}-{\stackrel{.}{R}}^{(r)}({\tau }_u){ℬ}_2^{(n)}(u)\right|=\underset{u\in [0,1]}{sup}\left|{\stackrel{.}{R}}^{(r)}({\tau }_u)[\sqrt{n}\{V_n(u)-V(u)\}-{ℬ}_2^{(n)}(u)]\right|$$

which is less than or equal to

$${\Vert {\stackrel{.}{R}}^{(r)}\Vert }_{\infty }{\Vert \sqrt{n}(V_n-V)-{ℬ}_2^{(n)}\Vert }_{\infty }=O(1)o(n^{-1/2}{(logn)}^2)=o(n^{-1/2}{(logn)}^2).$$

The O(1) term above comes from the assumption that the derivative of R^(r) is uniformly bounded for all r over [0, 1]. Likewise, the o(n^−1/2(log n)²) term comes from Equation (7.6). Therefore, because $lim{c}_{mn}/\{{(1-\lambda )}^{1/2}\sqrt{n}\}=1$ and because ${ℬ}_2^{(n)}$ is bounded with probability 1, we have shown the sup-norm of the right-hand side of Equation (7.4), that is, $c_{mn}{\Vert R^{(r)}{V}_n^{(r)}-R^{(r)}V\Vert }_{\infty }=O_P(1)$. Finally, from Equation (7.2), we have $c_{mn}{\Vert R_{mn}^{(r)}-R^{(r)}\Vert }_{\infty }=O_P(1)+O_P(1)=O_P(1)$, which establishes part (a).

To prove part (b), let $q_{\beta ,p}^{(r)}$ denote the 1 – β quantile of the finite-sample distribution of $c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p$; i.e., $q_{\beta ,p}^{(r)}$ solves ${\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p\le q_{\beta ,p}^{(r)})=1-\beta$. We have already shown $c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p=O_P(1)$, so ${sup}_r{q}_{\beta ,p}^{(r)}\equiv q_{\beta ,p}<\infty$. Therefore,

$$liminf{\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p<q_{\beta ,p})\ge 1-\beta .$$

Set η_p(β) = q_β,p +c_α,p. Whenever lim inf c_mn||𝒟R||_p ≥ η_p(β), it follows from the triangle inequality that $liminf{\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(M_{mn}^p\ge c_{\alpha ,p})$ is greater than or equal to

$$liminf{\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p<c_{mn}{\Vert \mathcal{D}{R}^{(r)}\Vert }_p-c_{\alpha ,p})\ge liminf{\text{pr}}_{R^{(r)}\in {\mathrm{\Theta }}_1}(c_{mn}{\Vert \mathcal{D}{R}_{mn}^{(r)}-\mathcal{D}{R}^{(r)}\Vert }_p<q_{\beta ,p})\ge 1-\beta .$$

This completes the proof of part (b).