A GENERAL ASYMPTOTIC THEORY FOR MAXIMUM LIKELIHOOD ESTIMATION IN SEMIPARAMETRIC REGRESSION MODELS WITH CENSORED DATA

Abstract

We establish a general asymptotic theory for nonparametric maximum likelihood estimation in semiparametric regression models with right censored data. We identify a set of regularity conditions under which the nonparametric maximum likelihood estimators are consistent, asymptotically normal, and asymptotically efficient with a covariance matrix that can be consistently estimated by the inverse information matrix or the profile likelihood method. The general theory allows one to obtain the desired asymptotic properties of the nonparametric maximum likelihood estimators for any specific problem by verifying a set of conditions rather than by proving technical results from first principles. We demonstrate the usefulness of this powerful theory through a variety of examples.

1. Introduction

Semiparametric regression models are highly useful in investigating the effects of covariates on potentially censored responses (e.g. failure times and repeated measures) in longitudinal studies. It is desirable to analyze such models by the nonparametric maximum likelihood approach, which generally yields consistent, asymptotically normal, and asymptotically efficient estimators. It is technically difficult to prove the asymptotic properties of the nonparametric maximum likelihood estimators (NPMLEs). Thus far, rigorous proofs exist only in some special cases.

In this paper, we develop a general asymptotic theory for the NPMLEs with right censored data. The theory is very encompassing in that it pertains to a generic form of likelihood rather than specific models. We prove that, under a set of mild regularity conditions, the NPMLEs are consistent, asymptotically normal, and asymptotically efficient with a limiting covariance matrix that can be consistently estimated by the inverse information matrix or the profile likelihood method.

This paper is the technical companion to Zeng and Lin (2007), in which several classes of models were proposed to unify and extend existing semiparametric regression models. The likelihoods for those models can all be written in the general form considered in this paper. For each class of models in Zeng and Lin (2007), we identify a set of conditions under which the regularity conditions for the general theory hold so that desired asymptotic properties are ensured.

2. Some Semiparametric Models

We describe briefly the three kinds of models considered in Zeng and Lin (2007). We assume that the censoring mechanism satisfies coarsening at random (Heitjan and Rubin (1991)).

2.1. Transformation Models for Counting Processes

Let N*(t) record the number of events that the subject has experienced by time t, and let Z(·) denote the corresponding covariate processes. Zeng and Lin (2007) proposed the following class of transformation models for the cumulative intensity function of N*(t)

$$\mathrm{\Lambda }(t\mid Z)=G\left[{\left\{1+{\int }_0^t{R}^{\ast }(s)e^{{\beta }^{T}Z(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T\stackrel{\sim }{Z}}}\right]-G(1),$$

where G is a continuously differentiable and strictly increasing function with G′(1) > 0 and G(∞) = ∞, R*(·) is an indicator process, Z̃ is a subset of Z, β and γ are regression parameters, and Λ(·) is an unspecified increasing function. The data consist of {N_i(t), R_i(t), Z_i(t); t ∈ [0, τ]} (i = 1, …, n), where $R_i(t)=I(C_i\ge t)R_i^{\ast }(t),N_i(t)=N_i^{\ast }(t\wedge C_i)$, C_i is the censoring time, and τ is a finite constant. The likelihood is

$$\prod _{i=1}^n\prod _{t\le \tau }{\{R_i(t)d\mathrm{\Lambda }(t\mid Z_i)\}}^{d{N}_i(t)}exp\left\{-{\int }_0^{\tau }{R}_i(t)d\mathrm{\Lambda }(t\mid Z_i)\right\},$$

where dN_i(t) = N_i(t) − N_i(t−).

2.2. Transformation Models With Random Effects for Dependent Failure Times

For i = 1, …, n, k = 1, …, K and l = 1, …, n_ik, let $N_{\mathit{ikl}}^{\ast }(·)$ denote the number of the kth type of event experienced by the lth individual in the ith cluster, and Z_ikl(·) the corresponding covariate processes. Zeng and Lin (2007) assumed that the cumulative intensity for $N_{\mathit{ikl}}^{\ast }(t)$ takes the form

$${\mathrm{\Lambda }}_k(t\mid Z_{\mathit{ikl}};b_i)=G_k\left\{{\int }_0^t{R}_{\mathit{ikl}}^{\ast }(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b_i^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{\mathrm{\Lambda }}_k(s)\right\},$$

where G_k, Λ_k, and $R_{\mathit{ikl}}^{\ast }$ are analogous to G, Λ, and R* of Section 2.1, Z̃_ikl is a subset of Z_ikl plus the unit component, and b_i is a vector of random effects with density f (b; γ). Let C_ikl, N_ikl, and R_ikl be defined analogously to C_i, N_i, and R_i of Section 2.1. The likelihood is

$$\prod _{i=1}^n{\int }_b\prod _{k=1}^K\prod _{l=1}^{n_{ik}}\prod _{t\le \tau }{\left[R_{\mathit{ikl}}(t)e^{{\beta }^T{Z}_{\mathit{ikl}}(t)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t)}d{\mathrm{\Lambda }}_k(t)G_k^{\prime }\left\{{\int }_0^t{R}_{\mathit{ikl}}(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{\mathrm{\Lambda }}_k(s)\right\}\right]}^{d{N}_{\mathit{ikl}}(t)}\times exp\left[-G_k\left\{{\int }_0^{\tau }{R}_{\mathit{ikl}}(t)e^{{\beta }^T{Z}_{\mathit{ikl}}(t)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t)}d{\mathrm{\Lambda }}_k(t)\right\}\right]f(b;\gamma )db.$$

2.3. Joint Models for Repeated Measures and Failure Times

For i = 1, …, n and j = 1, …, n_i, let Y_ij be the response variable at time t_ij for the ith subject, and X_ij the corresponding covariates. We assume that (Y_i1, …, Y_ini) follows a generalized linear mixed model with density f_y(y|X_ij; b_i), where b_i is a set of random effects with density f (b; γ). We define $N_i^{\ast }$ and Z_i as in Section 2.1, and assume that

$$\mathrm{\Lambda }(t\mid Z_i;b_i)=G\left\{{\int }_0^t{R}_i^{\ast }(s)e^{{\beta }^T{Z}_i(s)+{(\psi \circ b_i)}^T{\stackrel{\sim }{Z}}_i(s)}d\mathrm{\Lambda }(s)\right\},$$

where Z̃_i is a subset of Z_i plus the unit component, ψ is a vector of unknown constants, and v₁ ◦ v₂ is the component-wise product of two vectors v₁ and v₂. The likelihood is

$$\prod _{i=1}^n{\int }_b\prod _{t\le \tau }{\{R_i(t)d\mathrm{\Lambda }(t\mid Z_i;b)\}}^{d{N}_i(t)}exp\left\{-{\int }_0^{\tau }{R}_i(t)d\mathrm{\Lambda }(t\mid Z_i;b)\right\}\prod _{j=1}^{n_i}{f}_y(Y_{ij}\mid X_{ij};b)f(b;\gamma )db.$$

For continuous measures, Zeng and Lin (2007) proposed the semiparametric linear mixed model

$$\stackrel{\sim }{H}(Y_{ij})={\alpha }^T{X}_{ij}+b_i^T{\stackrel{\sim }{X}}_{ij}+{\epsilon }_{ij},$$

where H̃ is an unknown increasing function with H̃(−∞) = −∞, H̃(∞) = ∞, and H̃(0) = 0, α is a set of regression parameters, X̃_ij is typically a subset of X_ij, and ε_ij (i = 1, …, n; j = 1, …, n_ij) are independent with density f_ε. Write Λ̃(y) = e^H̃(y). The likelihood is

$$\prod _{i=1}^n{\int }_b\prod _{t\le \tau }{\{R_i(t)d\mathrm{\Lambda }(t\mid Z_i;b)\}}^{d{N}_i(t)}exp\left\{-{\int }_0^{\tau }{R}_i(t)d\mathrm{\Lambda }(t\mid Z_i;b)\right\}\times \prod _{j=1}^{n_i}{f}_{\epsilon }(log(\stackrel{\sim }{\mathrm{\Lambda }}(Y_{ij}))-{\alpha }^T{X}_{ij}-b_i^T{\stackrel{\sim }{X}}_{ij})\{dlog\stackrel{\sim }{\mathrm{\Lambda }}(Y_{ij})/dy\}f(b;\gamma )db.$$

3. Nonparametric Maximum Likelihood Estimation

All the likelihood functions given in Section 2 can be expressed as

$$\prod _{i=1}^n\prod _{k=1}^K\prod _{l=1}^{n_{ik}}\prod _{t\le \tau }{\lambda }_k{(t)}^{R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)}\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A}),$$

where ${\lambda }_k(t)={\mathrm{\Lambda }}_k^{\prime }(t)$, θ is a d-vector of regression parameters and variance components, = (Λ₁, …, Λ_K), pertains to the observation on the ith cluster, and Ψ is a functional of , θ, and . For nonparametric maximum likelihood estimation, we allow to be discontinuous with jumps at the observed failure times and maximize the modified likelihood function

$$\prod _{i=1}^n\prod _{k=1}^K\prod _{l=1}^{n_{ik}}\prod _{t\le \tau }{\mathrm{\Lambda }}_k{\left\{t\right\}}^{R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)}\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A}),$$

where Λ_k{t} denotes the jump size of the monotone function Λ_k at t. Equivalently, we maximize the logarithm of the above function

$${\mathcal{L}}_n(\theta ,\mathcal{A})=\sum _{i=1}^n\left[\sum _{k=1}^K\sum _{l=1}^{n_{ik}}{\int }_0^{\tau }{R}_{\mathit{ikl}}(t)log{\mathrm{\Lambda }}_k\left\{t\right\}d{N}_{\mathit{ikl}}^{\ast }(t)+log\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})\right].$$ (1)

We wish to establish an asymptotic theory for the resulting NPMLEs θ̂ and .

4. Regularity Conditions

We impose the following conditions on the model and data structures.

(C1) The true value θ₀ lies in the interior of a compact set Θ, and the true functions Λ_0k are continuously differentiable in [0, τ] with ${\mathrm{\Lambda }}_{0k}^{\prime }(t)>0$, k = 1, …, K.

(C2) With probability one, P(inf_s∈[0,t] R_ik·(s) ≥ 1|Z_ikl, l = 1, …, n_ik) > δ₀ > 0 for all t ∈ [0, τ], where $R_{ik·}(t)={\sum }_{l=1}^{n_{ik}}{R}_{\mathit{ikl}}(t)$.

(C3) There exist a constant c₁ > 0 and a random variable r₁() > 0 such that E[log r₁()] < ∞ and, for any θ ∈ Θ and any finite Λ₁, …, Λ_K,

$$\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})\le r_1({\mathcal{O}}_i)\prod _{k=1}^K\prod _{t\le \tau }{\left\{1+{\int }_0^t{R}_{ik·}(t)d{\mathrm{\Lambda }}_k(t)\right\}}^{-d{N}_{ik·}^{\ast }(t)}{\left\{1+{\int }_0^{\tau }{R}_{ik·}(t)d{\mathrm{\Lambda }}_k(t)\right\}}^{-c_1}$$

almost surely, where $N_{ik·}^{\ast }(t)={\sum }_{l=1}^{n_{ik}}{N}_{\mathit{ikl}}^{\ast }(t)$. In addition, for any constant c₂,

$$inf\{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A}):{\left|\right|{\mathrm{\Lambda }}_1\left|\right|}_{V[0,\tau ]}\le c_2,\dots ,{\left|\right|{\mathrm{\Lambda }}_K\left|\right|}_{V[0,\tau ]}\le c_2,\theta \in \mathrm{\Theta }\}>r_2({\mathcal{O}}_i)>0,$$

where ||h||_V[0,τ] is the total variation of h(·) in [0, τ], and r₂(), which may depend on c₂, is a finite random variable with E[|log r₂()|] < ∞.

We require certain smoothness of Ψ. Let Ψ̇_θ denote the derivative of Ψ(; θ, ) with respect to θ, and let Ψ̇_k[H_k] denote the derivative of Ψ(; θ, ) along the path (Λ_k + εH_k), where H_k belongs to the set of functions in which Λ_k + εH_k is increasing with bounded total variation.

(C4) For any (θ⁽¹⁾, θ⁽²⁾) ∈ Θ, and $({\mathrm{\Lambda }}_1^{(1)},{\mathrm{\Lambda }}_1^{(2)}),\dots ,({\mathrm{\Lambda }}_K^{(1)},{\mathrm{\Lambda }}_K^{(2)}),(H_1^{(1)},H_1^{(2)}),\dots ,(H_K^{(1)},H_K^{(2)})$ with uniformly bounded total variations, there exist a random variable ℱ() ∈ L₄(P) and K stochastic processes μ_ik(t; ) ∈ L₆(P), k = 1, …, K, such that

$$\begin{array}{l}\left|\mathrm{\Psi }({\mathcal{O}}_i;{\theta }^{(1)},{\mathcal{A}}^{(1)})-\mathrm{\Psi }({\mathcal{O}}_i;{\theta }^{(2)},{\mathcal{A}}^{(2)})\right|+\left|{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;{\theta }^{(1)},{\mathcal{A}}^{(1)})-{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;{\theta }^{(2)},{\mathcal{A}}^{(2)})\right|+\sum _{k=1}^K\left|{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }^{(1)},{\mathcal{A}}^{(1)})[H_k^{(1)}]-{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }^{(2)},{\mathcal{A}}^{(2)})[H_k^{(2)}]\right|+\sum _{k=1}^K\left|\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }^{(1)},{\mathcal{A}}^{(1)})[H_k^{(1)}]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }^{(1)},{\mathcal{A}}^{(1)})}-\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }^{(2)},{\mathcal{A}}^{(2)})[H_k^{(2)}]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }^{(2)},{\mathcal{A}}^{(2)})}\right|\\ \le \mathcal{F}({\mathcal{O}}_i)\left[\mid {\theta }^{(1)}-{\theta }^{(2)}\mid +\sum _{k=1}^K\left\{{\int }_0^{\tau }\mid {\mathrm{\Lambda }}_k^{(1)}(s)-{\mathrm{\Lambda }}_k^{(2)}(s)\mid d{\mu }_{ik}(s;{\mathcal{O}}_i)+{\int }_0^{\tau }\mid H_k^{(1)}(s)-H_k^{(2)}(s)\mid d{\mu }_{ik}(s;{\mathcal{O}}_i)\right\}\right].\end{array}$$

In addition, μ_ik(s; ) is non-decreasing, and E[ℱ()μ_ik(s; )] is left-continuous with uniformly bounded left- and right-derivatives for any s ∈ [0, τ]. Here, the right-derivative for a function f(x) is defined as lim_h→0+(f (x + h) − f (x+))/h.

The following condition ensures identifiability of parameters.

(C5) (First Identifiability Condition) If

$$\left[\prod _{k=1}^K\prod _{l=1}^{n_{ik}}\prod _{t\le \tau }{\lambda }_k^{\ast }{(t)}^{R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)}\right]\mathrm{\Psi }({\mathcal{O}}_i;{\theta }^{\ast },{\mathcal{A}}^{\ast })=\left[\prod _{k=1}^K\prod _{l=1}^{n_{ik}}\prod _{t\le \tau }{\lambda }_{0k}{(t)}^{R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)}\right]\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)$$

almost surely, then θ* = θ₀ and ${\mathrm{\Lambda }}_k^{\ast }(t)={\mathrm{\Lambda }}_{0k}(t)$ for t ∈ [0, τ], k = 1, …, K.

The next assumption is more technical and will be used in proving the weak convergence of the NPM-LEs. For any fixed (θ, ) in a small neighborhood of (θ₀, ) in R^d × {BV[0, τ]}^K, where BV[0, τ] denotes the space of functions with bounded total variations in [0, τ], (C4) implies that the linear functional

$$H_k\mapsto E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;\theta ,\mathcal{A})[H_k]}{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})}\right]$$

is continuous from BV[0, τ] to R. Thus, there exists a bounded function η_0k(s; θ, ) such that

$$E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;\theta ,\mathcal{A})[H_k]}{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})}\right]={\int }_0^{\tau }{\eta }_{0k}(s;\theta ,\mathcal{A})d{H}_k(s).$$

(C6) There exist functions ζ_0k(s; θ₀, ) ∈ BV[0, τ], k = 1, …, K, and a matrix ζ_0θ(θ₀, ) such that

$$\left|E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;\theta ,\mathcal{A})}{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})}-\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]-{\zeta }_{0\theta }({\theta }_0,{\mathcal{A}}_0)(\theta -{\theta }_0)-\sum _{k=1}^K{\int }_0^{\tau }{\zeta }_{0k}(s;{\theta }_0,{\mathcal{A}}_0)d({\mathrm{\Lambda }}_k-{\mathrm{\Lambda }}_{0k})\right|=o\left(\mid \theta -{\theta }_0\mid +\sum _{k=1}^K{\left|\right|{\mathrm{\Lambda }}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right).$$

In addition, for k = 1, …, K,

$$\sum _{k=1}^K\underset{s\in [0,\tau ]}{sup}\left|\left\{{\eta }_{0k}(s;\theta ,\mathcal{A})-{\eta }_{0k}(s;{\theta }_0,{\mathcal{A}}_0)\right\}-{\eta }_{0k\theta }(s;{\theta }_0,{\mathcal{A}}_0)(\theta -{\theta }_0)-{\int }_0^{\tau }\sum _{m=1}^K{\eta }_{0km}(s,t;{\theta }_0,{\mathcal{A}}_0)d({\mathrm{\Lambda }}_m-{\mathrm{\Lambda }}_{0m})(t)\right|=o\left(\mid \theta -{\theta }_0\mid +\sum _{k=1}^K{\left|\right|{\mathrm{\Lambda }}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right),$$

where η_0km is a bounded bivariate function and η_0kθ is a d-dimensional bounded function. Furthermore, there exists a constant c₃ such that |η_0km(s, t₁; θ₀, ) − η_0km(s, t₂; θ₀, )| ≤ c₃|t₁ − t₂| for any s ∈ [0, τ] and any t₁, t₂ ∈ [0, τ].

The final assumption ensures that the Fisher information matrix along any finite-dimensional sub-model is non-singular.

(C7) (Second Identifiability Condition) If with probability one,

$$\sum _{k=1}^K\sum _{l=1}^{n_{ik}}\int h_k(t)R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)+\frac{{\stackrel{.}{\mathrm{\Psi }}}_{\theta }{({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}^{T}v+{\sum }_{k=1}^K{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[\int h_{k}d{\mathrm{\Lambda }}_{0k}]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}=0$$

for some constant vector v ∈ R^d and h_k ∈ BV[0, τ], k = 1, …, K, then v = 0 and h_k = 0 for k = 1, …, K.

Remark 1

(C1)–(C2) are standard assumptions in any analysis of censored data. (C3) pertains to the model structure, and (C4) and (C6) essentially impose the smoothness of this structure. Although they appear technical, these conditions are easy to verify in practice. (C5) and (C7) usually require some work to verify, but can be translated to simple conditions in specific cases.

5. Some Useful Lemmas

Lemma 1

For any constant c, the following classes of functions are P-Donsker:

$$\begin{array}{l}{\mathcal{F}}_1=\left\{log\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A}):{\left|\right|{\mathrm{\Lambda }}_k\left|\right|}_{V[0,\tau ]}\le c,k=1,\dots ,K,\theta \in \mathrm{\Theta }\right\},\\ {\mathcal{F}}_2=\left\{\frac{{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;\theta ,\mathcal{A})}{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})}:{\left|\right|{\mathrm{\Lambda }}_k\left|\right|}_{V[0,\tau ]}\le c,k=1,\dots ,K,\theta \in \mathrm{\Theta }\right\},\\ {\mathcal{F}}_{3k}=\left\{\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;\theta ,\mathcal{A})[H]}{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})}:{\left|\right|{\mathrm{\Lambda }}_m\left|\right|}_{V[0,\tau ]}\le c,m=1,\dots ,K,\theta \in \mathrm{\Theta }{\left|\right|H\left|\right|}_{V[0,\tau ]\le c}\right\},k=1,\dots ,K.\end{array}$$

Proof

We only prove that ℱ_3k is P-Donsker; the proofs for the other two classes are similar. For k = 1, …, K, we define a measure μ̃_k in [0, τ] such that, for any Borel set A ⊂ [0, τ],

$${\stackrel{\sim }{\mu }}_k(A)={\int }_0^{\tau }I(t\in A)E[\mathcal{F}{({\mathcal{O}}_i)}^2{({\mu }_{ik}(\tau ;{\mathcal{O}}_i)-{\mu }_{ik}(0;{\mathcal{O}}_i))}^{2}d{\mu }_{ik}(t;{\mathcal{O}}_i)].$$

Condition (C4) implies that μ̃_k([0, τ]) ≤ ||ℱ()||_L4(P)||μ_ik(τ; ) − μ_ik(0; )||_L6(P). Thus, μ̃_k is a finite measure. According to Theorem 2.7.5 of van der Vaart and Wellner (1996), the bracket covering number for any bounded set in BV[0, τ] is of order exp{O(1/ε)} in L₂(μ̃_k), k = 1, …, K. Thus, we can construct N_ε ≡ (1/ε)^d × exp{O(K/ε)} × exp{O(1/ε)} brackets for the set of (θ, , H) in ℱ_3k, denoted by

$$[{\theta }_p^L,{\theta }_p^U]\times [{\mathrm{\Lambda }}_{1p}^L,{\mathrm{\Lambda }}_{1p}^U]\times \cdots \times [{\mathrm{\Lambda }}_{Kp}^L,{\mathrm{\Lambda }}_{Kp}^U]\times [H_p^L,H_p^U],p=1,\dots ,N_{\epsilon },$$

such that $\mid {\theta }_p^U-{\theta }_p^L\mid <\epsilon$ and

$$\int \mid {\mathrm{\Lambda }}_{kp}^U-{\mathrm{\Lambda }}_{kp}^L{\mid }^{2}d{\stackrel{\sim }{\mu }}_k<{\epsilon }^2,\int \mid H_p^U-H_p^L{\mid }^{2}d{\stackrel{\sim }{\mu }}_k<{\epsilon }^2,k=1,\dots ,K.$$

Any (θ, , H) must belong to one of these brackets. Obviously, the bracket functions

$$\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_p^L,{\mathcal{A}}_p^L)[H^L]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_p^L,{\mathcal{A}}_p^L)}\pm \mathcal{F}({\mathcal{O}}_i)\left\{\mid {\theta }_p^U-{\theta }_p^L\mid +\sum _{m=1}^K\int \mid {\mathrm{\Lambda }}_{mp}^U(s)-{\mathrm{\Lambda }}_{mp}^L(s)\mid d{\mu }_{im}(s;{\mathcal{O}}_i)+\int \mid H_p^U(s)-H_p^L(s)\mid d{\mu }_{im}(s;{\mathcal{O}}_i)\right\},p=1,\dots ,N_{\epsilon },$$

cover all the functions in ℱ_3k. Since

$$\begin{array}{l}{\Vert \mathcal{F}({\mathcal{O}}_i)\left\{\mid {\theta }_p^U-{\theta }_p^L\mid +\sum _{m=1}^K\int \mid {\mathrm{\Lambda }}_{mp}^U(s)-{\mathrm{\Lambda }}_{mp}^L(s)\mid d{\mu }_{im}(s;{\mathcal{O}}_i)+\sum _{m=1}^K\int \mid H_p^U(s)-H_p^L(s)\mid d{\mu }_{im}(s;{\mathcal{O}}_i)\right\}\Vert }_{L_2(P)}\\ \le c\left[\mid {\theta }_p^U-{\theta }_p^L\mid +\sum _{m=1}^K{\left\{E{\left(\int \mid {\mathrm{\Lambda }}_{mp}^U(s)-{\mathrm{\Lambda }}_{mp}^L(s)\mid d{\stackrel{\sim }{\mu }}_{im}\mathcal{F}({\mathcal{O}}_i)\right)}^2\right\}}^{1/2}+\sum _{m=1}^K{\left\{{\int }_0^{\tau }\mid H_p^U(s)-H_p^L(s){\mid }^{2}d{\stackrel{\sim }{\mu }}_m\right\}}^{1/2}\right]\\ \le c\left[\mid {\theta }_p^U-{\theta }_p^L\mid +\sum _{m=1}^K{\left\{\int \mid {\mathrm{\Lambda }}_{mp}^U(s)-{\mathrm{\Lambda }}_{mp}^L(s){\mid }^{2}d{\stackrel{\sim }{\mu }}_m\right\}}^{1/2}+\sum _{m=1}^K{\left\{{\int }_0^{\tau }\mid H_p^U(s)-H_p^L(s){\mid }^{2}d{\stackrel{\sim }{\mu }}_m\right\}}^{1/2}\right],\end{array}$$

where c is a constant depending on K, the L₂(P)-distance within each bracket pair is O(ε). Hence, the bracket entropy integral of ℱ_3k is finite, so that ℱ_3k is P-Donsker.

Lemma 2

For any bounded random variable (θ, Λ) in Θ × BV[0, τ], the function g(s) ≡ |E[Ψ̇_k(; θ, ) [I(· ≥ s)]/Ψ (; θ, )]| is left-continuous and satisfies that, for any s ∈ [0, τ], there exist δ_s, c_s > 0 such that |g(s̃) − g(s)| ≤ c_s|s̃ − s| for s̃ ∈ (s − δ_s, s) and |g(s̃) − g(s+)| ≤ c_s|s̃ − s| for s̃ ∈ (s, s + δ_s).

Proof

Since μ_ik(t; ) is non-decreasing in t, it follows from (C4) that for any s₁ and s₂,

$$\begin{array}{l}\mid g(s_1)-g(s_2)\mid \le E\left[\mathcal{F}({\mathcal{O}}_i)\left\{\int \mid I(t\ge s_1)-I(t\ge s_2)\mid d{\mu }_{ik}(t;{\mathcal{O}}_i)\right\}\right]\\ \le \mid E[\mathcal{F}({\mathcal{O}}_i){\mu }_{ik}(s_1;{\mathcal{O}}_i)]-E[\mathcal{F}({\mathcal{O}}_i){\mu }_{ik}(s_2;{\mathcal{O}}_i)]\mid .\end{array}$$

Thus, g(s) is in BV[0, τ] and is left-continuous. In addition, the left- and right-differentiability of E[ℱ()μ_ik(s; )] in (C4) implies that the second part of the lemma holds.

Lemma 3

For any h(s) ∈ BV[0, τ], the linear map $h\mapsto {\int }_0^{\tau }h(t){\eta }_{0km}(t,s;{\theta }_0,{\mathcal{A}}_0)d{\mathrm{\Lambda }}_{0k}(t)$ is a bounded compact operator from BV[0, τ] to BV[0, τ].

Proof

It is clear from (C6) that this function maps any bounded set in BV[0, τ] into a bounded set consisting of Lipschitz-continuous functions. The result thus follows since any bounded and Lipschitz-continuous functions consist of a totally bounded set in BV[0, τ] and the linear map is continuous.

6. Consistency

The following theorem states the consistency of θ̂ and Λ̂_k, k = 1, …, K.

Theorem 1

Under (C1)–(C5), $\mid \widehat{\theta }-{\theta }_0\mid +{\sum }_{k=1}^K{sup}_{t\in [0,\tau ]}\mid {\widehat{\mathrm{\Lambda }}}_k(t)-{\mathrm{\Lambda }}_{0k}(t)\mid {\to }_{a.s.}0$.

Proof

We fix a random sample in the probability space and assume that (C1)–(C5) hold for this sample. The set of such samples has probability one. We prove the result for this fixed sample. The entire proof consists of three steps.

Step 1

We show that the NPMLEs exist or, equivalently, Λ̂_k(τ) < ∞ (k = 1, …, K) for large n. By (C3), the likelihood function is bounded by

$$\begin{array}{l}\prod _{i=1}^n{r}_1({\mathcal{O}}_i)\prod _{k=1}^K\prod _{t\le \tau }{\left[{\mathrm{\Lambda }}_k\{t\}{R}_{ik·}(t){\left\{1+{\int }_0^t{R}_{ik·}(s)d{\mathrm{\Lambda }}_k(s)\right\}}^{-1}\right]}^{d{N}_{ik·}^{\ast }(t)}{\left\{1+{\int }_0^{\tau }{R}_{ik·}(s)d{\mathrm{\Lambda }}_k(s)\right\}}^{-c_1}\\ \le \prod _{i=1}^n{r}_1({\mathcal{O}}_i)\prod _{k=1}^K{\left\{1+{\int }_0^{\tau }{R}_{ik·}(s)d{\mathrm{\Lambda }}_k(s)\right\}}^{-c_1}.\end{array}$$

If Λ_k(τ) = ∞ for some k, then (C2) implies that, with probability one, inf_t∈[0,τ] R_ik·(t) ≥ 1 for some i, so that the above function is equal to zero. Thus, the maximum of the likelihood function can only be attained for Λ̂_k(τ) < ∞.

Step 2

We show that lim sup_n Λ̂_k(τ) < ∞ almost surely, i.e., Λ̂_k(τ) is bounded uniformly for all large n. By differentiating the objective function (1) with respect to Λ_k{Y_ikl} for which $d{N}_{\mathit{ikl}}^{\ast }(Y_{\mathit{ikl}})=1$ and R_ikl(Y_ikl) = 1, we note that Λ̂_k{Y_ikl} satisfies

$$\frac{1}{{\widehat{\mathrm{\Lambda }}}_k\left\{Y_{\mathit{ikl}}\right\}}=-\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})[I(·\ge Y_{\mathit{ikl}})]}{\mathrm{\Psi }({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})}.$$

In other words,

$${\widehat{\mathrm{\Lambda }}}_k(t)=-\sum _{i=1}^n\sum _{m=1}^{n_{ik}}{\int }_0^t{\left\{\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})}\right\}}^{-1}{R}_{\mathit{ikm}}(s)d{N}_{\mathit{ikm}}^{\ast }(s).$$

To prove the boundedness of Λ̂_k(τ), we construct another step function Λ̃_k with jumps only at the Y_ikl for which $d{N}_{\mathit{ikl}}^{\ast }(Y_{\mathit{ikl}})=1$and R_ikl(Y_ikl) = 1,

$$\frac{1}{{\stackrel{\sim }{\mathrm{\Lambda }}}_k\left\{Y_{\mathit{ikl}}\right\}}=-\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge Y_{\mathit{ikl}})]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)},$$

that is,

$${\stackrel{\sim }{\mathrm{\Lambda }}}_k(t)=-\sum _{i=1}^n\sum _{m=1}^{n_{ik}}{\int }_0^t{\left\{\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)}\right\}}^{-1}{R}_{\mathit{ikm}}(s)d{N}_{\mathit{ikm}}^{\ast }(s).$$

We show that Λ̃_k uniformly converges to Λ_0k. By Lemma 1,

$$n^{-1}\left\{\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)}\right\}\to E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]$$ (2)

uniformly in s ∈ [0, τ]. Since the score function along the path Λ_k = Λ_0k + εI(· ≥ s) with the other parameters fixed at their true values has zero expectation,

$$\begin{array}{l}0=E\left[\sum _{l=1}^{n_{ik}}\int \frac{\delta (t=s)}{{\lambda }_{0k}(t)}{R}_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)\right]+E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]\\ =E\left[\sum _{l=1}^{n_{ik}}{R}_{\mathit{ikl}}(s)d{N}_{\mathit{ikl}}^{\ast }(s)/ds\right]/{\lambda }_{0k}(s)+E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right],\end{array}$$ (3)

where δ(t = s) is the Dirac function. The submodel is not in the parameter space; however, we can always choose a sequence of submodels in the parameter space which approximates this submodel. Thus, the uniform limit of Λ̃_k(t) is

$$E\left[\sum _{m=1}^{n_{ik}}{\int }_0^t{\left\{E\left[\sum _{l=1}^{n_{ik}}{R}_{\mathit{ikl}}(s)d{N}_{\mathit{ikl}}^{\ast }(s)/ds\right]/{\lambda }_{0k}(s)\right\}}^{-1}{R}_{\mathit{ikm}}(s)d{N}_{\mathit{ikm}}^{\ast }(s)\right]={\mathrm{\Lambda }}_{0k}(t).$$

That is, Λ̃_k(t) uniformly converges to Λ_0k(t).

We next show that the difference between the log-likelihood functions evaluated at (θ̂, ) and (θ₀, ), where = (Λ̃₁, …, Λ̃_K), is negative eventually if some Λ̂_k(τ) diverges, which will induce a contradiction. The key arguments are based on (C3). Clearly, n⁻¹ℒ_n(θ̂, ) ≥ n⁻¹ℒ_n(θ₀, ). It follows from (2) and (3) that nΛ̃_k{t} converges to ${\lambda }_{0k}(t)/E[{\sum }_{l=1}^{n_{ik}}{R}_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)/dt]$, and is thus uniformly bounded away from zero, where t is an observed failure time. Therefore,

$$n^{-1}{\mathcal{L}}_n({\theta }_0,\stackrel{\sim }{\mathcal{A}})+n^{-1}\sum _{i=1}^n\sum _{k=1}^K\sum _{l=1}^{n_{ik}}\int R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)logn=n^{-1}\sum _{i=1}^n\sum _{k=1}^K\sum _{l=1}^{n_{ik}}\int log(n{\stackrel{\sim }{\mathrm{\Lambda }}}_k\{t\})R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)+n^{-1}\sum _{i=1}^{n}log\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0),$$

which is bounded away from − ∞ when n is large. That is,

$$n^{-1}{\mathcal{L}}_n({\theta }_0,\stackrel{\sim }{\mathcal{A}})+n^{-1}\sum _{i=1}^n\sum _{k=1}^K\sum _{l=1}^{n_{ik}}\int R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)logn=O(1),$$

where O(1) denotes a finite constant. On the other hand, (C3) implies that

$$\begin{array}{l}{n}^{-1}{\mathcal{L}}_n(\widehat{\theta },\widehat{\mathcal{A}})\le n^{-1}\sum _{i=1}^n\sum _{k=1}^K\sum _{l=1}^{n_{ik}}\int R_{\mathit{ikl}}(t)log{\widehat{\mathrm{\Lambda }}}_k\left\{t\right\}d{N}_{\mathit{ikl}}^{\ast }(t)+n^{-1}\sum _{i=1}^{n}log\mathrm{\Psi }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})\\ \le n^{-1}\sum _{i=1}^{n}log{r}_1({\mathcal{O}}_i)+n^{-1}\sum _{i=1}^n\sum _{k=1}^K\int I(R_{ik·}(t)>0)log{\widehat{\mathrm{\Lambda }}}_k\left\{t\right\}d{N}_{ik·}(t)-n^{-1}\sum _{i=1}^n\sum _{k=1}^K\int log\left\{1+{\int }_0^t{R}_{ik·}(s)d{\widehat{\mathrm{\Lambda }}}_k(s)\right\}d{N}_{ik·}(t)-n^{-1}\sum _{i=1}^n\sum _{k=1}^K{c}_{1}log\left\{1+{\int }_0^{\tau }{R}_{ik·}(s)d{\widehat{\mathrm{\Lambda }}}_k(s)\right\},\end{array}$$

where $d{N}_{ik·}(t)={\sum }_{l=1}^{n_{ik}}{R}_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)$. Thus,

$$O(1)\le n^{-1}\sum _{i=1}^n\sum _{k=1}^K\int I(R_{ik·}(t)>0)log(n{\widehat{\mathrm{\Lambda }}}_k\{t\})d{N}_{ik·}(t)-n^{-1}\sum _{i=1}^n\sum _{k=1}^K\int log\left\{1+{\int }_0^t{R}_{ik·}(s)d{\widehat{\mathrm{\Lambda }}}_k(s)\right\}d{N}_{ik·}(t)-n^{-1}\sum _{i=1}^n\sum _{k=1}^K{c}_{1}log\left\{1+{\int }_0^{\tau }{R}_{ik·}(s)d{\widehat{\mathrm{\Lambda }}}_k(s)\right\}.$$ (4)

We now show that the right-hand side diverges to − ∞ if Λ̂_k(τ) diverges for some k. The proof is based on the partitioning idea of Murphy (1994). Specifically, we construct a sequence t_0k = τ > t_1k > t_2k > … in the following manner. First, we define

$$t_{1k}=\text{argmin}\left\{t\in [\text{0},t_{0k}\text{)}:\frac{c_{\text{1}}}{2}E[I({\overline{R}}_{ik·}(\tau )>0)]\ge E\left[I({\overline{R}}_{ik·}(t)>0,{\overline{R}}_{ik·}(\tau )=0){\int }_t^{t_{0k}}d{N}_{ik·}(t)\right]\right\},$$

where R̄_ik·(t) = inf_s∈[0,t] R_ik·(s). Clearly, such a t_1k exists, and the above inequality becomes an equality if t_1k > 0. If t_1k > 0, we choose a small constant ε₀ such that

$$\frac{{\epsilon }_0}{1-{\epsilon }_0}<\frac{c_{1}E[I({\overline{R}}_{ik·}(\tau )=0,{\overline{R}}_{ik·}(t_{1k})>0)]}{E[I({\overline{R}}_{ik·}(t_{1k})=0,{\overline{R}}_{ik·}(0)>0){\int }_0^{\tau }d{N}_{ik·}(t)]},$$

and define

$$t_{2k}=\text{argmin}\left\{t\in [0,t_{1k}):(1-{\epsilon }_0)E\left[\left\{c_1+{\int }_{t_{1k}}^{t_{0k}}d{N}_{ik·}(t)\right\}I({\overline{R}}_{ik·}(t_{0k})=0,{\overline{R}}_{ik·}(t_{1k})>0)\right]\ge E\left[I({\overline{R}}_{ik·}(t_{1k})=0,{\overline{R}}_{ik·}(t)>0){\int }_t^{t_{1k}}d{N}_{ik·}(t)\right]\right\}.$$

Such a t_2k exists. If t_2k > 0, the inequality is an equality, and we define

$$t_{3k}=\text{argmin}\left\{t\in [0,t_{1k}):(1-{\epsilon }_0)E\left[\left\{c_1+{\int }_{t_{2k}}^{t_{1k}}d{N}_{ik·}(t)\right\}I({\overline{R}}_{ik·}(t_{1k})=0,{\overline{R}}_{ik·}(t_{2k})>0)\right]\ge E\left[I({\overline{R}}_{ik·}(t_{2k})=0,{\overline{R}}_{ik·}(t)>0){\int }_t^{t_{2k}}d{N}_{ik·}(t)\right]\right\}.$$

We continue this process. The sequence eventually stops at some t_Nk,k = 0. If this is not true, then the sequence is infinite and strictly decreases to some t* ≥ 0. Since all the inequalities are equalities, we sum all the equations except the first one to obtain

$$(1-{\epsilon }_0)E\left[\left\{c_1+{\int }_{t^{\ast }}^{t_{0k}}d{N}_{ik·}(t)\right\}I({\overline{R}}_{ik·}(t^{\ast })>0,{\overline{R}}_{ik·}(\tau )=0)\right]=E\left[I({\overline{R}}_{ik·}(t_{1k})=0,{\overline{R}}_{ik·}(t^{\ast })>0){\int }_{t\ast }^{t_{1k}}d{N}_{ik·}(t)\right],$$

which implies that

$$c_1(1-{\epsilon }_0)E[I({\overline{R}}_{ik·}(\tau )=0,{\overline{R}}_{ik·}(t_{1k})>0)]\le {\epsilon }_{0}E\left[I({\overline{R}}_{ik·}(t_{1k})=0,{\overline{R}}_{ik·}(0)>0){\int }_0^{\tau }d{N}_{ik·}(t)\right].$$

This contradicts the choice of ε₀. Thus, the sequence stops at some t_Nk,k = 0.

If we write I_qk = [t_q+1,k, t_qk), then the right-hand side of (4) can be bounded by

$$\sum _{k=1}^K\left[n^{-1}\sum _{i=1}^n\sum _{q=0}^{N_k-1}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}log\left(n{\widehat{\mathrm{\Lambda }}}_k\{t\}\right)d{N}_{ik·}-n^{-1}\sum _{i=1}^n\sum _{q=0}^{N_k-1}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}log\left\{1+{\widehat{\mathrm{\Lambda }}}_k\left\{t_{q+1,k}\right\}\right\}-n^{-1}\sum _{i=1}^n\sum _{q=0}^{N_k-1}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0)c_{1}log\left\{1+{\widehat{\mathrm{\Lambda }}}_k\left\{t_{q+1,k}\right\}\right\}-n^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{0k})>0)log\left\{1+{\widehat{\mathrm{\Lambda }}}_k\left\{\tau \right\}\right\}\right].$$ (5)

Since log x is a concave function,

$$\begin{array}{l}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}log\left(n{\widehat{\mathrm{\Lambda }}}_k\{t\}\right)d{N}_{ik·}(t)\\ \le \left\{\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}\right\}\times log\left[\frac{{\sum }_{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}n{\widehat{\mathrm{\Lambda }}}_k\left\{t\right\}d{N}_{ik·}(t)}{{\sum }_{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}(t)}\right]\\ \le \left\{\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}\right\}\times log\left[\frac{n{\widehat{\mathrm{\Lambda }}}_k\left\{t_{qk}\right\}}{{\sum }_{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}(t)}\right].\end{array}$$

Therefore, (5) can be further bounded by

$$O(1)\le \sum _{k=1}^K[\sum _{q=0}^{N_k-1}{n}^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}\times log\left\{\frac{n}{{\sum }_{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}}\right\}+\sum _{q=0}^{N_k-1}log{\widehat{\mathrm{\Lambda }}}_k(t_{qk})\left\{n^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}\right\}-n^{-1}\sum _{i=1}^n\sum _{q=0}^{N-1}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}log\left\{1+{\widehat{\mathrm{\Lambda }}}_k(t_{q+1,k})\right\}-\sum _{q=0}^{N_k-1}{n}^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0)c_{1}log\left\{1+{\widehat{\mathrm{\Lambda }}}_k(t_{q+1,k})\right\}-n^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{0k})>0)log\left\{1+{\widehat{\mathrm{\Lambda }}}_k(\tau )\right\}].$$

By (C2),

$$\frac{n}{{\sum }_{i=1}^{n}I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}}{\to }_{a.s.}{\left(E\left[I({\overline{R}}_{ik·}(t_{qk})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}\right]\right)}^{-1}<\infty ,$$

so that

$$O(1)\le \sum _{k=1}^K(-n^{-1}\sum _{i=1}^n\frac{c_1}{2}I({\overline{R}}_{ik·}(t_{0k})>0)log\left\{1+{\widehat{\mathrm{\Lambda }}}_k(\tau )\right\}-\left\{n^{-1}\sum _{i=1}^n\frac{c_1}{2}I({\overline{R}}_{ik·}(t_{0k})>0)-n^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{0k})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{0k}}d{N}_{ik·}\right\}\times log\left\{1+{\widehat{\mathrm{\Lambda }}}_k(t_{0k})\right\}-\sum _{q=1}^{N_k-1}[n^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{q-1,k})=0,{\overline{R}}_{ik·}(t_{q,k})>0)\left\{c_1+{\int }_{t\in I_{qk}}d{N}_{ik·}\right\}-n^{-1}\sum _{i=1}^{n}I({\overline{R}}_{ik·}(t_{q,k})=0,{\overline{R}}_{ik·}(t_{q+1,k})>0){\int }_{t\in I_{qk}}d{N}_{ik·}]\left\{1+log{\widehat{\mathrm{\Lambda }}}_k(t_{q,k})\right\}).$$

According to the construction of the t_qk’s, the coefficients in front of log Λ̂_k(t_qk) are all negative when n is large enough. Therefore, the corresponding terms cannot diverge to ∞. However, if Λ̂_k(τ) → ∞, the first term in the summation goes to −∞. We conclude that for all n large enough, Λ̂_k(τ) < ∞. Thus, lim sup_n Λ̂_k(τ) < ∞.

Step 3

We obtain the consistency result from (C5). Since Λ̂_k is bounded and monotone, Λ̂_k is weakly compact. Helly’s Selection Theorem implies that, for any subsequence, we can always choose a further subsequence such that Λ̂_k point-wise converges to some monotone function ${\mathrm{\Lambda }}_k^{\ast }$. Without loss of generality, we also assume that θ̂ converges to some θ^*. The consistency will hold if we can show that ${\mathrm{\Lambda }}_k^{\ast }={\mathrm{\Lambda }}_{0k}$ and θ^* = θ₀. Since Λ_0k is continuous, the weak convergence of Λ̂_k to Λ_0k can be strengthened to the uniform convergence of Λ̂_k to Λ_0k in [0, τ].

Note that

$${\widehat{\mathrm{\Lambda }}}_k(t)={\int }_0^t\frac{\mid n^{-1}{\sum }_{j=1}^n{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]/\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)\mid }{\mid n^{-1}{\sum }_{j=1}^n{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})[I(·\ge s)]/\mathrm{\Psi }({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})\mid }d{\stackrel{\sim }{\mathrm{\Lambda }}}_k(s).$$ (6)

Clearly, Λ̂_k is absolutely continuous with respect to Λ̃_k. By condition (C3),

$$\underset{s\in [0,\tau ]}{sup}\left|n^{-1}\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})}-n^{-1}\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }^{\ast },{\mathcal{A}}^{\ast })[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }^{\ast },{\mathcal{A}}^{\ast })}\right|\le n^{-1}\sum _{j=1}^n\mathcal{F}({\mathcal{O}}_j)\left\{\mid \widehat{\theta }-{\theta }^{\ast }\mid +\sum _{k=1}^K\int \mid {\widehat{\mathrm{\Lambda }}}_k(t)-{\mathrm{\Lambda }}_k^{\ast }(t)\mid d{\mu }_{jk}(t;{\mathcal{O}}_j)\right\}\to 0$$

since Λ̂_k converges to ${\mathrm{\Lambda }}_k^{\ast }$ and is bounded and {ℱ()μ_jk(t; ): t ∈ [0, τ]} is a P-Glivenko-Cantelli class. By Lemma 1 and the Glivenko-Cantelli Theorem,

$$\begin{array}{l}{n}^{-1}\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }^{\ast },{\mathcal{A}}^{\ast })[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }^{\ast },{\mathcal{A}}^{\ast })}\to E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }^{\ast },{\mathcal{A}}^{\ast })[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }^{\ast },{\mathcal{A}}^{\ast })}\right]\text{uniformly}\text{in}s\in [0,\tau ],\\ n^{-1}\sum _{j=1}^n\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)}\to E\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]}{\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)}\right]\text{uniformly}\text{in}s\in [0,\tau ].\end{array}$$

The numerator and denominator in the integrand of (6) converge uniformly to deterministic functions, denoted by g_1k(s) and g_2k(s), respectively. It follows from (3) that $g_{1k}(s)\equiv E[{\sum }_{l=1}^{n_{ik}}{R}_{\mathit{ikl}}(s)d{N}_{ik{l}^{\ast }(s)}/ds]/{\lambda }_{ik}(s)$ is bounded away from zero. We claim that inf _s∈[0,τ] g_2k(s) > 0. If this is not true, then there exists some s^* ∈ [0, τ] such that g_2k(s^*+) = 0 or g_2k(s^*) = 0. By Lemma 2, there exist δ^* and c^* such that |g_2k(s)| ≤ c^*|s − s^*| for s ∈ (s^*, s^* + δ ^*) or s ∈ (s^* − δ ^*, s^*]. On the other hand, for any ε > 0,

$${\widehat{\mathrm{\Lambda }}}_k(\tau )\ge {\int }_0^{\tau }\frac{\mid n^{-1}{\sum }_{j=1}^n{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)[I(·\ge s)]/\mathrm{\Psi }({\mathcal{O}}_j;{\theta }_0,{\mathcal{A}}_0)\mid }{\epsilon +\mid n^{-1}{\sum }_{j=1}^n{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})[I(·\ge s)]/\mathrm{\Psi }({\mathcal{O}}_j;\widehat{\theta },\widehat{\mathcal{A}})\mid }d{\stackrel{\sim }{\mathrm{\Lambda }}}_k(s).$$

Taking limits on both sides, we obtain $O(1)\ge {\int }_0^{\tau }{\{\epsilon +g_{2k}(s)\}}^{-1}{g}_{1k}(s)d{\mathrm{\Lambda }}_{0k}(s)$. Let ε → 0. By the Monotone Convergence Theorem, $O(1)\ge {\int }_{s^{\ast }}^{s^{\ast }+{\delta }^{\ast }}{\{c^{\ast }\mid s-s^{\ast }\mid \}}^{-1}{g}_{1k}(s){\lambda }_{0k}(s)ds$, or $O(1)\ge {\int }_{s^{\ast }-{\delta }^{\ast }}^{s^{\ast }}{\{c^{\ast }\mid s-s^{\ast }\mid \}}^{-1}{g}_{1k}(s){\lambda }_{0k}(s)ds$. This is a contradiction since the right-hand side is infinite. The contradiction implies that the limit g_2k(s) is uniformly positive. We can take limits on both sides of (6) to obtain ${\mathrm{\Lambda }}_k^{\ast }(t)={\int }_0^t{g}_{2k}^{-1}(s)g_{1k}(s)d{\mathrm{\Lambda }}_{0k}(s)$. Thus, ${\mathrm{\Lambda }}_k^{\ast }$ is also absolutely continuous with respect to Λ_0k and $d{\mathrm{\Lambda }}_k^{\ast }/d{\mathrm{\Lambda }}_{0k}=g_{1k}/g_{2k}$. Since Λ_0k(t) is differentiable with respect to t, so is ${\mathrm{\Lambda }}_k^{\ast }(t)$. Write ${\left\{{\mathrm{\Lambda }}_k^{\ast }\right\}}^{\prime }(t)={\lambda }_k^{\ast }(t)$. The forgoing arguments show that dΛ̂_k(t)/dΛ̃_k(t) uniformly converges to ${\lambda }_k^{\ast }(t)/{\lambda }_{0k}(t)$, which is uniformly positive in [0, τ].

It follows from the inequality n⁻¹ℒ_n(θ̂, ) ≥ n⁻¹ℒ_n(θ₀, ) that

$$n^{-1}\sum _{i=1}^n\sum _{k=1}^K\sum _{l=1}^{n_{ik}}\int log\frac{d{\widehat{\mathrm{\Lambda }}}_k(t)}{d{\stackrel{\sim }{\mathrm{\Lambda }}}_k(t)}{R}_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)+n^{-1}\sum _{i=1}^{n}log\frac{\mathrm{\Psi }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,\stackrel{\sim }{\mathcal{A}})}\ge 0.$$

In view of Lemma 1, the Glivenko-Cantelli Theorem and the uniform convergence of dΛ̂_k/dΛ̃_k, taking limits on both sides of the above inequality yields

$$E\left[log\frac{{\prod }_{k=1}^K{\prod }_{l=1}^{n_{ik}}{\prod }_{t\le \tau }{\{{\lambda }_k^{\ast }(t)\}}^{R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)}\mathrm{\Psi }({\mathcal{O}}_i;{\theta }^{\ast },{\mathcal{A}}^{\ast })}{{\prod }_{k=1}^K{\prod }_{l=1}^{n_{ik}}{\prod }_{t\le \tau }{\{{\lambda }_0(t)\}}^{R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)}\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]\ge 0.$$

The left-hand side is the negative Kullback-Leibler distance of the density indexed by (θ^*, ). Thus, (C5) entails that θ^* = θ₀ and Λ^* = Λ₀.

7. Weak Convergence and Asymptotic Efficiency

Define = {v ∈ R^d, |v| ≤ 1}, and = {h(t): ||h(t)||_{V [0,τ]} ≤ 1 }. We identify (θ̂ − θ₀, − ) as a random element in l^∞( × ) through the definition ${(\widehat{\theta }-{\theta }_0)}^{T}v+{\sum }_{k=1}^K{\int }_0^{\tau }{h}_k(s)d({\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k})(s)$.

Theorem 2

Under (C1)–(C7), n^1/2(θ̂ − θ₀, − ) →_d ℊ in l^∞( × ), where ℊ is a continuous zero-mean Gaussian process. Furthermore, the limiting covariance matrix of n^1/2(θ̂ − θ₀) attains the semiparametric efficiency bound.

Proof

The proof is based on the likelihood equation and follows the arguments of van der Vaart (1998, pp. 419–424). Let ℒ(θ, ) be the log-likelihood function from a single cluster, ℒ̇_θ(θ, ) be the derivative of ℒ(θ, ) with respect to θ, and ℒ̇_k(θ, )[H_k] be the path-wise derivative along the path Λ_k + εH_k. We sometimes omit the arguments in these derivatives when θ = θ₀ and = . Let be the empirical measure based on n i.i.d. observations, and be its expectation.

Let = (h₁, …, h_K) ∈ . The likelihood equation for (θ̂, ) along the path (θ̂+εv, +ε∫d), where v ∈ R^d and h_k ∈ BV [0, τ], is given by

$$0={\mathcal{P}}_n\left[v^T{\stackrel{.}{\mathcal{L}}}_{\theta }(\theta ,\mathcal{A})+\sum _{k=1}^K{\stackrel{.}{\mathcal{L}}}_k(\theta ,\mathcal{A})\left[\int h_{k}d{\mathrm{\Lambda }}_k\right]\right].$$

To be specific,

$$0={\mathcal{P}}_n\left[\frac{v^T{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;\theta ,\mathcal{A})}{\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})}\right]+\sum _{k=1}^K{\mathcal{P}}_n\left[\sum _{l=1}^{n_{ik}}\int h_k(t)R_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)+{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;\theta ,\mathcal{A})\left[\int h_{k}d{\mathrm{\Lambda }}_k\right]\right].$$

Since (θ₀, ) maximizes [ℒ(θ, )],

$$0=\mathcal{P}\left[v^T{\stackrel{.}{\mathcal{L}}}_{\theta }({\theta }_0,{\mathcal{A}}_0)\right],0=\mathcal{P}\left[{\stackrel{.}{\mathcal{L}}}_k({\theta }_0,{\mathcal{A}}_0)\left[\int h_{k}d{\mathrm{\Lambda }}_{0k}\right]\right],h_k\in \mathcal{Q},k=1,\dots ,K.$$

These equations, combined with the likelihood equation for (θ̂, ), yield

$$n^{1/2}({\mathcal{P}}_n-\mathcal{P})\left[v^T{\stackrel{.}{\mathcal{L}}}_{\theta }(\widehat{\theta },\widehat{\mathcal{A}})+\sum _{k=1}^K{\stackrel{.}{\mathcal{L}}}_k(\widehat{\theta },\widehat{\mathcal{A}})\left[\int h_{k}d{\widehat{\mathrm{\Lambda }}}_k\right]\right]=-n^{1/2}\mathcal{P}\left[\frac{v^T{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}{\mathrm{\Psi }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}-\frac{v^T{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]-\sum _{k=1}^K{n}^{1/2}\mathcal{P}\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})[\int h_{k}d{\widehat{\mathrm{\Lambda }}}_k]}{\mathrm{\Psi }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}-\frac{{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[\int h_{k}d{\widehat{\mathrm{\Lambda }}}_{0k}]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right].$$

Define ${\mathcal{N}}_0=\left\{(\theta ,A):\mid \theta -{\theta }_0\mid +{\sum }_{k=1}^K{\left|\right|{\mathrm{\Lambda }}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}<{\delta }_0\right\}$, where δ₀ is a small positive constant. When n is large enough, (θ̂, ) belongs to with probability one. By Lemma 1 and the Donsker Theorem,

$$\begin{array}{l}{o}_p(1)+n^{1/2}({\mathcal{P}}_n-\mathcal{P})\left[v^T{\stackrel{.}{\mathcal{L}}}_{\theta }({\theta }_0,{\mathcal{A}}_0)+\sum _{k=1}^K{\mathcal{L}}_k({\theta }_0,{\mathcal{A}}_0)\left[\int h_{k}d{\mathrm{\Lambda }}_{0k}\right]\right]\\ =-n^{1/2}\mathcal{P}\left[\frac{v^T{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}{\mathrm{\Psi }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}-\frac{v^T{\stackrel{.}{\mathrm{\Psi }}}_{\theta }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]-\sum _{k=1}^K{n}^{1/2}\mathcal{P}\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})[\int h_{k}d{\widehat{\mathrm{\Lambda }}}_k]}{\mathrm{\Psi }({\mathcal{O}}_i;\widehat{\theta },\widehat{\mathcal{A}})}-\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[\int h_{k}d{\widehat{\mathrm{\Lambda }}}_{0k}]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right],\end{array}$$ (7)

where o_p(1) represents some random element converging in probability to zero in l^∞( × ).

Under (C6), the first term on the right-hand side of (7) is

$$n^{-1/2}\left\{\sum _{k=1}^K{\int }_0^{\tau }{v}^T{\zeta }_{0k}(s)d({\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k})+v^T{\zeta }_{0\theta }(\widehat{\theta }-{\theta }_0)\right\}+o\left(n^{1/2}\mid \widehat{\theta }-{\theta }_0\mid +n^{1/2}\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right).$$

The second term is $-{\sum }_{k=1}^K{n}^{1/2}\left\{{\int }_0^{\tau }{h}_k(t){\eta }_{0k}(t;\widehat{\theta },\widehat{A})d{\widehat{\mathrm{\Lambda }}}_k(t)-{\int }_0^{\tau }{h}_k(y){\eta }_{0k}(t;{\theta }_0,{\mathcal{A}}_0)d{\mathrm{\Lambda }}_{0k}(t)\right\}$. It follows from (C6) that the above expression is

$$\begin{array}{l}-\sum _{k=1}^K{n}^{1/2}\left[{\int }_0^{\tau }{h}_k(t)\left\{{\eta }_{0k\theta }(t;{\theta }_0,{\mathcal{A}}_0)(\widehat{\theta }-{\theta }_0)+\sum _{m=1}^K{\int }_0^{\tau }{\eta }_{0km}(s,t;{\theta }_0,{\mathcal{A}}_0)d({\widehat{\mathrm{\Lambda }}}_m-{\mathrm{\Lambda }}_{0m})(s)\right\}d{\mathrm{\Lambda }}_{0k}(t)+{\int }_0^{\tau }{h}_k(t){\eta }_{0k}(t;{\theta }_0,{\mathcal{A}}_0)d({\widehat{\mathrm{\Lambda }}}_k(t)-{\mathrm{\Lambda }}_{0k}(t))\right]+o\left(n^{1/2}\mid \widehat{\theta }-{\theta }_0\mid +n^{1/2}\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right)\\ =-\sum _{k=1}^K{n}^{1/2}\left[{(\widehat{\theta }-{\theta }_0)}^T{\int }_0^{\tau }{h}_k(t){\eta }_{0k\theta }(t;{\theta }_0,{\mathcal{A}}_0)d{\mathrm{\Lambda }}_{0k}(t)+\sum _{m=1}^K{\int }_0^{\tau }\left\{I(m=k)h_m(t){\eta }_{0m}(t;{\theta }_0,{\mathcal{A}}_0)+{\int }_0^{\tau }{\eta }_{0km}(s,t;{\theta }_0,{\mathcal{A}}_0)h_k(s)d{\mathrm{\Lambda }}_{0k}(s)\right\}d({\widehat{\mathrm{\Lambda }}}_m(t)-{\mathrm{\Lambda }}_{0m}(t))\right]+o\left(n^{1/2}\mid \widehat{\theta }-{\theta }_0\mid +n^{1/2}\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right).\end{array}$$

Thus, the right-hand side of (7) can be written as

$$-n^{-1/2}\left\{B_1{[v,\mathcal{W}]}^T(\widehat{\theta }-{\theta }_0)+\sum _{k=1}^K\int B_{2k}[v,\mathcal{W}]d({\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k})\right\}+o\left(n^{1/2}\mid \widehat{\theta }-{\theta }_0\mid +n^{1/2}\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right),$$

where (B₁, B₂₁, …, B_2K) are linear operators in R^d × {BV [0, τ]}^K, and

$$B_1[v,\mathcal{W}]=v^T{\zeta }_{0\theta }({\theta }_0,{\mathcal{A}}_0)+\sum _{k=1}^K{\int }_0^{\tau }{h}_k(t){\eta }_{0k\theta }(t;{\theta }_0,{\mathcal{A}}_0)d{\mathrm{\Lambda }}_{0k}(t),$$ (8)

$$B_{2k}[v,\mathcal{W}]=v^T{\zeta }_{0k}(s;{\theta }_0,{\mathcal{A}}_0)+h_k(t){\eta }_{0k}(t;{\theta }_0,{\mathcal{A}}_0)+\sum _{m=1}^K{\int }_0^{\tau }{\eta }_{0mk}(s,t;{\theta }_0,{\mathcal{A}}_0)h_m(s)d{\mathrm{\Lambda }}_{0k}(s),k=1,\dots ,K.$$ (9)

It follows from the above derivation that

$$\begin{array}{l}{B}_1{[v,\mathcal{W}]}^T\stackrel{\sim }{v}+\sum _{m=1}^K\int B_{2k}[v,\mathcal{W}]{\stackrel{\sim }{W}}_{k}d{\mathrm{\Lambda }}_{0k}\\ ={\frac{d}{d\epsilon }|}_{\epsilon =0}\mathcal{P}\left[v^T{\mathcal{L}}_{\theta }({\theta }_0+\epsilon \stackrel{\sim }{v},{\mathcal{A}}_0+\epsilon \int \stackrel{\sim }{\mathcal{W}}d{\mathcal{A}}_0)+\sum _{k=1}^K{\mathcal{L}}_k({\theta }_0+\epsilon \stackrel{\sim }{v},{\mathcal{A}}_0+\epsilon \int \stackrel{\sim }{\mathcal{W}}d{\mathcal{A}}_0)[\int h_{k}d{\mathrm{\Lambda }}_{0k}]\right].\end{array}$$ (10)

We can write (B₁, B₂₁, …, B_2K)[v, ] as

$$\left(\begin{array}{c}v\\ {\eta }_{01}(t;{\theta }_0,{\mathcal{A}}_0)\times h_1(t)\\ ⋮\\ {\eta }_{0K}(t;{\theta }_0,{\mathcal{A}}_0)\times h_K(t)\end{array}\right)+\left(\begin{array}{c}{v}^T{\zeta }_{0\theta }({\theta }_0,{\mathcal{A}}_0)+{\sum }_{k=1}^K{\int }_0^{\tau }{h}_k(t){\eta }_{0k\theta }(t;{\theta }_0,{\mathcal{A}}_0)d{\mathrm{\Lambda }}_{0k}(t)-v\\ v^T{\zeta }_{01}(t;{\theta }_0,{\mathcal{A}}_0)+{\sum }_{k=1}^K{\int }_0^{\tau }{\eta }_{0m1}(s,t;{\theta }_0,{\mathcal{A}}_0)h_m(s)d{\mathrm{\Lambda }}_{0m}(s)\\ ⋮\\ v^T{\zeta }_{0K}(t;{\theta }_0,{\mathcal{A}}_0)+{\sum }_{k=1}^K{\int }_0^{\tau }{\eta }_{0mK}(s,t;{\theta }_0,{\mathcal{A}}_0)h_m(s)d{\mathrm{\Lambda }}_{0m}(s)\end{array}\right).$$

We wish to prove that (B₁, B₂₁, …, B_2K) is invertible. As shown at the end of this section, η_0k(t; θ₀, ) < 0, so that the first term of (B₁, B₂₁, …, B_2K) is an invertible operator. It follows from Lemma 3 that the second term is a compact operator. Thus, (B₁, B₂₁, …, B_2K) is a Fredholm operator, and the invertibility of (B₁, …, B_2K) is equivalent to the operator being one-to-one (Rudin (1973, pp. 99–103)). Suppose that B₁[v, ] = 0, …, and B_2K[v, ] = 0. It is easy to see from (10) that the derivative of $\mathcal{P}[v^T{\mathcal{L}}_{\theta }({\theta }_0,{\mathcal{A}}_0)+{\sum }_{k=1}^K{\mathcal{L}}_k({\theta }_0,{\mathcal{A}}_0)[\int h_{k}d{\mathrm{\Lambda }}_{0k}]]$ along the path (θ₀ + εv, + ε∫d) is zero. That is, the information along this path is zero, or $v^T{\mathcal{L}}_{\theta }({\theta }_0,{\mathcal{A}}_0)+{\sum }_{k=1}^K{\mathcal{L}}_k({\theta }_0,{\mathcal{A}}_0)\left[\int h_{k}d{\mathrm{\Lambda }}_{0k}\right]=0$ almost surely. By (C7), v = 0 and = 0, so that (B₁, B₂₁, …, B_2K) is one-to-one and invertible.

It follows from (7) that, for any (v, ) ∈ × ,

$$n^{1/2}\left\{v^T(\widehat{\theta }-{\theta }_0)+\sum _{k=1}^K{\int }_0^{\tau }{h}_k(t)d({\widehat{\mathrm{\Lambda }}}_k(t)-{\mathrm{\Lambda }}_{0k}(t))\right\}=-n^{1/2}({\mathcal{P}}_n-\mathcal{P})\left[{\widehat{v}}^T{\stackrel{.}{\mathcal{L}}}_{\theta }({\theta }_0,{\mathcal{A}}_0)+\sum _{k=1}^K{\stackrel{.}{\mathcal{L}}}_k({\theta }_0,{\mathcal{A}}_0)\left[\int {\stackrel{\sim }{h}}_{k}d{\mathrm{\Lambda }}_{0k}\right]\right]+o\left(n^{1/2}\mid \widehat{\theta }-{\theta }_0\mid +n^{1/2}\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right),$$

where (ṽ, h̃₁, …, h̃_K) = (B₁, B₂₁, …, B_2K)⁻¹(v, h₁, …, h_K). Since

$$\mid \widehat{\theta }-{\theta }_0\mid +\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}=\underset{(v,h_1,\dots ,h_K)\in \mathcal{V}\times {\mathcal{Q}}^K}{sup}\left|v^T(\widehat{\theta }-{\theta }_0)+\sum _{k=1}^K{\int }_0^{\tau }{h}_k(t)d({\widehat{\mathrm{\Lambda }}}_k(t)-{\mathrm{\Lambda }}_{0k}(t))\right|,$$

we have

$$n^{1/2}\left\{\mid \widehat{\theta }-{\theta }_0\mid +\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right\}=O_p(1)+o\left(n^{1/2}\mid \widehat{\theta }-{\theta }_0\mid +n^{1/2}\sum _{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\right).$$

Thus, $n^{1/2}\{\mid \widehat{\theta }-{\theta }_0\mid +{\sum }_{k=1}^K{\left|\right|{\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k}\left|\right|}_{V[0,\tau ]}\}=O_p(1)$. Consequently,

$$n^{1/2}\left\{v^T(\widehat{\theta }-{\theta }_0)+\sum _{k=1}^K{\int }_0^{\tau }{h}_k(t)d({\widehat{\mathrm{\Lambda }}}_k(t)-{\mathrm{\Lambda }}_{0k}(t))\right\}=-n^{1/2}({\mathcal{P}}_n-\mathcal{P})\left[{\stackrel{\sim }{v}}^T{\stackrel{.}{\mathcal{L}}}_{\theta }({\theta }_0,{\mathcal{A}}_0)+\sum _{k=1}^K{\stackrel{.}{\mathcal{L}}}_k({\theta }_0,{\mathcal{A}}_0)\left[\int {\stackrel{\sim }{h}}_{k}d{\mathrm{\Lambda }}_{0k}\right]\right]+o_p(1).$$

We have proved that n^1/2(θ̂− θ₀, − ) converges weakly to a Gaussian process in l^∞( × ). By choosing h_k = 0 for k = 1, …, K, we see that v^Tθ̂ is an asymptotically linear estimator of v^T θ₀ with influence function ${\stackrel{\sim }{v}}^T{\stackrel{.}{\mathcal{L}}}_{\theta }({\theta }_0,{\mathcal{A}}_0)+{\sum }_{k=1}^K{\stackrel{.}{\mathcal{L}}}_k({\theta }_0,{\mathcal{A}}_0)[\int {\stackrel{\sim }{h}}_{k}d{\mathrm{\Lambda }}_{0k}]$. Since the influence function lies in the space spanned by the score functions, θ̂ is an efficient estimator for θ₀.

It remains to verify that η_0k(t; θ₀, ) < 0. Under (C6), $\mathcal{P}\left[{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[H_k]/\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)\right]={\int }_0^{\tau }{\eta }_{0k}(s;{\theta }_0,{\mathcal{A}}_0)d{H}_k(s)$. The choice of H_k(s) = I(s ≥ t) yields [Ψ̇_k(;θ₀,)[I(· ≥ t)]/Ψ(;θ₀,)] = η_0k(t;θ₀,). On the other hand, the score function along the path Λ_0k + εI(· ≥ t) with the other parameters fixed at their true values has zero expectation. We expand this expectation to obtain

$$\mathcal{P}\left[\frac{{\stackrel{.}{\mathrm{\Psi }}}_k({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)[I(·\ge t)]}{\mathrm{\Psi }({\mathcal{O}}_i;{\theta }_0,{\mathcal{A}}_0)}\right]=-{\lambda }_k^{-1}(t)dE[I(R_{ik·}(t)>0)N_{ik·}^{\ast }(t)]/dt<0.$$

Thus, η_0k(t; θ₀, ) < 0.

8. Information Matrix

Theorem 2 implies that the functional parameter can be estimated at the same rate as the Euclidean parameter θ. Thus, we may treat (1) as a parametric log-likelihood with θ and the jump sizes of Λ_k, k = 1, …, K, at the observed failure times as the parameters and estimate the asymptotic covariance matrix of the NPMLEs for these parameters by inverting the information matrix. This result is formally stated in Theorem 3. We impose an additional assumption.

(C8) There exists a neighborhood of (θ₀, ) such that for (θ, ) in this neighborhood, the first and second derivatives of log Ψ(; θ, ) with respect to θ and along the path Λ_k + εH_k with respect to ε satisfy the inequality in (C4).

For any v ∈ and h₁, …, h_K ∈ , we consider the vector ${(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T$, where h⃗_k is the vector consisting of the values of h_k(·) at the observed failure times. Let ℐ_n be the negative Hessian matrix of (1) with respect to θ̂ and the jump sizes of (Λ̂₁, …, Λ̂_K).

Theorem 3

Assume (C1)–(C8). Then ℐ_n is invertible for large n, and

$$\underset{v\in \mathcal{V},h_1,\dots ,h_K\in \mathcal{Q}}{sup}\left|n(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T){\mathcal{I}}_n^{-1}{(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T-\text{AVar}\left[n^{1/2}\left\{v^T(\widehat{\theta }-{\theta }_0)+\sum _{k=1}^K\int h_{k}d({\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k})\right\}\right]\right|\to 0$$

in probability, where AVar denotes the asymptotic variance.

Proof

The proof is similar to that of Theorem 3 in Parner (1998); see also van der Vaart (1998, pp. 419–424). First, (10) implies that, for any v ∈ and h₁, …, h_K ∈ ,

$$-\mathcal{P}\left(\left(\begin{array}{cccc}{\ddot{\mathcal{L}}}_{\theta \theta }& {\ddot{\mathcal{L}}}_{\theta 1}& \dots & {\ddot{\mathcal{L}}}_{\theta K}\\ ⋮& ⋮& \ddots & ⋮\\ {\ddot{\mathcal{L}}}_{K\theta }& {\mathcal{L}}_{K1}& \dots & {\mathcal{L}}_{KK}\end{array}\right)\left[\left(\begin{array}{c}v\\ \int h_{1}d{\mathrm{\Lambda }}_{01}\\ ⋮\\ \int h_{K}d{\mathrm{\Lambda }}_{0K}\end{array}\right),\left(\begin{array}{c}v\\ \int h_{1}d{\mathrm{\Lambda }}_{01}\\ ⋮\\ \int h_{K}d{\mathrm{\Lambda }}_{0K}\end{array}\right)\right]\right)=v^T{B}_1(v,h_1,\dots ,h_K)+\sum _{k=1}^K\int B_{2k}(v,h_1,\dots ,h_K)h_{k}d{\mathrm{\Lambda }}_{0k},$$ (11)

where ℒ̈ pertains to the second-order derivative of the log-likelihood function.

On the right-hand side of (10), we replace by to obtain two new linear operators B_n1 and B_n2k. It is easy to show that B_n1 and B_n2k converge uniformly to B₁ and B_2k, respectively. Under (C8), the results of Lemma 1 apply to the second-order derivatives ℒ̈ and the operators (B₁, B₂₁, …, B_2K). By replacing θ₀, Λ_0k and on both sides of (11) with θ̂, Λ̂_0k and , we obtain

$$(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T){\mathcal{I}}_n{(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T=v^T{B}_{n1}(\stackrel{\sim }{v},{\stackrel{\sim }{h}}_1,\dots ,{\stackrel{\sim }{h}}_K)+\sum _{k=1}^K\int B_{n2k}(\stackrel{\sim }{v},{\stackrel{\sim }{h}}_1,\dots ,{\stackrel{\sim }{h}}_K)h_{k}d{\stackrel{\sim }{\mathrm{\Lambda }}}_k+o_p(1).$$

According to the proof of Theorem 2, (B₁, B₂₁, …, B_2K) is invertible, and so is (B_n1, …, B_n2k) for large n. Note that $v^T{B}_{n1}(\stackrel{\sim }{v},{\stackrel{\sim }{h}}_1,\dots ,{\stackrel{\sim }{h}}_K)+{\sum }_{k=1}^K\int B_{n2k}(\stackrel{\sim }{v},{\stackrel{\sim }{h}}_1,\dots ,{\stackrel{\sim }{h}}_K)h_{k}d{\stackrel{\sim }{\mathrm{\Lambda }}}_k$ can be written as $(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)\times {\mathcal{B}}_n{(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T$ for some matrix ℬ_n. Therefore ℬ_n is invertible, and so is ℐ_n. Furthermore,

$$\underset{v\in \mathcal{V},h_1,\dots ,h_K\in \mathcal{Q}}{sup}\left|(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T){\mathcal{I}}_n{(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T-(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T){\mathcal{B}}_n{(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T\right|\to 0.$$

According to Theorem 2, the asymptotic variance of $n^{1/2}\left\{v^T(\widehat{\theta }-{\theta }_0)+{\sum }_{k=1}^K\int h_{k}d({\widehat{\mathrm{\Lambda }}}_k-{\mathrm{\Lambda }}_{0k})\right\}$ is

$$\mathcal{P}\left[{\left\{{\stackrel{.}{\mathcal{L}}}_{\theta }^T\stackrel{\sim }{v}+\sum _{k=1}^K{\stackrel{.}{\mathcal{L}}}_k\left[\int {\stackrel{\sim }{h}}_{k}d{\mathrm{\Lambda }}_{0k}\right]\right\}}^2\right]-\mathcal{P}\left\{\left(\begin{array}{cccc}{\ddot{\mathcal{L}}}_{\theta \theta }& {\ddot{\mathcal{L}}}_{\theta 1}& \dots & {\ddot{\mathcal{L}}}_{\theta K}\\ ⋮& ⋮& \ddots & ⋮\\ {\ddot{\mathcal{L}}}_{K\theta }& {\mathcal{L}}_{K1}& \dots & {\mathcal{L}}_{KK}\end{array}\right)\left[\left(\begin{array}{c}\stackrel{\sim }{v}\\ \int {\stackrel{\sim }{h}}_{1}d{\mathrm{\Lambda }}_{01}\\ ⋮\\ \int {\stackrel{\sim }{h}}_{K}d{\mathrm{\Lambda }}_{0K}\end{array}\right),\left(\begin{array}{c}\stackrel{\sim }{v}\\ \int {\stackrel{\sim }{h}}_{1}d{\mathrm{\Lambda }}_{01}\\ ⋮\\ \int {\stackrel{\sim }{h}}_{K}d{\mathrm{\Lambda }}_{0K}\end{array}\right)\right]\right\},$$

where (ṽ, h̃₁, …, h̃_K) is (B₁, B₂₁, …, B_2K)⁻¹(v, h₁, …, h_K ), which can be approximated by (B_n1, B_n21, …, B_n2K)⁻¹(v, h₁, …, h_K). Hence, the asymptotic variance can be approximated uniformly in v and h_k’s by its empirical counterpart $(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T){\mathcal{B}}_n^{-1}{\mathcal{I}}_n{\mathcal{B}}_n^{-1}{({\stackrel{\sim }{v}}^T,{\overrightarrow{\stackrel{\sim }{h}}}_1^T,\dots ,{\overrightarrow{\stackrel{\sim }{h}}}_K^T)}^T$, which is further approximately by $(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T){\mathcal{I}}_n^{-1}{(v^T,{\overrightarrow{h}}_1^T,\dots ,{\overrightarrow{h}}_K^T)}^T$.

9. Profile Likelihood

Theorem 4

Let pl_n(θ) be the profile log-likelihood function for θ, and assume (C1)–(C8). For any ε_n = O_p(n^−1/2) and any vector v,

$$-\frac{p{l}_n(\widehat{\theta }+{\epsilon }_{n}v)-2{pl}_n(\widehat{\theta })+p{l}_n(\widehat{\theta }-{\epsilon }_n)}{n{\epsilon }_n^2}{\to }_p{v}^T{\mathrm{\sum }}^{-1}v,$$

where Σ is the limiting covariance matrix of n^1/2(θ̂ − θ₀). Furthermore, $2\left\{p{l}_n(\widehat{\theta })-p{l}_n({\theta }_0)\right\}{\to }_d{\chi }_d^2$.

Proof

We appeal to Theorem 1 of Murphy and van der Vaart (2000). Specifically, we construct the least favorable submodel for θ₀ and verify all the conditions in their Theorem 1. For notational simplicity, we assume that K = 1. It is straightforward to extend to K > 1.

It follows from the proof of Theorem 2 that

$${\int }_0^{\tau }{B}_2(0,h)h^{\ast }d{\mathrm{\Lambda }}_0=-E\left[{\ddot{\mathcal{L}}}_{\mathrm{\Lambda }\mathrm{\Lambda }}\left[\int h^{\ast }d{\mathrm{\Lambda }}_0,\int hd{\mathrm{\Lambda }}_0\right]\right],$$

where B₂ stands for the operator (B₂₁, …, B_2K), and ℒ̈_ΛΛ[H₁, H₂] denotes the second-order derivative of ℒ(θ, A) with respect to Λ along the bi-directions H₁ and H₂. On the other hand,

$$E\left[{\stackrel{.}{\mathcal{L}}}_{\mathrm{\Lambda }}\left[\int h^{\ast }d{\mathrm{\Lambda }}_0\right]{\stackrel{.}{\mathcal{L}}}_{\theta }\right]=-{\int }_0^{\tau }{h}^{\ast }(s){\stackrel{.}{\mathcal{L}}}_{\mathrm{\Lambda }}^{\ast }{\stackrel{.}{\mathcal{L}}}_{\theta }d{\mathrm{\Lambda }}_0(s),$$

where ${\mathcal{L}}_{\mathrm{\Lambda }}^{\ast }$ is the dual operator of ℒ_Λ in L₂[0, τ]. Thus, if we choose h such that $B_2(0,h)=-{\stackrel{.}{\mathcal{L}}}_{\mathrm{\Lambda }}^{\ast }{\stackrel{.}{\mathcal{L}}}_{\theta }$, then

$$E\left[{\stackrel{.}{\mathcal{L}}}_{\mathrm{\Lambda }}\left[\int h^{\ast }d{\mathrm{\Lambda }}_0\right]{\stackrel{.}{\mathcal{L}}}_{\theta }\right]=-E\left[{\ddot{\mathcal{L}}}_{\mathrm{\Lambda }\mathrm{\Lambda }}\left[\int h^{\ast }d{\mathrm{\Lambda }}_0,\int hd{\mathrm{\Lambda }}_0\right]\right].$$

By definition, ∫hdΛ₀ is the least favorable direction for θ₀ and ℒ̇_θ − ℒ̇_Λ [∫hd Λ₀] is the efficient score function. Such an h exists since B₂(0, ·) is invertible. In addition, h ∈ BV [0, τ]. Hence, we can construct the least favorable submodel at (θ, Λ) by ε ↦ (ε, Λ_ε) with dΛ_ε (θ, Λ) = {1 + (ε − θ) · h} dΛ. Clearly, Λ_θ (θ, Λ) = Λ and

$${\frac{\partial \mathcal{L}(\epsilon ,{\mathrm{\Lambda }}_{\epsilon })}{\partial \epsilon }|}_{\epsilon ={\theta }_0,\theta ={\theta }_0,\mathrm{\Lambda }={\mathrm{\Lambda }}_0}={\stackrel{.}{\mathcal{L}}}_{\theta }-{\stackrel{.}{\mathcal{L}}}_{\mathrm{\Lambda }}\left[\int hd{\mathrm{\Lambda }}_0\right].$$

If θ̃ →_p θ₀ and Λ̂_θ̃ maximizes the objective function with θ̂ replaced by θ̃, we can use the arguments in the proof of Theorem 1 to show that Λ̂_θ̃ is consistent. In the likelihood equation for Λ̂_θ̃, we can use the arguments for the linearization of (7) to show that, uniformly in h ∈ ,

$$o_p(1)+n^{1/2}({\mathcal{P}}_n-\mathcal{P})\left[{\stackrel{.}{\mathcal{L}}}_{\mathrm{\Lambda }}({\theta }_0,{\mathrm{\Lambda }}_0)\left[\int hd{\mathrm{\Lambda }}_0\right]\right]=-n^{1/2}{\int }_0^{\tau }{B}_2(0,h)d({\widehat{\mathrm{\Lambda }}}_{\stackrel{\sim }{\theta }}-{\mathrm{\Lambda }}_0)+O_p(n^{1/2}\mid \stackrel{\sim }{\theta }-{\theta }_0\mid )+o_p(n^{1/2}{\left|\right|{\widehat{\mathrm{\Lambda }}}_{\stackrel{\sim }{\theta }}-{\mathrm{\Lambda }}_0\left|\right|}_{V[0,\tau ]}).$$

The arguments for proving the invertibility of (B₁, B₂) show that h ↦ B₂(0, h) is invertible. Thus,

$${\left|\right|{\widehat{\mathrm{\Lambda }}}_{\stackrel{\sim }{\theta }}-{\mathrm{\Lambda }}_0\left|\right|}_{V[0,\tau ]}=O_p(\mid \stackrel{\sim }{\theta }-{\theta }_0\mid +n^{1/2}).$$

By condition (C6), we obtain the no-bias condition, i.e.,

$$E\left[{\frac{\partial \mathcal{L}(\epsilon ,{\mathrm{\Lambda }}_{\epsilon })}{\partial \epsilon }|}_{\epsilon ={\theta }_0,\theta =\stackrel{\sim }{\theta },\mathrm{\Lambda }={\widehat{\mathrm{\Lambda }}}_{\stackrel{\sim }{\theta }}}\right]=O_p(\mid \stackrel{\sim }{\theta }-{\theta }_0\mid +n^{-1/2}).$$

We have verified conditions (8)–(11) of Murphy and van der Vaart (2000).

Condition (C4), together with Lemma 1, implies that the class

$$\left\{\frac{\partial \mathcal{L}(\epsilon ,{\mathrm{\Lambda }}_{\epsilon })}{\partial \epsilon }:\mid \epsilon -{\theta }_0\mid <{\delta }_0,(\theta ,\mathrm{\Lambda })\in {\mathcal{N}}_0\right\}$$

is P-Donsker and that the functions in the class are continuous at (θ₀, Λ₀) almost surely, while condition (C8) implies that the class

$$\left\{\frac{{\partial }^2\mathcal{L}(\epsilon ,{\mathrm{\Lambda }}_{\epsilon })}{\partial {\epsilon }^2}:\mid \epsilon -{\theta }_0\mid <{\delta }_0,(\theta ,\mathrm{\Lambda })\in {\mathcal{N}}_0\right\}$$

is P-Glivenko-Cantelli and is bounded in L₂(P). Therefore, all the conditions in Murphy and van der Vaart (2000) hold, so that the desired results follows from their Theorem 1.

10. Applications

In this section, we apply the general results to the problems described in Section 2. We identify a set of conditions for each problem under which regularity conditions (C1)–(C8) are satisfied so that the desired asymptotic properties hold. These applications not only provide the theoretical justifications for the work of Zeng and Lin (2007), but also illustrate how the general theory can be applied to specific problems.

10.1. Transformation Models With Random Effects for Dependent Failure Times

We assume the following conditions.

(D1) The parameter value ${({\beta }_0^T,{\gamma }_0^T)}^T$ belongs to the interior of a compact set Θ in R^d, and ${\mathrm{\Lambda }}_{0k}^{\prime }(t)>0$ for all t ∈ [0, τ], k = 1, …, K.

(D2) With probability one, Z_ikl(·) and Z̃_ikl(·) are in BV [0, τ] and are left-continuous with bounded left-and right-derivatives in [0, τ].

(D3) With probability one, P (C_ikl ≥ τ|Z_ikl) > δ₀ > 0 for some constant δ₀.

(D4) With probability one, n_ik is bounded by some integer n₀. In addition, E[N_ik·(τ)] < ∞.

(D5) For k = 1, …, K, G_k(x) is four-times differentiable such that G_k(0) = 0, $G_k^{\prime }(x)>0$, and for any integer m ≥0 and any sequence 0 < x₁ < … < x_m ≤ y,

$$\prod _{l=1}^m\{(1+x_l)G_k^{\prime }(x_l)\}exp\{-G_k(y)\}\le {\mu }_{0k}^m{(1+y)}^{-{\kappa }_{0k}}$$

for some constants μ_0k and κ_0k > 0. In addition, there exists a constant ρ_0k such that

$$\underset{x}{sup}\left\{\frac{\mid G_k^{″}(x)\mid +\mid G^{(3)}(x)\mid +\mid G^{(4)}(x)\mid }{G^{\prime }(x){(1+x)}^{{\rho }_{0k}}}\right\}<\infty .$$

(D6) For any constant a₁ > 0,

$$\underset{\gamma }{sup}E\left[{\int }_{b}exp\{a_1(N_{ik·}^{\ast }(\tau )+1)\mid b\mid \}f(b;\gamma )db\right]<\infty ,$$

and there exists a constant a₂ > 0 such that for any γ,

$$\left|\frac{{\stackrel{.}{f}}_{\gamma }(b;\gamma )}{f(b;\gamma )}\right|+\left|\frac{{\ddot{f}}_{\gamma }(b;\gamma )}{f(b;\gamma )}\right|+\left|\frac{f_{\gamma }^{(3)}(b;\gamma )}{f(b;\gamma )}\right|\le O(1)exp\{a_2(1+\mid b\mid )\}.$$

(D7) Consider two types of events: k ∈ indicates that event k is recurrent and k ∈ indicates that event k is survival time. For k ∈ ∪ , if there exist c_k(t) and v such that with probability 1, c_k(t) + v^TZ_ikl(t) = 0 for k ∈ and c_k(0) + v^TZ_ikl(0) = 0 for k ∈ , then v = 0.

(D8) If there exist constants α_k and α_0k such that for any subset L_k ⊂ {1, …, n_ik} and for any ω_kl and t_kl,

$$\begin{array}{l}{\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{b}^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t_{kl})\}\prod _{k\in {\mathcal{K}}_2}\prod _{l\in L_k}exp\{{\alpha }_k+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(0)\}f(b;\mathit{\gamma })db\\ ={\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{b}^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t_{kl})\}\prod _{k\in {\mathcal{K}}_2}\prod _{l\in L_k}exp\{{\alpha }_{0k}+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(0)\}f(b;{\mathit{\gamma }}_0)db,\end{array}$$

then γ = γ₀. In addition, if for k ∈ and for any t,

$${\int }_{b}exp\left\{-G_k({\int }_0^t{e}^{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{\mathrm{\Lambda }}_1(s))\right\}f(b;{\mathit{\gamma }}_0)db={\int }_{b}exp\left\{-G_k({\int }_0^t{e}^{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{\mathrm{\Lambda }}_2(s))\right\}f(b;{\mathit{\gamma }}_0)db,$$

then Λ₁ = Λ₂. Furthermore, if for some vector v and constant α_k,

$$I(k\in {\mathcal{K}}_1){\int }_b{e}^{2b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(0)}{f}^{\prime }{(b;{\gamma }_0)}^T\mathit{vdb}+(k\in {\mathcal{K}}_2){\int }_b{e}^{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(0)}({\alpha }_{k}f(b;{\gamma }_0)-f^{\prime }{(b;{\gamma }_0)}^{T}v)db=0,$$

then v = 0.

(D1)–(D4) are standard conditions for this type of problem. We show that (D5) holds for all commonly used transformations. We first consider the class of logarithmic transformations G(x) = ρ log(1 + rx) (ρ > 0, r > 0). Clearly,

$$\begin{array}{l}\prod _{k=1}^m\{(1+x_k)G^{\prime }(y)\}exp\{-G(y)\}\le \prod _{k=1}^m\left\{\frac{\rho r(1+x_k)}{1+r{x}_k}\right\}{(1+ry)}^{-\rho }\\ \le {\{\rho r(1+1/r)\}}^{m}min{(1,r)}^{-\rho }{(1+y)}^{-\rho }.\end{array}$$

Thus, in (D5), we can set μ₀ to ρr(1 + 1/r) min(1, r)^−ρ and κ₀ to ρ. We can verify the polynomial bounds for G″(x)/G(x), G⁽³⁾(x)/G(x) and G⁽⁴⁾(x)/G(x) by direct calculations. We next consider the class of Box-Cox transformations G(x) = {(1 + x)^ρ − 1}/ρ. Clearly,

$$\begin{array}{l}\prod _{k=1}^m\{(1+x_k)G^{\prime }(x_k)\}exp\{-G(y)\}\le \prod _{k=1}^m{(1+x_k)}^{\rho }exp[-\{{(1+y)}^{\rho }-1\}/\rho ]\\ \le {(1+y)}^{m\rho }exp\{-{(1+y)}^{\rho }/2\rho \}exp\{-{(1+y)}^{\rho }/2\rho \}exp(1/\rho )\\ \le {\{4\rho +exp(1/\rho )\}}^m{(1+y)}^{-\rho }.\end{array}$$

Thus, we can set μ₀ to 4ρ + exp(1/ρ) and κ₀ to ρ. The polynomial bounds for G″ (x)/G(x), G⁽³⁾(x)/G(x) and G⁽⁴⁾(x)/G(x) hold naturally. Finally, we consider the linear transformation model: H(T) = β ^T Z + ε, where ε is standard normal. In this case, G(x) = − log{1 − Φ(log x)}, where Φ is the standard normal distribution function. We claim that there exists a constant ν₀ > 0 such that φ (x) − ν₀{1 − Φ(x)}(1 + |x|). If x < 0, then φ(x) ≤ (2π)^−1/2 ≤ 2(2π)^−1/2 {1 − Φ(x)}(1 + |x|). If x ≥ 0,

$$\underset{x\to 0}{lim}\frac{\phi (x)}{\{1-\mathrm{\Phi }(x)\}(1+x)}=2{(2\pi )}^{-1/2}.$$

By the L’Hospital rule,

$$\begin{array}{c}\underset{x\to \infty }{lim}\frac{1-\mathrm{\Phi }(x)}{\phi (x)}=\underset{x\to \infty }{lim}\frac{\phi (x)}{\phi (x)x}=0,\\ \underset{x\to \infty }{lim}\frac{\phi (x)}{\{1-\mathrm{\Phi }(x)\}(1+x)}=\underset{x\to \infty }{lim}\frac{-\phi (x)x}{-\phi (x)(1+x)+\{1-\mathrm{\Phi }(x)\}}=\underset{x\to \infty }{lim}\frac{1}{(1+x)/x-\{1-\mathrm{\Phi }(x)\}/x\phi (x)}=1.\end{array}$$

Therefore, φ(x)/[{1 − Φ(x)}(1 + x)] is bounded for x ≥ 0. Without loss of generality, assume that y > 1. Clearly,

$$\prod _{k=1}^m\{(1+x_k)G^{\prime }(x_k)\}exp\{-G(y)\}=\prod _{k=1}^m\left\{\frac{(1+x_k)\phi (log(x_k))/x_k}{1-\mathrm{\Phi }(log(x_k))}\right\}\{1-\mathrm{\Phi }(logy)\}.$$

Since (1 + x) φ(log(x))/[x{1 − Φ(log x)}] is bounded when x is close to zero and it is bounded by a multiplier of (1 + log x) when x is close to ∞, (1 + x)φ(log(x))/x{1 − Φ(log x)} ≤ ν₀₁ + ν₀₂ log(1 + x) for two constants ν₀₁ and ν₀₂. Therefore,

$$\prod _{k=1}^m\{(1+x_k)G^{\prime }(x_k)\}exp\{-G(y)\}\le {\{{\nu }_{01}+{\nu }_{02}log(1+y)\}}^m\{1-\mathrm{\Phi }(logy)\}.$$

Since 1 − Φ(x) ≤ 2^1/2 exp(−x²/4) when x > 0, the above expression is bounded by

$$\begin{array}{l}{2}^{1/2}{\{{\nu }_{01}+{\nu }_{02}log(1+y)\}}^{m}exp\{-{(logy)}^2/4\}\\ \le {\nu }_{03}{\{{\nu }_{01}+{\nu }_{02}log(1+y)\}}^{m}exp\{-{\nu }_{04}{(log(1+y))}^2\}\\ \le {\nu }_{05}^m{(1+y)}^{-{\nu }_{04}/2},\end{array}$$

where all the ν’s are positive constants. The polynomial bounds for G″ (x)/G(x), G⁽³⁾(x)/G(x) and G⁽⁴⁾(x)/G(x) follow from the fact that φ(x)={1 − Φ (x)} ≤ O(1 + |x|).

Condition (D6) pertains to the tail property of the density function for the random effects f(b; γ). For survival data, $N_{ik·}^{\ast }(\tau )\le 1$, so that the first half of condition (D6) is tantamount to that the moment generating function of b exists everywhere. This condition holds naturally when b has a compact support or a Gaussian density tail. The second half of condition (D6) clearly holds for Gaussian density functions.

(D7) and (D8) are sufficient conditions to ensure parameter identifiability and non-singularity of the Fisher information matrix. In most applications, these conditions are tantamount to the linear independence of covariates and the unique parametrization of the random-effects distribution. Specifically, if Z̃_ikl is time-independent, then the second condition in (D8) is not necessary; if Z̃_ikl does not depend on k and l, and b has a normal distribution, then the other two conditions in (D8) hold as well provided that Z̃_ikl is linearly independent with positive probability; if Z̃_ikl is time-independent and is non-empty (i.e., at least one event is recurrent), then (D8) can be replaced by the linear independence of Z̃_ikl for some k ∈ and the unique parametrization of f (b; γ).

We wish to show that (D1)–(D8) imply (C1)–(C8), so that the desired asymptotic properties hold. Conditions (C1) and (C2) follow naturally from (D1)–(D4). To verify (C3), we note that

$$\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathcal{A})={\int }_b\prod _{k=l}^K\prod _{l=1}^{n_{ik}}{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)f(b;\gamma )db,$$

where

$${\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)=\prod _{t\le \tau }{\left\{R_{\mathit{ikl}}(t)e^{{\beta }^T{Z}_{\mathit{ikl}}(t)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t)}{G}_k^{\prime }(q_{\mathit{ikl}}(t))\right\}}^{d{N}_{\mathit{ikl}}^{\ast }(t)}exp\{-G_k(q_{\mathit{ikl}}(\tau ))\},$$

and $q_{\mathit{ikl}}(t)={\int }_0^t{R}_{\mathit{ikl}}(s)exp\{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)\}d{\mathrm{\Lambda }}_k(s)$.

If ||Λ_k||_{V [0,τ]} are bounded, then ${\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\ge exp\{O(1)N_{\mathit{ikl}}^{\ast }(\tau )\}I(\mid b\mid \le B_0)$ for any fixed constant B₀ such that P(|b| ≤ B₀) > 0. Thus, Ψ(; θ, ) is bounded from below by $exp\{O(1)N_{\mathit{ikl}}^{\ast }(\tau )\}$, so that the second half of (C3) holds. It follows from (D5) that

$${\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\le O(1)\prod _{t\le \tau }{\left\{R_{\mathit{ikl}}(t)e^{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t)}\right\}}^{d{N}_{\mathit{ikl}}^{\ast }(t)}{\mu }_{0k}^{N_{\mathit{ikl}}^{\ast }(\tau )}\prod _{t\le \tau }{\{1+q_{\mathit{ikl}}(t)\}}^{-d{N}_{\mathit{ikl}}^{\ast }(t)}{\{1+q_{\mathit{ikl}}(\tau )\}}^{-{\kappa }_{0k}}.$$

Since exp{β^T Z_ikl(s) + b^T Z̃_ikl(s)} ≥ exp{−O(1 + |b|)}, we have $1+q_{\mathit{ikl}}(t)\ge e^{-O(1+\mid b\mid )}\left\{1+\int R_{ik·}(s)d{\mathrm{\Lambda }}_k(s)\right\}$, so that

$${\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\le O(1){\mu }_{0k}^{N_{\mathit{ikl}}^{\ast }(\tau )}{e}^{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))\mid b\mid }\prod _{t\le \tau }{\left\{1+{\int }_0^t{R}_{ik·}(s)d{\mathrm{\Lambda }}_k(s)\right\}}^{-d{N}_{\mathit{ikl}}^{\ast }(t)}{\left\{1+{\int }_0^{\tau }{R}_{\mathit{ikl}}(s)d{\mathrm{\Lambda }}_k(s)\right\}}^{-{\kappa }_{0k}}.$$

Thus, the first half of (C3) holds as well.

We now verify (C4). Under (D5),

$$\begin{array}{l}\mid {\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\mid \le exp\{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))\mid b\mid \},\\ \left|\frac{\partial }{\partial \beta }{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\right|=\left|{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\left[\left\{\int R_{\mathit{ikl}}(t)Z_{\mathit{ikl}}(t)d{N}_{\mathit{ikl}}^{\ast }(t)+\int R_{\mathit{ikl}}(t)\frac{G_k^{″}(q_{\mathit{ikl}}(t)){\int }_0^t{R}_{\mathit{ikl}}(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}{Z}_{\mathit{ikl}}(s)d{\mathrm{\Lambda }}_k(s)}{G_k^{\prime }(q_{\mathit{ikl}}(t))}d{N}_{\mathit{ikl}}^{\ast }(t)\right\}-G_k^{\prime }(q_{\mathit{ikl}}(\tau ))\left\{{\int }_0^{\tau }{R}_{\mathit{ikl}}(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}{Z}_{\mathit{ikl}}(s)d{\mathrm{\Lambda }}_k(s)\right\}\right]\right|\\ \le exp\{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))(1+\mid b\mid )\},\end{array}$$

$$\begin{array}{l}\left|\frac{\partial }{\partial {\mathrm{\Lambda }}_k}{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)[H_k]\right|=\left|{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)\times \left[\left\{\int R_{\mathit{ikl}}(t)\frac{G_k^{″}(q_{\mathit{ikl}}(t)){\int }_0^t{R}_{\mathit{ikl}}(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{H}_k(s)}{G_k^{\prime }(q_{\mathit{ikl}}(t))}d{N}_{\mathit{ikl}}^{\ast }(t)\right\}-G_k^{\prime }(q_{\mathit{ikl}}(\tau ))\left\{{\int }_0^{\tau }{R}_{\mathit{ikl}}(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{H}_k(s)\right\}\right]\right|\\ \le exp\{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))(1+\mid b\mid )\}.\end{array}$$

Thus, it follows from the Mean-Value Theorem that

$$\begin{array}{l}\left|{\mathrm{\Omega }}_{\mathit{ikl}}(b;{\beta }^{(1)},{\mathrm{\Lambda }}_k)-{\mathrm{\Omega }}_{\mathit{ikl}}(b;{\beta }^{(2)},{\mathrm{\Lambda }}_k)\right|=\left|\frac{\partial }{\partial \beta }{\mathrm{\Omega }}_{\mathit{ikl}}(b;{\beta }^{\ast },{\mathrm{\Lambda }}_k)\right|\mid {\beta }^{(1)}-{\beta }^{(2)}\mid \\ \le exp\{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))\mid b\mid \}\mid {\beta }^{(1)}-{\beta }^{(2)}\mid ,\end{array}$$

$$\begin{array}{l}\mid {\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k^{(1)})-{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k^{(2)})\mid =\left|\frac{\partial }{\partial {\mathrm{\Lambda }}_k}{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k^{\ast })[{\mathrm{\Lambda }}_k^{(1)}-{\mathrm{\Lambda }}_k^{(2)}]\right|\\ \le exp\{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))\mid b\mid \}\times \left\{\int R_{\mathit{ikl}}(t)\left|{\int }_0^t{e}^{{\beta }^{\ast T}{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d({\mathrm{\Lambda }}_k^{(1)}-{\mathrm{\Lambda }}_k^{(2)})(s)\right|d{N}_{\mathit{ikl}}^{\ast }(t)+\left|{\int }_0^{\tau }{R}_{\mathit{ikl}}(t)e^{{\beta }^{\ast T}{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d({\mathrm{\Lambda }}_k^{(1)}-{\mathrm{\Lambda }}_k^{(2)})(s)\right|\right\}\\ \le exp\{O(1+N_{\mathit{ikl}}^{\ast }(\tau ))(1+\mid b\mid )\}\times \left\{\int R_{\mathit{ikl}}(t)\mid {\mathrm{\Lambda }}_k^{(1)}(t)-{\mathrm{\Lambda }}_k^{(2)}(t)\mid d{N}_{\mathit{ikl}}^{\ast }(t)+{\int }_0^{\tau }\mid {\mathrm{\Lambda }}_k^{(1)}(s)-{\mathrm{\Lambda }}_k^{(2)}(s)\mid ds\right\},\end{array}$$

where the last inequality follows from integration by parts and the fact that Z_ikl(t) and Z̃_ikl(t) have bounded variations. It then follows from (D6) that |Ψ (; θ⁽¹⁾, ) − Ψ (; θ⁽²⁾, )| is bounded by the right-hand side of the inequality in (C4). By the same arguments, we can verify the bounds for the other three terms in (C4).

To verify (C6), we calculate that

$${\eta }_{0k}(s;\theta ,\mathcal{A})=E\left[{\int }_b\frac{{\prod }_{m=1}^K{\prod }_{l=1}^{n_{im}}{\mathrm{\Omega }}_{\mathit{iml}}(b;\beta ,{\mathrm{\Lambda }}_m)f(b;\gamma )}{{\int }_b{\prod }_{m=1}^K{\prod }_{l=1}^{n_{im}}{\mathrm{\Omega }}_{\mathit{iml}}(b;\beta ,{\mathrm{\Lambda }}_m)f(b;\gamma )db}\times \left\{{\int }_{t\ge s}\frac{G_k^{″}(q_{\mathit{ikl}}(t))}{G_k^{\prime }(q_{\mathit{ikl}}(t))}d{N}_{\mathit{ikl}}^{\ast }(t)-G_k^{\prime }(q_{\mathit{ikl}}(\tau ))\right\}{R}_{\mathit{ikl}}(s)e^{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}db\right].$$

For (θ, ) in a neighborhood of (θ₀, ),

$$\left|{\eta }_{0k}(s;\theta ,\mathcal{A})-{\eta }_{0k}(s;{\theta }_0,{\mathcal{A}}_0)-\frac{\partial }{\partial \theta }{\eta }_{0k}{(s;{\theta }_0,{\mathcal{A}}_0)}^T(\theta -{\theta }_0)-\sum _{m=1}^K\frac{\partial {\eta }_{0k}}{\partial {\mathrm{\Lambda }}_m}(s;{\theta }_0,{\mathcal{A}}_0)[{\mathrm{\Lambda }}_m-{\mathrm{\Lambda }}_{0m}]\right|=0\left(\mid \theta -{\theta }_0\mid +\sum _{m=1}^K{\left|\right|{\mathrm{\Lambda }}_m-{\mathrm{\Lambda }}_{0m}\left|\right|}_{V[0,\tau ]}\right).$$

Thus, for the second equation in (C6), η_0km(s, t; θ₀, ) is obtained from the derivative of η_0k with respect to Λ_m along the direction Λ_m − Λ_0m, and η_0kθ is the derivative of η_0k with respect to θ. Likewise, we can obtain the first equation in (C6). It is straightforward to verify the Lipschitz continuity of η_0km.

The verification of (C8) is similar to that of (C4), relying on the explicit expressions of Ψ̈ _θθ (; θ, ) and the first and second derivatives of Ψ(; θ, + εℋ) with respect to ε.

It remains to verify the two identifiability conditions under (D7) and (D8). To verify (C5), suppose that (β, γ, Λ₁, …, Λ_k) yields the same likelihood as (β₀, γ₀, Λ₁₀, …, Λ_k0). That is,

$$\underset{b}{\int }\prod _{k=1}^K\prod _{l=1}^{n_{ik}}{\lambda }_k{(t)}^{d{N}_{\mathit{ikl}}^{\ast }(t)}{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)f(b;\gamma )db=\underset{b}{\int }\prod _{k=1}^K\prod _{l=1}^{n_{ik}}{\lambda }_{k0}{(t)}^{d{N}_{\mathit{ikl}}^{\ast }(t)}{\mathrm{\Omega }}_{\mathit{ikl}}(b;{\beta }_0,{\mathrm{\Lambda }}_{k0})f(b;{\gamma }_0)db.$$

We perform the following operations on both sides sequentially for k = 1, …, K and l = 1, …, n_ik.

(a) If the kth type of event pertains to survival time, for the lth subject of this type of event, the first equation is obtained with R_ikl(t) = 1 and $d{N}_{\mathit{ikl}}^{\ast }(t)=0$ for any t ≤ τ, i.e., the subject does not experience any event in [0, τ]. The second equation is obtained by integrating t from t_kl to τ on both sides under the scenario that R_ikl(t) = 1 and $N_{\mathit{ikl}}^{\ast }(t)$ has a jump at t, i.e, the subject experiences the event at time t_kl. We then take the difference between these two equations. In the resulting equation, the terms ${\lambda }_k{(t)}^{d{N}_{\mathit{ikl}}^{\ast }(t)}{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)$ and ${\lambda }_{k0}{(t)}^{d{N}_{\mathit{ikl}}^{\ast }(t)}{\mathrm{\Omega }}_{\mathit{ikl}}(b;{\beta }_0,{\mathrm{\Lambda }}_{k0})$ are replaced by $exp\{-G_k({\int }_0^{t_{kl}}exp\{{\beta }^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{ik}(s)\}d{\mathrm{\Lambda }}_k)\}$ and $exp\{-G_k({\int }_0^{t_{kl}}exp\{{\beta }_0^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{ik}(s)\}d{\mathrm{\Lambda }}_{k0})\}$, respectively.

(b) If the kth type of event is recurrent, for the lth subject of this type of event, we let R_ikl(t) = 1 and let $N_{\mathit{ikl}}^{\ast }(t)$ have jumps at s₁, s₂, …, s_m and $s_1^{\prime },\dots ,s_{m^{\prime }}^{\prime }$ for any arbitrary (m + m′) times in [0, τ]. We integrate s₁, …, s_m from 0 to t_kl and integrate $s_1^{\prime },\dots ,s_{m^{\prime }}^{\prime }$ from 0 to τ. In the obtained equation, ${\lambda }_k{(t)}^{d{N}_{\mathit{ikl}}^{\ast }(t)}{\mathrm{\Omega }}_{\mathit{ikl}}(b;\beta ,{\mathrm{\Lambda }}_k)$ is replaced by {G_k(q_ikl(t_kl))}^m {G_k(q_ikl(τ))}^m′ on both sides. Note that m and m′ are arbitrary. We then multiple both sides by {(iω_kl)^m/m!}/m′! and sum over m, m′ = 0, 1, … On both sides of the resulting equation, the terms associated with k and l are replaced by exp{iω_klG_k(q_ikl(t_kl))}.

After these sequential operations, we obtain

$${\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{G}_k(q_{\mathit{ikl}}(t_{kl}))\}\prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}exp\{-G_k(q_{\mathit{ikl}}(t_{kl}))\}f(b;\mathit{\gamma })db={\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{G}_k(q_{\mathit{ikl}}(t_{kl}))\}\prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}exp\{-G_k(q_{\mathit{ikl}0}(t_{kl}))\}f(b;{\mathit{\gamma }}_0)db.$$

For survival time, we can let any subject from the n_ik subjects have t_kl = 0, which results in

$${\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{G}_k(q_{\mathit{ikl}}(t_{kl}))\}\prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}\left[\frac{1}{{\xi }_{kl}}+exp\{-G_k(q_{\mathit{ikl}}(t_{kl}))\}\right]f(b;\mathit{\gamma })db={\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{G}_k(q_{\mathit{ikl}0}(t_{kl}))\}\prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}\left[\frac{1}{{\xi }_{kl}}+exp\{-G_k(q_{\mathit{ikl}0}(t_{kl}))\}\right]f(b;{\mathit{\gamma }}_0)db,$$

where ξ_kl is any positive variable.

The above expression implies that {G_k(q_ikl(t)), k ∈ } as a function of

$$b_1\sim \prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}\left[\frac{1}{{\xi }_{kl}}+exp\{-G_k(q_{\mathit{ikl}}(t_{kl}))\}\right]f(b;\mathit{\gamma })$$

has the same distribution as {G_k(q_ikl0(t)), k ∈ } as a function of

$$b_2\sim \prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}\left[\frac{1}{{\xi }_{kl}}+exp\{-G_k(q_{\mathit{ikl}0}(t_{kl}))\}\right]f(b;{\mathit{\gamma }}_0);$$

so this is true between {q_ikl(t)} and {q_ikl0(t)} because of the one-to-one mapping. Thus, the distributions of { $log{q}_{\mathit{ikl}}^{\prime }(t)$} and { $log{q}_{\mathit{ikl}0}^{\prime }(t)$} should also agree and they have the same expectation. Now let t_kl = 0 for k ∈ . Since E[b₁] = E[b₂] = 0, we obtain $log{\lambda }_k(t)+{\beta }^T{Z}_{\mathit{ikl}}(t)=log{\lambda }_{k0}(t)+{\beta }_0^T{Z}_{\mathit{ikl}}(t)$ for k ∈ . The above arguments also yield

$${\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t_{kl})\}\prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}\left[\frac{1}{{\xi }_{kl}}+exp\{-G_k(q_{\mathit{ikl}}(t_{kl}))\}\right]f(b;\mathit{\gamma })db={\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t_{kl})\}\prod _{k\in {\mathcal{K}}_2}\prod _{l=1}^{n_{ik}}\left[\frac{1}{{\xi }_{kl}}+exp\{-G_k(q_{\mathit{ikl}0}(t_{kl}))\}\right]f(b;{\mathit{\gamma }}_0)db.$$

We compare the coefficients of ξ_kl for k ∈ . This yields that for any subset L_k ⊂ {1, …, n_ik},

$${\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{b}^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t_{kl})\}\prod _{k\in {\mathcal{K}}_2}\prod _{l\in L_k}exp\{-G_k(q_{\mathit{ikl}}(t))\}f(b;\mathit{\gamma })db={\int }_b\prod _{k\in {\mathcal{K}}_1}\prod _{l=1}^{n_{ik}}exp\{\text{i}{\omega }_{kl}{b}^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(t_{kl})\}\prod _{k\in {\mathcal{K}}_2}\prod _{l\in L_k}exp\{-G_k(q_{\mathit{ikl}0}(t))\}f(b;{\mathit{\gamma }}_0)db.$$

We differentiate the above expression with respect to t_kl at 0 for k ∈ . It then follows from (D8) that log λ_k(0) − log λ_0k(0) + (β − β₀)^T Z_ikl(0) = 0 and γ = γ₀. Thus, (D7) implies that β = β₀ and λ_k(t) = λ_0k(t) for k ∈ . On the other hand, for any fixed k ∈ , we let t_k′l′ = 0 if k′ ≠ k or l′ ≠ l. Thus, ∫_b exp{− G_k(q_ikl(t_kl))}f (b; γ₀)db = ∫_b exp{− G_k(q_0ikl(t_kl))}f (b; γ₀)db. Therefore, Λ_k = Λ_0k for k ∈ according to (D8).

To verify (C7), we write v = (v_β, v_γ). We perform operations (a) and (b) on the score equation in (C7). The arguments used in proving the identifiability yield

$${\int }_b\left[\sum _{k\in {\mathcal{K}}_1}\sum _{l=1}^{n_{ik}}\text{i}{\omega }_{kl}{A}_{\mathit{ikl}}(t_{kl})G_k(q_{\mathit{ikl}0}(t_{kl}))-\sum _{k\in {\mathcal{K}}_2}\sum _{l\in L_k}{A}_{\mathit{ikl}}(t_{kl})+\frac{f^{\prime }{(b;{\gamma }_0)}^T{v}_{\gamma }}{f(b;{\gamma }_0)}\right]\times exp\left\{\sum _{k\in {\mathcal{K}}_1}\sum _{l=1}^{n_{\mathit{ikl}}}\text{i}{\omega }_{kl}{G}_k(q_{\mathit{ikl}0}(t_{kl}))-\sum _{k\in {\mathcal{K}}_2}\sum _{l\in L_k}{G}_k(q_{\mathit{ikl}0}(t_{kl}))\right\}f(b;{\gamma }_0)db=0,$$ (12)

where $A_{\mathit{ikl}}(t)={\int }_0^t(h_k(s)+Z_{\mathit{ikl}}{(s)}^T{v}_{\beta })e^{{\beta }_0^T{Z}_{\mathit{ikl}}(s)+b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(s)}d{\mathrm{\Lambda }}_{k0}(s)G_k^{\prime }(q_{\mathit{ikl}0}(t))$. We differentiate (12) with respect to t_kl twice at 0 for k ∈ . Comparison of the coefficients for ω_kl yields ∫_b e^{2bTZ̃_ikl(0)} f′(b; γ₀)^T v_γdb = 0. We also differentiate (12) with respect to t_kl at 0 for k ∈ . Thus, for each k ∈ and l = 1, …, n_ik, ${\int }_b(h_k(0)+Z_{\mathit{ikl}}{(0)}^T{v}_{\beta })e^{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(0)}f(b;{\gamma }_0)db=-G_k^{\prime }(0){\int }_b{e}^{b^T{\stackrel{\sim }{Z}}_{\mathit{ikl}}(0)}{f}^{\prime }{(b;{\gamma }_0)}^T{v}_{\gamma }db$. It then follows from (D8) that v_γ = 0. For fixed k₀ and l₀, with the fact of v_γ = 0, the score equation under operations (a) and (b), where in (a) we let $d{N}_{\mathit{ikl}}^{\ast }(t)=0$ for any t ≤ τ and in (b) we let m = 0 whenever k ≠ k₀ or l ≠ l₀, becomes a homogeneous integral equation for h_k0 (t) + Z_ik0l0 (t)^T v_β. The equation has a trivial solution, so h_k0 (t) + Z_ik0l0 (t)^T v_β= 0. Since k₀ and l₀ are arbitrary, (D7) implies that h_k = 0 and v_β= 0.

Remark 2

For survival time, (D5) is required to hold only for m = 0 and m = 1.

Remark 3

The above results do not apply directly to the proportional hazards model with gamma frailty because (D6) does not hold when b has a gamma distribution. It is mathematically convenient to handle this model because the marginal hazard function has an explicit form. The likelihood is a special case of ours with

$$\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathrm{\Lambda })=\prod _{j=1}^{n_i}\prod _{t\le \tau }{Y}_{ij}{(t;\beta )}^{d{N}_{ij}(t)}\prod _{t\le \tau }{\{1+\theta N_{i·}(u-)\}}^{d{N}_{i·}(t)}{\left\{1+\theta {\int }_0^{\tau }{Y}_{i·}(u;\beta )d\mathrm{\Lambda }(u)\right\}}^{-(1/\theta +N_{i·}(\tau ))}$$

in the notation of Parner (1998). Clearly, Ψ satisfies (C3) when θ > 0. The other conditions can be verified in the same manner as before.

Remark 4

Our theory does not cover the case in which the true parameter values lie on the boundary of Θ. It is delicate to deal with the boundary problem. One possible solution is to follow the idea of Parner (1998) by extending the definition of the likelihood function outside Θ and verifying (C2)–(C8) for the extended likelihood function.

Remark 5

We have assumed known transformations. We may allow G_k to belong to a parametric family of distributions, say G_k(·; φ), where φ is a parameter in a compact set. Then θ contains φ. Our results and proofs apply to this situation if (D5) holds uniformly in φ and the two identifiability conditions are satisfied.

10.2. Joint Models for Repeated Measures and Failure Times

For the (parametric) generalized linear mixed model, the likelihood can be viewed as a special case of that of Section 10.1 except that there is an additional parameter α in f(y|x; b). We assume that (D1)–(D8) hold but with (D6) replaced by the following condition.

(D6′) For any constant a₁ > 0,

$$\underset{\alpha ,\gamma }{sup}E\left[{\int }_{b}exp\{a_1(N_i^{\ast }(\tau )+1)\mid b\mid \}\prod _{j=1}^{n_i}f(Y_{ij}\mid X_{ij};b)f(b;\gamma )db\right]<\infty ,$$

and there exists a constant a₂ > 0 such that for any γ and α,

$$\sum _{k=1}^3\left|\frac{f_{\alpha }^{(k)}(Y_{ij}\mid X_{ij},b)}{f(Y_{ij}\mid X_{ij},b)}\right|+\left|\frac{f_{\gamma }^{(k)}(b;\gamma )}{f(b;\gamma )}\right|\le r_3({\mathcal{O}}_i)exp\{a_2(1+\mid b\mid )\}$$

almost surely, where r₃() is a random variable in L₂(P).

Under these conditions, the desired asymptotic properties follow from the arguments of Section 10.1.

Under the semiparametric linear transformation model for continuous repeated measures, the likelihood is in the form of that of Section 2.2 with K = 2 and n_i2 = n_i, where the time to the second type of failure is defined by Y_ij (assuming without loss of generality that Y_ij ≥ 0). Thus, if we regard Y_ij as a right-censored observation when it is greater than a very large value (i.e., the upper limit of detection), then the asymptotic results given in Section 10.1 hold. When such an upper limit does not exist, the estimator for Λ̃ can be unbounded when sample size goes to infinity. Then our proof of Theorem 1 does not apply.

10.3. Transformation Models for Counting Processes

We verify (C1)–(C8) under the following conditions.

(E1) The parameter value ${({\beta }_0^T,{\gamma }_0^T)}^T$ belongs to the interior of a compact set Θ in R^d, and ${\mathrm{\Lambda }}_0^{\prime }(t)>0$ for all t ∈ [0, τ].

(E2) With probability one, P (C ≥ τ|Z) > δ₀ > 0 for some constant δ₀.

(E3) Condition (D5) holds.

(E4) With probability one, Z(·) and Z̃ are in BV [0, τ] and are left-continuous with bounded left- and right-derivatives in [0, τ].

(E5) If γ^T Z̃ is equal to a constant with probability one, then γ = 0. In addition, if β^T Z(t) = c(t) for a deterministic function c(t) with probability one, then β = 0.

In this case,

$$\mathrm{\Psi }({\mathcal{O}}_i;\theta ,\mathrm{\Lambda })=\prod _{t\le \tau }{\left(R_i(t)e^{{\beta }^T{Z}_i(t)+{\gamma }^T{\stackrel{\sim }{Z}}_i}{\left\{1+{\int }_0^t{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T{\stackrel{\sim }{Z}}_{i-1}}}\times G^{\prime }\left[{\left\{1+{\int }_0^t{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T{\stackrel{\sim }{Z}}_i}}\right]\right)}^{d{N}_i^{\ast }(t)}exp\left(-G\left[{\left\{1+{\int }_0^{\tau }{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T{\stackrel{\sim }{Z}}_i}}\right]\right).$$

By (D5),

$$\begin{array}{l}\prod _{t\le \tau }{\left(R_i(t)e^{{\beta }^T{Z}_i(t)+{\gamma }^T{\stackrel{\sim }{Z}}_i}{\left\{1+{\int }_0^t{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T{\stackrel{\sim }{Z}}_{i-1}}}\times G^{\prime }\left[{\left\{1+{\int }_0^t{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T{\stackrel{\sim }{Z}}_i}}\right]\right)}^{d{N}_i^{\ast }(t)}exp\left(-G\left[{\left\{1+{\int }_0^{\tau }{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{e^{{\gamma }^T{\stackrel{\sim }{Z}}_i}}\right]\right)\\ \le {\mu }_1^{N_i^{\ast }(\tau )}\prod _{t\le \tau }{\left\{1+{\int }_0^t{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{-d{N}_i^{\ast }(t)}{\left\{1+{\int }_0^{\tau }{R}_i(s)e^{{\beta }^T{Z}_i(s)}d\mathrm{\Lambda }(s)\right\}}^{-\kappa e^{{\gamma }^T{\stackrel{\sim }{Z}}_i}}\end{array}$$

for some constant μ₁. Thus, (C3) follows from the boundedness of γ^T Z̃_i. We can verify the other conditions by using the arguments of Section 10.1.

To verify the first identifiability condition, we assume that $N_i^{\ast }(t)$ has jumps at x, x₁, …, x_m for some integer m. After integrating both sides of the equation in (C5) over x₁, …, x_m from 0 to τ and integrating x from x to τ, we obtain

$$\left(G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]-G\left[{\left\{1+{\int }_0^x{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]\right)\times {\left(G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]-G(1)\right)}^{m}exp\left(-G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]+G(1)\right)=\left(G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right]-G\left[{\left\{1+{\int }_0^x{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right]\right)\times {\left(G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right]-G(1)\right)}^{m}exp\left(-G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right]+G(1)\right).$$

Multiplying both sides of this equation by 1/m! and summing over m ≥ 0, we obtain

$$G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]-G\left[{\left\{1+{\int }_0^x{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]=G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right]-G\left[{\left\{1+{\int }_0^x{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right].$$

Setting $N_i^{\ast }(\tau )=0$ in the likelihood function yields

$$G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}\right]=G\left[{\left\{1+{\int }_0^{\tau }{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}\right].$$

Thus

$${\left\{1+{\int }_0^x{e}^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)\right\}}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}={\left\{1+{\int }_0^x{e}^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)\right\}}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}.$$

Then Λ^* (t) is absolutely continuous with respect to t. Differentiating both sides with respect to x and letting x = 0 yield λ^* (0) > 0. When x converges to zero, the left-hand side is ${[exp\{{\beta }_0^T{Z}_i(0)\}{\lambda }_0(0)x]}^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}+o(x^{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}})$ while the right-hand side is ${[exp\{{\beta }^{\ast T}{Z}_i(0)\}{\lambda }^{\ast }(0)x]}^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}}+o(x^{e^{{\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i}})$. Thus, ${\gamma }_0^T{\stackrel{\sim }{Z}}_i={\gamma }^{\ast T}{\stackrel{\sim }{Z}}_i$. By (E5), γ₀ = γ ^*. Furthermore, $e^{{\beta }_0^T{Z}_i(t)}d{\mathrm{\Lambda }}_0(t)/dt=e^{{\beta }^{\ast T}{Z}_i(t)}d{\mathrm{\Lambda }}^{\ast }(t)/dt$. It follows from (E5) that β₀ = β^* and Λ₀ = Λ^*.

To verify (C7), we assume that the score function along (β₀ + εh_β, γ₀ + εh_γ, dΛ₀ + εhdΛ₀) is zero. Equivalently, if we let $g_0(t)={\{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)\}}^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}$, then we obtain

$$0=\int h(t)R_i(t)d{N}_i^{\ast }(t)+\int R_i(t)\left\{h_{\beta }^T{Z}_i(t)+h_{\gamma }^T{\stackrel{\sim }{Z}}_i\right\}d{N}_i^{\ast }(t)+\int \frac{R_i(t)(e^{{\gamma }^T{\stackrel{\sim }{Z}}_i}-1)}{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)}\left[{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}\{h_{\beta }^T{Z}_i(s)+h(s)\}d{\mathrm{\Lambda }}_0(s)\right]d{N}_i^{\ast }(t)+\int R_i(t)h_{\gamma }^T{\stackrel{\sim }{Z}}_i{e}^{{\gamma }^T{\stackrel{\sim }{Z}}_i}log\left\{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)\right\}d{N}_i^{\ast }(t)+\int R_i(t)\frac{G^{″}(g_0(t))}{G^{\prime }(g_0(t))}{g}_0(t)h_{\gamma }^T{\stackrel{\sim }{Z}}_i{e}^{{\gamma }^T{\stackrel{\sim }{Z}}_i}log\left\{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)\right\}d{N}_i^{\ast }(t)+\int R_i(t)\frac{G^{″}(g_0(t))}{G^{\prime }(g_0(t))}{g}_0(t)\left[\frac{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}\left\{h_{\beta }^T{Z}_i(s)+h(s)\right\}d{\mathrm{\Lambda }}_0(s)}{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)}\right]d{N}_i^{\ast }(t)-G^{\prime }(g_0(\tau ))g_0(\tau )h_{\gamma }^T{\stackrel{\sim }{Z}}_i{e}^{{\gamma }^T{\stackrel{\sim }{Z}}_i}log\left\{1+{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)\right\}-G^{\prime }(g_0(\tau ))g_0(\tau )\frac{e^{{\gamma }_0^T{\stackrel{\sim }{Z}}_i}}{1+{\int }_0^T{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)}{\int }_0^{\tau }{e}^{{\beta }_0^T{Z}_i(s)}\{h_{\beta }^T{Z}_i(s)+h(s)\}d{\mathrm{\Lambda }}_0(s).$$

We multiply both sides by the likelihood function and let $N_i^{\ast }(t)$ have jumps at times t₁, t₂, …, t_m. We integrate t₁ from 0 to t and t_l, 1 < l ≤ m from 0 to τ. By multiplying the resulting equation by 1/(m − k)! and summing over m = 1, 2, …, we obtain

$$h_{\gamma }^T{\stackrel{\sim }{Z}}_{i}log\left\{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)\right\}+\frac{{\int }_0^t{e}^{{\beta }_0^T{\stackrel{\sim }{Z}}_i(s)}\{h_{\beta }^T{Z}_i(s)+h(s)\}d{\mathrm{\Lambda }}_0(s)}{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)}=0.$$

Differentiation with respect to t then yields

$$h_{\gamma }^T{\stackrel{\sim }{Z}}_i+\{h_{\beta }^T{Z}_i(t)+h(t)\}-\frac{{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}\left\{h_{\beta }^T{Z}_i(s)+h(s)\right\}d{\mathrm{\Lambda }}_0(s)}{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)}=0.$$

Combining the above two equations, we have

$$\{h_{\beta }^T{Z}_i(t)+h(t)\}-\frac{{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}\left\{h_{\beta }^T{Z}_i(s)+h(s)\right\}d{\mathrm{\Lambda }}_0(s)}{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)}\left[1+\frac{1}{log\{1+{\int }_0^t{e}^{{\beta }_0^T{Z}_i(s)}d{\mathrm{\Lambda }}_0(s)\}}\right]=0.$$

This is a homogeneous integral equation for $h_{\beta }^T{Z}_i(t)+h(t)$ and has zero solution. That is, $h_{\beta }^T{Z}_i(t)+h(t)=0$. It follows from (E5) that h(t) = 0 and h_β= 0. Thus, h_γ = 0.

11. Concluding Remarks

We have developed a general asymptotic theory for the NPMLEs with right censored data and shown that this theory applies to the models considered by Zeng and Lin (2007). This theory can also be used to establish the desired asymptotic properties for other existing semiparametric models, particularly the models mentioned in Sections 7.1–7.4 of Zeng and Lin (2007), as well as those that may be invented in the future. It is much simpler to verify the set of sufficient conditions identified in this paper than to prove the asymptotic results from scratch. Conditions (C1) and (C2) are standard conditions required in all censored-data regression; (C3), (C4) and (C6) are certain smoothness conditions that can be verified directly, as demonstrated in Section 10; (C5) and (C7) are two minimal identifiability conditions that need to be verified for any specific problem.

Although the basic structures of our proofs mimic those of Murphy (1994; 1995) and Parner (1998), our technical arguments are innovative and substantially more difficult because we deal with a very general form of likelihood function rather than specific problems. In all previous work, verification of the Donsker property relies on the specific expressions of the functions, whereas our Lemma 1 provides a universal way to verify this property. In verifying the invertibility of the information operator, all previous work requires an explicit expression of the information operator that is identified as the sum of an invertible operator and a compact operator, whereas we allow a very generic form of information operator obtained from the likelihood function (1). Murphy and van der Vaart (2001) stated that the consistency of NPMLEs needs to be proved on a case-by-case basis; however, we were able to prove the consistency for a very general likelihood function. Although we borrowed the partitioning idea of Murphy (1994), our technical arguments are very different because of the generic form of the likelihood.

In some applications, the failure times are subject to left truncation in addition to right censoring. To accommodate general censoring/truncation patterns, we define N(t) as the number of events observed by time t and R(t) as the at-risk indicator at time t, reflecting both left truncation and right censoring. Assume that the truncation time has positive mass at time 0, so that (C2) is satisfied. Then all the results continue to hold.

This paper is concerned with the theoretical aspect of the NPMLEs and complements the work of Zeng and Lin (2007). The interested readers are referred to the latter for the calculations of the NPMLEs and for the use of the semiparametric regression models and NPMLEs in practice. The latter also provides rationale for the kind of model considered in Sections 2 and 10 of this paper. Although the latter contains some theoretical elements, this paper presents the theory (especially the regularity conditions) in a more rigorous manner and provides all the proofs.