Asymptotic results for fitting marginal hazards models from stratified case-cohort studies with multiple disease outcomes

Abstract

In stratified case-cohort designs, samplings of case-cohort samples are conducted via a stratified random sampling based on covariate information available on the entire cohort members. In this paper, we extended the work of Kang & Cai (2009) to a generalized stratified case-cohort study design for failure time data with multiple disease outcomes. Under this study design, we developed weighted estimating procedures for model parameters in marginal multiplicative intensity models and for the cumulative baseline hazard function. The asymptotic properties of the estimators are studied using martingales, modern empirical process theory, and results for finite population sampling.

1 Introduction

The case-cohort study design (Prentice, 1986) has been proposed to reduce the cost and effort arising in large cohort studies with time-to-event data. The amount of reduction could be substantial especially if the main disease of interest is rare and the main covariate of interest (exposure) is expensive to measure since the case-cohort design requires measurement of the exposure only on a subset of the whole cohort. Specifically, the sampling in the case-cohort design is comprised of the following two steps. First, a subset called subcohort from the entire cohort is sampled randomly regardless of failure status. Second, remaining cases outside the subcohort are sampled. Information on the exposure is obtained only on these sampled subjects.

While the exposure will be available only for the case-cohort sample, some less expensive covariates such as age, gender, or a correlate of the exposure might be easily obtained for all the cohort members. In such cases, the subcohort could be sampled via a stratified simple random sampling based on strata defined by some of these covariates. This stratified case-cohort design could lead to a large efficiency gain compared to the unstratified counterpart since the latter ignores the available information.

For a single disease outcome assuming independent failure times among subjects in Cox models, many statistical methods have been proposed and studied for data from unstratified case-cohort studies (Prentice, 1986; Self & Prentice, 1988; Barlow, 1994; Chen & Lo, 1999; Chen, 2001) and stratified case-cohort designs (Borgan et al., 2000; Kulich & Lin, 2004).

The case-cohort design, among several other study designs which have been proposed for the similar purpose, is known to have advantage since the same subcohort can be utilized for different disease outcomes (Langholz & Thomas, 1990; Wacholder et al., 1991). When more than one disease outcomes from a subject are of interest, failure time data from the same subject constitute multivariate failure time data wherein correlations among the failure times within the same subject should be accounted for. Such multivariate failure time data are frequently encountered in many biomedical studies. One interesting example is a study of relationship between serum ferritin and coronary heart disease and stroke events in the Busselton Health Study (Cullen, 1972). In order to reduce costs and preserve stored serum, case-cohort sampling was used. It is of scientific interest to compare the effects of serum ferritin on coronary heart disease and on stroke. A subject can experience both coronary heart disease and stroke, and times to coronary heart disease and stroke events observed from the same subject are obviously not independent. In this case, methods developed for single disease outcome assuming univariate failure time data can no longer be directly applied.

Statistical methods which address this problem have been somewhat limited. Recently, we proposed weighted estimating equation methods for failure time data with multiple disease outcomes from case-cohort studies assuming marginal hazards models (Kang & Cai, 2009). In that paper, the generalized case-cohort design which allows sampling of cases outside the subcohort was considered. It is more realistic and flexible when considering multiple disease outcomes since not all disease outcomes need to be rare or number of cases need to be small (Breslow & Wellner, 2007).

The main purpose of this article is to extend the study design considered in Kang & Cai (2009) to a stratified case-cohort design, propose estimation procedure under such study designs, and provide a detailed derivation for the asymptotic properties of the proposed estimators. The model and the estimating procedures for regression coefficients and cumulative baseline hazards function are presented in Section 2. The corresponding asymptotic properties are stated and proven in Section 3. A brief summary and discussion is provided in Section 4.

2 Model, study design, and estimating procedure

2.1 Model

Suppose a cohort with n subjects can be divided into L mutually exclusive strata using some information available for all the cohort members. Let T_lik be the failure time for the kth type of disease outcome (k = 1, …, K) of the ith subject (i = 1, …, n_l) within the lth stratum (l = 1, …, L). Due to right censoring, what one actually observes for the kth type of disease outcome within the lth stratum is X_lik = min(T_lik, C_lik) where C_lik is the potential censoring time. Given p-vector of covariates Z_lik(t), T_lik and C_lik are assumed to be independent. We assume that all the time-dependent covariates in Z_lik(t) are “external”, i.e., they are not affected by the disease processes, as described by Kalbfleisch & Prentice (2002). Let Δ_lik = I(T_lik ≤ C_lik), N_lik(t) = Δ_likI (X_lik ≤ t), and Y_lik(t) = I (X_lik ≥ t) where I (·) is an indicator function. Let λ_lik(t) denote the corresponding marginal hazards function and let τ denote the study end time.

For the kth type of disease outcome of the ith subject within the lth stratum, the marginal hazards function λ_lik(t) is assumed to be associated with the covariate Z_lik(t) by

$${\lambda }_{\mathit{lik}}\{t\mid Z_{\mathit{lik}}(t)\}=Y_{\mathit{lik}}(t){\lambda }_{0k}(t)e^{{\beta }_0^T{Z}_{\mathit{lik}}(t)},$$ (1)

where λ_0k(t) is an unspecified baseline hazard function for the kth disease outcome and β₀ = (β₀₁, …, β_0K )^T is a p × 1 vector of regression parameters.

2.2 Study designs

Let V denote the discrete random variable for indicating the stratum. We consider sampling procedures depending on V. We assume that T_lik is independent of V_ik given Z_lik(·), i.e., V_ik affects the failure time only through the covariates (Kulich & Lin, 2004).

First, we consider a direct extension of the stratified case-cohort design for a single disease outcome to multiple disease outcomes and refer to this design as the “original” stratified case-cohort design. Under the original stratified case-cohort design for multiple disease outcomes, the subcohort is selected by a stratified random sampling. Specifically, for the lth stratum, we select a fixed size ñ_l subjects from n_l subjects in the entire cohort via simple random sampling without replacement. Thus, each subject in the lth stratum has the same probability pr(ξ_li = 1) = α̃_l = ñ_l/n_l of being selected into the subcohort where ξ_li is subcohort sampling indicator for the ith subject in the lth stratum. We obtain covariate measurements only on the subcohort members and all the remaining cases outside the subcohort. For the kth type of disease outcome, complete data, {X_lik, Δ_lik, Z_lik(t), 0 ≤ t ≤ X_ik, V_ik} are available for the subcohort members (ξ_li = 1) or cases (Δ_lik = 1). Note that, for cases, information on V_ik do not need to be available. For the non-subcohort controls (ξ_li = 0 and Δ_lik = 0), only partial data, {X_lik, Δ_lik V_ik} are available.

Since we consider more than one disease outcome, it might be more realistic that some of diseases outcomes are not rare or the numbers of cases are not small. In this situation, obtaining covariate information on all the cases outside the subcohort might not be feasible. Thus, we consider a stratified case-cohort design which allows the sampling of cases outside the subcohort to be different for different stratum and refer to this design as the “generalized” stratified case-cohort design.

Under the generalized stratified case-cohort design for multiple disease outcomes, sampling of the subcohort members follows the same routine as before: for the lth stratum, sampling a fixed size ñ_l subjects from n_l subjects in the entire cohort via simple random sampling without replacement. After the sampling of a subcohort, instead of sampling all the cases outside the subcohort, we allow sampling a fraction of cases for each of the disease outcomes. Specifically, for the kth type of disease (k = 1, …, K) within the lth stratum (l = 1, …, L), we select a fixed number of $m_l^k$ cases who are outside the subcohort via simple random sampling without replacement. Then, each case outside the subcohort has the same probability ${\stackrel{\sim }{q}}_{lk}=pr({\eta }_{\mathit{lik}}=1\mid {\mathrm{\Delta }}_{\mathit{lik}}=1,{\xi }_{li}=0)=m_l^k/(n_l^k-{\stackrel{\sim }{n}}_l^k)$ of being sampled where η_lik is the case sampling indicator, $n_l^k$ is the number of the kth type of disease cases within the lth stratum in the cohort and ${\stackrel{\sim }{n}}_l^k$ is the number of kth disease cases within the lth stratum in the subcohort.

Note that due to the sampling scheme, (η_11k, …, η_lnlk) are correlated, however, (η_l1k, …, η_lnlk) and (η _l′1k′, …, η_l′nlk′) are not correlated for k ≠ k′ or l ≠ l′. We obtain covariate measurements only on the sampled subject. Thus complete data, {X_lik, Δ_lik, Z_lik(t), 0 ≤ t ≤ X_lik, V_ik}, are available for the sub-cohort members (ξ_li = 1) or sampled cases outside the subcohort (η_lik = 1). Only partial data, {X_lik, Δ_lik, V_ik} are available for all others (ξ_li = 0 and η_lik = 0). Note that the generalized stratified case-cohort design includes the original stratified case-cohort design as a special case since if q̃_lk = 1 for all k and l, it reduces to the original stratified case-cohort design which samples all the cases outside the subcohort. Also, if we do not consider the strata for the cohort, i.e., L = 1, then it reduces to the generalized case-cohort design considered by Kang & Cai (2009).

2.3 Estimation of regression parameters under the original stratified case-cohort design

For full cohort data, Wei et al. (1989) proposed the following pseudo-likelihood score equations for the estimation of the hazards regression parameter β₀:

$$U(\beta )=\sum _{i=1}^n\sum _{k=1}^K{\int }_0^{\tau }\left\{Z_{ik}(u)-\frac{S_k^{(1)}(\beta ,u)}{S_k^{(0)}(\beta ,u)}\right\}{dN}_{ik}(u),$$ (2)

where $S_k^{(d)}(\beta ,t)=n^{-1}{\sum }_{i=1}^n{Y}_{ik}(t)Z_{ik}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{ik}(t)}$ for d = 0 and 1, and a^⊗2 = aa^T, a^⊗1 = a, a^⊗0 = 1 for a vector a. Since these estimating equations do not have analytical solutions, they need to be solved iteratively, for example, by Newton-Raphson method (Thisted, 1988).

For data from the original stratified case-cohort studies, $S_k^{(d)}(\beta ,t)(d=0,1)$ in (2) cannot be calculated due to the incompleteness of the data. In order to handle this problem, we consider the idea of weighting the incomplete data by the inverse selection probability (Horvitz & Thompson, 1951). Specifically, we consider ${\widehat{S}}_k^{(d)}(\beta ,t)=n^{-1}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{\rho }_{\mathit{lik}}(t)Y_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}$ in place of $S_k^{(d)}(\beta ,t)$ for d = 0 and 1 where ρ_lik(t) is a possibly time-varying weight function, incorporated to account for the sampling scheme and has the following form:

$${\rho }_{\mathit{lik}}(t)={\mathrm{\Delta }}_{\mathit{lik}}+(1-{\mathrm{\Delta }}_{\mathit{lik}}){\xi }_{li}/{\widehat{\alpha }}_{lk}(t)\text{where}{\widehat{\alpha }}_{lk}(t)=\frac{{\sum }_{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}}){\xi }_{li}{Y}_{\mathit{lik}}(t)}{{\sum }_{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)}.$$

Then, for the estimation of β₀, we propose the following pseudo-partial-likelihood score equations Û(β) = 0_p×1, where

$$\widehat{U}(\beta )=\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\int }_0^{\tau }\left\{Z_{\mathit{lik}}(u)-\frac{{\widehat{S}}_k^{(1)}(\beta ,u)}{{\widehat{S}}_k^{(0)}(\beta ,u)}\right\}{dN}_{\mathit{lik}}(u),$$ (3)

A time-invariant version of the weight function which uses α̃_l in place of α̂_lk(t) may also be considered.

The solution to Û(β) = 0_p×1 is defined to be the estimator of the hazards regression parameter β₀. We will denote the estimator which uses time-invariant weight functions as β̂_I and time-varying weight functions as β̂_II. The corresponding pseudo-partial-likelihood functions will be denoted by Û_I(β) and Û_II (β), respectively.

2.4 Estimation of regression parameters under the generalized stratified case-cohort design

For the generalized stratified case-cohort design, the weight function needs to be modified to appropriately account for the sampling of cases outside the subcohort. Specifically, cases outside the subcohort who are sampled are weighted by ${\widehat{q}}_{lk}^{-1}(t)$ where q̂_lk(t) denotes the number of sampled non-subcohort cases with the kth type of disease outcome in the lth stratum divided by the number of non-subcohort cases with the kth type of disease outcome in the lth stratum remaining in the risk set at time t. Then, the weight function ω_lik(t) has the following form:

$${\omega }_{\mathit{lik}}(t)=(1-{\mathrm{\Delta }}_{\mathit{lik}})\frac{{\xi }_{li}}{{\widehat{\alpha }}_{lk}(t)}+{\mathrm{\Delta }}_{\mathit{lik}}{\xi }_{li}+{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\frac{{\eta }_{\mathit{lik}}}{{\widehat{q}}_{lk}(t)}\text{where}{\widehat{q}}_{lk}(t)=\frac{{\sum }_{i=1}^{n_l}{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li}){\eta }_{\mathit{lik}}{Y}_{\mathit{lik}}(t)}{{\sum }_{i=1}^{n_l}{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})Y_{\mathit{lik}}(t)}.$$

Note that the proposed weight functions reduce to the ones for the original stratified case-cohort design if all cases outside the subcohort are sampled since q̃_lk = 1 for all k and l.

For the estimation of β₀ under the generalized stratified case-cohort design, the following weighted estimating functions with the weight function ω_lik(t) is considered:

$$\stackrel{\sim }{U}(\beta )=\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\int }_0^{\tau }{\omega }_{\mathit{lik}}(u)\left\{Z_{\mathit{lik}}(u)-\frac{{\stackrel{\sim }{S}}_k^{(1)}(\beta ,u)}{{\stackrel{\sim }{S}}_k^{(0)}(\beta ,u)}\right\}{dN}_{\mathit{lik}}(u),$$ (4)

where ${\stackrel{\sim }{S}}_k^{(d)}(\beta ,t)=n^{-1}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{\omega }_{\mathit{lik}}(t)Y_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}$ for d = 0 and 1.

The solution to the equations Ũ(β) = 0_p×1 is the estimator for the hazards regression parameter β₀. We will denote the estimator which uses time-invariant weight functions as β̃_I and time-varying weight functions as β̃_II, respectively. The corresponding weighted estimating functions are Ũ_I (β) and Ũ_II (β), respectively.

2.5 Estimation of the cumulative baseline hazard function

Let Λ_0k(t) denote the cumulative baseline hazard function for the kth type of disease outcome at time t where ${\mathrm{\Lambda }}_{0k}(t)={\int }_0^t{\lambda }_{0k}(u)du$. Then for the estimator of Λ_0k(t), we consider the following Breslow-Aalen type estimators Λ̂_0k(β̂, t) for the original stratified case-cohort design and Λ̂_0k(β̃, t) for the generalized stratified case-cohort design where

$${\widehat{\mathrm{\Lambda }}}_{0k}(\beta ,t)={\int }_0^t\frac{{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{\rho }_{\mathit{lik}}(u){dN}_{\mathit{lik}}(u)}{n{\widehat{S}}_k^{(0)}(\beta ,u)}\text{and}$$ (5)

$${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}(\beta ,t)={\int }_0^t\frac{{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{\omega }_{\mathit{lik}}(u){dN}_{\mathit{lik}}(u)}{n{\stackrel{\sim }{S}}_k^{(0)}(\beta ,u)},$$ (6)

respectively.

We, again, use the superscript $\text{I}({\widehat{\mathrm{\Lambda }}}_{0k}^I(·,t),{\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^I(·,t))$ and $\text{II}({\widehat{\mathrm{\Lambda }}}_{0k}^{II}(·,t),{\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}(·,t))$ to denote the estimator using time-invariant and time-varying weight functions, respectively.

3 Asymptotic properties

We will focus on the asymptotic properties of the estimators under the generalized stratified case-cohort design, β̃_I, β̃_II, ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^I(·,t)$, and ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}(·,t)$. This is because the estimators under the original stratified case-cohort studies, β̂_I, β̂_II, ${\widehat{\mathrm{\Lambda }}}_{0k}^I(·,t)$, and ${\widehat{\mathrm{\Lambda }}}_{0k}^{II}(·,t)$, are special cases of those under the generalized stratified case-cohort studies. Thus, their asymptotic properties can be directly drawn from those under the generalized case-cohort studies.

3.1 Conditions

In order to establish the consistency and asymptotic normality of the estimators for the generalized stratified case-cohort studies, the following sets of conditions are needed:

• (A)(T_li, C_li, Z_li), i = 1, …, n_l and l = 1, …, L are independent and identically distributed where T_li = (T_li1, …, T_liK)^T, C_li = (C_li1, …, C_liK)^T and Z_li = (Z_li1, …, Z_liK)^T.
• (B)pr{Y_lik(τ) > 0} > 0 for all i = 1, …, n_l, k = 1, …, K and l = 1, …, L.
• (C) $\mid Z_{\mathit{likj}}(0)\mid +{\int }_0^{\tau }\mid {dZ}_{\mathit{likj}}(u)\mid <C_z<\infty$ almost surely for all i = 1, …, n_l, k = 1, …, K, and l = 1, …, L where Z_likj is the jth component of Z_lik and C_z is some constant.
• (D)The matrix $A_k({\beta }_0)={\int }_0^{\tau }{v}_k({\beta }_0,u)s_k^{(0)}({\beta }_0,u){\lambda }_{0k}(u)du$ is positive definite.
• (E) ${\int }_0^{\tau }{\lambda }_{0k}(u)du<\infty$, for all k = 1, …, K.
• (F)There exists a neighborhood of β₀ that satisfies the following conditions, as n → ∞: for all k = 1, …, K, and d = 0, 1, 2, $\underset{\begin{array}{c}t\in [0,\tau ]\\ \beta \in \mathcal{B}\end{array}}{sup}\left|\right|S_k^{(d)}(\beta ,t)-s_k^{(d)}(\beta ,t)\left|\right|\stackrel{p}{\to }0$ where $E\{S_k^{(d)}(\beta ,t)\}=s_k^{(d)}(\beta ,t)$ are continuous functions of β ∈ , uniformly in t ∈ [0, τ] and are bounded on × [0, τ] and $s_k^{(0)}$ is bounded away from zero on × [0, τ].

The following additional conditions are also needed to ensure the desired asymptotic convergence of case-cohort samples:

• (G)As n → ∈,

  1. For all l = 1, …, L, α̃_l converges to a constant α_l ∈ (0, 1];

  2. For all k = 1, …, K, and l = 1, …, L, q̃_lk converges to a constant q_lk in (0, 1].

• (H)n_l/n converges to a constant p_l ∈ [0, 1] for all l = 1, …, L as n → ∈.

Here and hereafter the norms for the vector a, matrix A, and function f are defined as the following:

$$\left|\right|a\left|\right|=\underset{i}{max}\mid a_i\mid ,\left|\right|A\left|\right|=\underset{i,j}{max}\mid A_{ij}\mid ,\left|\right|f\left|\right|=\underset{t}{sup}\mid f(t)\mid$$

3.2 Asymptotic properties of β̃_I and ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^I({\stackrel{\sim }{\beta }}_I,t)$

The key component of the derivation of the asymptotic results involves a decomposition of the proposed estimating function into three asymptotically uncorrelated pieces and some negligible terms. These three components represent, respectively, the whole cohort counterpart, one arising from sampling of a subcohort, and one arising from sampling of cases outside the subcohort.

Let us provide some lemmas which will be frequently used in proving the theorems.

Lemma 1

Let W_n(t) and G_n(t) be two sequences of bounded processes. For some constant τ, assume that the following conditions (a) – (c) hold where

1. ${sup}_{0\le t\le \tau }\left|\right|W_n(t)-W(t)\left|\right|\stackrel{p}{\to }0$ for some bounded process W (t),

2. W_n(t) is monotone on [0, τ] and

3. G_n(t) converges to a zero-mean process with continuous sample paths. Under Conditions (a) – (c),

   $$\underset{0\le t\le \tau }{sup}\Vert {\int }_0^t\{W_n(u)-W(u)\}{dG}_n(u)\Vert \stackrel{p}{\to }0,\underset{0\le t\le \tau }{sup}\Vert {\int }_0^t{G}_n(u)d\{W_n(u)-W(u)\}\Vert \stackrel{p}{\to }0.$$

PROOF

Lemma 1 is given as a lemma in Lin (2000). Its proof follows from the strong embedding theorem (Shorack & Wellner, 1986, p47–48), lemma 1 of Lin et al. (2000) and the triangular argument of a norm.

Lemma 2 is an extension of the proposition in Kulich & Lin (2000).

Lemma 2

Let ξ = (ξ₁, …, ξ_n) be a random vector containing ñ ones and n–ñ zeros, with each permutation equally likely. Let B_i(t), i = 1, …, n, be i.i.d. real-valued random processes on [0, τ] with E{B_i(t)} = μ_B (t), Var{B_i(0)} < ∞ and Var{B_i(τ)} < ∞. Let B(t) = {B₁(t), …, B_n(t)} and ξ be independent. Suppose that almost all paths of B_i(t) have finite variation. Then,

$$n^{-1/2}\sum _{i=1}^n{\xi }_i\{B_i(t)-{\mu }_B(t)\}$$

converges weakly in ℓ^∞ [0, τ] to a zero-mean Gaussian process and therefore

$$n^{-1}\sum _{i=1}^n{\xi }_i\{B_i(t)-{\mu }_B(t)\}$$

converges in probability to 0 uniformly in t.

PROOF

This lemma is an extension of the proposition in Kulich & Lin (2000). The proof of this lemma follows from Hájek (1960)’s central limit theorem for finite population sampling and Example 3.6.14 of van der Vaart & Wellner (1996). Specifically, suppose first that the B_i(t)’s have nondecreasing sample paths then the finite-dimensional convergence follows from Hájek (1960)’s central limit theorem for finite population sampling while the tightness follows from Example 3.6.14 of van der Vaart & Wellner (1996). In the general case, since almost every path b(t) of B(t) have finite variation, b(t) can be written as $b_1^{\ast }(t)-b_2^{\ast }(t)$, where $b_1^{\ast }(t)$ and $b_2^{\ast }(t)$ are nonnegative, nondecreasing in t. Hence $B_i(t)=B_{i1}^{\ast }(t)-B_{i2}^{\ast }(t)$, where $B_{i1}^{\ast }(t)$ and $B_{i2}^{\ast }(t)$ are marginally tight since they meet the condition of Example 3.6.14 of van der Vaart & Wellner (1996). This implies that they are jointly tight. The joint finite-dimensional convergence of the normalized $n^{-1/2}{\sum }_{i=1}^n{\xi }_i\{B_{i1}^{\ast }(t)-{\mu }_{B_{i1}^{\ast }}(t)\}$ and $n^{-1/2}{\sum }_{i=1}^n{\xi }_i\{B_{i2}^{\ast }(t)-{\mu }_{B_{i1}^{\ast }}(t)\}$ follows again from Hájek (1960)’s central limit theorem for finite population sampling. Therefore, $n^{-1/2}{\sum }_{i=1}^n{\xi }_i\{B_i(t)-{\mu }_{B_i}(t)\}$ converges weakly in ℓ^∞ [0, τ] to zero mean Gaussian processes. It then follows that $n^{-1}{\sum }_{i=1}^n{\xi }_i\{B_i(t)-{\mu }_{B_i}(t)\}$ converges to 0 in probability uniformly in t.

Note that ξ_li is the subcohort membership indicator and η_lik is the sampling indicator for the ith subject with the kth disease within the lth stratum outside the subcohort where both the sampling of the subcohort and the cases outside the subcohort were conducted by simple random sampling without replacement. Thus, it is clear that our ξ_li’s and η_lik’s satisfy the conditions in lemma 2.

Theorem 1

Under Conditions (A) – (H), β̃_I solving Ũ_I (β) = 0 is a consistent estimator of β₀. Also, n^1/2(β̃_I – β₀) is asymptotically normally distributed with mean zero and with variance matrix Σ_I (β₀) of the following form

$$A{({\beta }_0)}^{-1}\sum _{l=1}^L{p}_l\left\{Q_l({\beta }_0)+\frac{1-{\alpha }_l}{{\alpha }_l}{V}_l^{I,(1)}({\beta }_0)+(1-{\alpha }_l)\sum _{k=1}^{K}pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right)V_{lk}^{I,(2)}({\beta }_0)\right\}A{({\beta }_0)}^{-1}$$

where

$$\begin{array}{l}A(\beta )=\sum _{k=1}^K{A}_k(\beta ),Q_l(\beta )=E_l{\left\{\sum _{k=1}^K{\mathit{M}}_{\stackrel{\sim }{z},l1k}(\beta ,\tau )\right\}}^{\otimes 2},\\ V_l^{I,(1)}(\beta )=Va{r}_l\left\{\sum _{k=1}^K{\int }_0^{\tau }(1-{\mathrm{\Delta }}_{l1k})R_{l1k}(\beta ,u)d{\mathrm{\Lambda }}_{0k}(u)\right\},\\ V_{lk}^{I,(2)}(\beta )=Va{r}_l\{{\mathit{M}}_{\stackrel{\sim }{z},l1k}(\beta ,\tau )\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\},R_{\mathit{lik}}(\beta ,t)=Y_{\mathit{lik}}(t){\stackrel{\sim }{Z}}_{\mathit{lik}}(\beta ,t)e^{{\beta }^T{Z}_{\mathit{lik}}(t)},\\ e_k(\beta ,t)=\frac{s_k^{(1)}(\beta ,t)}{s_k^{(0)}(\beta ,t)},v_k(\beta ,t)=\frac{s_k^{(2)}(\beta ,t)s_k^{(0)}(\beta ,t)-s_k^{(1)}{(\beta ,t)}^{\otimes 2}}{s_k^{(0)}{(\beta ,t)}^2},\\ A_k(\beta )={\int }_0^{\tau }{v}_k(\beta ,u)s_k^{(0)}(\beta ,u){\lambda }_{0k}(u)du,\\ {\stackrel{\sim }{Z}}_{\mathit{lik}}(\beta ,t)=Z_{\mathit{lik}}(t)-e_k(\beta ,t),{\mathit{M}}_{\stackrel{\sim }{z},\mathit{lik}}(\beta ,t)={\int }_0^t{\stackrel{\sim }{Z}}_{\mathit{lik}}(\beta ,u){dM}_{\mathit{lik}}(\beta ,u),\\ \mathit{and}{M}_{\mathit{lik}}(\beta ,t)=N_{\mathit{lik}}(t)-{\int }_0^t{Y}_{\mathit{lik}}(u){\lambda }_{0k}(u)e^{{\beta }^T{Z}_{\mathit{lik}}(u)}du.\end{array}$$

Note that E_l, Var_l, and Cov_l denote the expectation, the variance and the covariance within the lth stratum, respectively.

We now study the asymptotic properties of ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^I({\stackrel{\sim }{\beta }}_I,t)(k=1,\dots ,K)$. Let $W^I(t)=n^{1/2}{[\{{\stackrel{\sim }{\mathrm{\Lambda }}}_{01}^I({\stackrel{\sim }{\beta }}_I,t)-{\mathrm{\Lambda }}_{01}(t)\},\dots ,\{{\stackrel{\sim }{\mathrm{\Lambda }}}_{0K}^I({\stackrel{\sim }{\beta }}_I,t)-{\mathrm{\Lambda }}_{0K}(t)\}]}^T$ and ${\mathcal{W}}^I(t)={\{{\mathcal{W}}_1^I(t),\dots ,{\mathcal{W}}_K^I(t)\}}^T$ where (t) is a zero-mean Gaussian process with the covariance function between ${\mathcal{W}}_j^I(t_1)$ and ${\mathcal{W}}_k^I(t_2)$ (1 ≤ j, k ≤ K and 0 ≤ t₁, t₂ ≤ τ) is ${\phi }_{jk}^I(t_1,t_2)({\beta }_0)$ where

$$\begin{array}{l}{\phi }_{jk}^I(t_1,t_2)({\beta }_0)=\sum _{l=1}^L{p}_l({\text{E}}_l\{{\nu }_{l1j}({\beta }_0,t_1){\nu }_{l1k}({\beta }_0,t_2)\}+\frac{1-{\alpha }_l}{{\alpha }_l}{\text{E}}_l\{{\psi }_{l1j}^I({\beta }_0,t_1){\psi }_{l1k}^I({\beta }_0,t_2)\}+(1-{\alpha }_l)\\ \times [I(j=k)pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right){\text{Cov}}_l\left\{{\int }_0^{t_1}\frac{{dM}_{l1k}(u)}{s_k^{(0)}({\beta }_0,u)},{\int }_0^{t_2}\frac{{dM}_{l1k}(u)}{s_k^{(0)}({\beta }_0,u)}|{\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}\\ +pr({\mathrm{\Delta }}_{l1j}=1)\left(\frac{1-q_{lj}}{q_{lj}}\right){\text{Cov}}_l\left\{{\int }_0^{t_1}\frac{{dM}_{l1j}(u)}{s_j^{(0)}({\beta }_0,u)},r_k{({\beta }_0,t_2)}^{T}A{({\beta }_0)}^{-1}{\mathbf{M}}_{\stackrel{\sim }{z},l1j}({\beta }_0)|{\mathrm{\Delta }}_{l1j}=1,{\xi }_{l1}=0\right\}\\ +pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right){\text{Cov}}_l\left\{{\int }_0^{t_2}\frac{{dM}_{l1k}(u)}{s_k^{(0)}({\beta }_0,u)},r_j{({\beta }_0,t_1)}^{T}A{({\beta }_0)}^{-1}{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0)|{\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}\\ +\sum _{m=1}^{K}pr({\mathrm{\Delta }}_{l1m}=1)\left(\frac{1-q_{lm}}{q_{lm}}\right)r_j{({\beta }_0,t_1)}^{T}A{({\beta }_0)}^{-1}{\text{Var}}_l\{{\mathbf{M}}_{\stackrel{\sim }{z},l1m}({\beta }_0)\mid {\mathrm{\Delta }}_{l1m}=1,{\xi }_{l1}=0\}\\ \times A{({\beta }_0)}^{-1}{r}_k({\beta }_0,t_2)],{\nu }_{\mathit{lik}}(\beta ,t)=r_k{(\beta ,t)}^{T}A{(\beta )}^{-1}\sum _{m=1}^K{\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lim}}(\beta )+{\int }_0^t{\{s_k^{(0)}(\beta ,u)\}}^{-1}{dM}_{\mathit{lik}}(u),\\ {\psi }_{\mathit{lik}}^I(\beta ,t)=r_k{(\beta ,t)}^{T}A{(\beta )}^{-1}\sum _{m=1}^K(1-{\mathrm{\Delta }}_{\mathit{lim}}){\int }_0^{\tau }{R}_{\mathit{lim}}(\beta ,u)d{\mathrm{\Lambda }}_{0m}(u)\\ +(1-{\mathrm{\Delta }}_{\mathit{lik}}){\int }_0^t\frac{Y_{\mathit{lik}}(u)e^{{\beta }^T{Z}_{\mathit{lik}}(u)}}{s_k^{(0)}(\beta ,u)}d{\mathrm{\Lambda }}_{0k}(u)\text{and}{r}_k(\beta ,t)=-{\int }_0^t{e}_k(\beta ,u)d{\mathrm{\Lambda }}_{0k}(u).\end{array}$$

Also, let D[0, τ]^K be a metric space consisting of right-continuous functions f (t) with left-hand limits where f (t) = {f₁(t), …, f_K (t)}^T and f_k(t): [0, τ] → . This metric space is equipped with a uniform metric d_k(f, g) = sup_k,t∈[0,τ]{|f_k(t) − g_k(t)|: 1 ≤ k ≤ K} for f, g ∈ D[0, τ]^K.

Theorem 2

Under Conditions (A) – (H), for each k = 1, …, K, ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^I({\stackrel{\sim }{\beta }}_I,t)$ converges in probability to Λ_0k(t) uniformly in t ∈ [0, τ]. Also, W^I(t) converges weakly to (t) in D[0, τ]^K

Proofs for Theorems 1 and 2 can be derived from those for their time-varying counterparts which will be provided in the next subsection. More detailed explanation on this is deferred to Section 3.4

3.3 Asymptotic properties of β̃_II and ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}({\stackrel{\sim }{\beta }}_{II},t)$

In order to establish the asymptotic properties of the proposed estimators with time-varying weight functions, we need the following lemma on the asymptotic properties of time-varying sampling probability estimators α̂_lk(t) and q̂_k(t).

Lemma 3

For all l = 1, …, L and k = 1, …, K,

1. α̂_lk(t) and α̃_l converge to the same limit uniformly in t and

   $$n^{1/2}\left\{{\widehat{\alpha }}_{lk}{(t)}^{-1}-{\stackrel{\sim }{\alpha }}_l^{-1}\right\}=\frac{1}{p_l{\stackrel{\sim }{\alpha }}_l{E}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)\}}{n}^{-1/2}\left\{\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)\right\}+o_p(1).$$
2. q̂_lk(t) and q̃_lk converge to the same limit uniformly in t and

   $$n^{1/2}\left\{{\widehat{q}}_{lk}{(t)}^{-1}-{\stackrel{\sim }{q}}_{lk}^{-1}\right\}=\frac{1}{{\stackrel{\sim }{q}}_{lk}{p}_l(1-{\stackrel{\sim }{\alpha }}_l)E_l\{{\mathrm{\Delta }}_{l1k}{Y}_{l1k}(t)\}}{n}^{-1/2}\left\{\sum _{i=1}^{n_l}\left(1-\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}\right){\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})Y_{\mathit{lik}}(t)\right\}+o_p(1).$$

PROOF

For each l and k, it follows from the Taylor expansion of α̂_lk(t)⁻¹ around α̃_l,

$$\begin{array}{l}{\widehat{\alpha }}_{lk}{(t)}^{-1}-{\stackrel{\sim }{\alpha }}_l^{-1}=-\frac{1}{{\alpha }_{\ast }{(t)}^2}\{{\widehat{\alpha }}_{lk}(t)-{\stackrel{\sim }{\alpha }}_l\}\\ =\frac{{\stackrel{\sim }{\alpha }}_l}{{\alpha }_{\ast }{(t)}^2}·\frac{1}{{\sum }_{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)}\left\{\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)\right\}\end{array}$$

where α_* (t) is on the line segment between α̂_lk(t) and α̃_l. Then,

$$n^{1/2}\left\{{\widehat{\alpha }}_{lk}{(t)}^{-1}-{\stackrel{\sim }{\alpha }}_l^{-1}\right\}=\frac{{\stackrel{\sim }{\alpha }}_l}{{\alpha }_{\ast }{(t)}^2}·\frac{n}{n_l}·\frac{n_l}{{\sum }_{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)}{n}^{-1/2}\left\{\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)\right\}$$

By Glivenko-Cantelli lemma, $n_l^{-1}{\sum }_{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)$ converges to E_l {(1 − Δ_l1k)Y_l1k(t)} in probability uniformly in t. In view of lemma 2, $n^{-1/2}{\sum }_{i=1}^{n_l}(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)$ converges to a zero-mean Gaussian process since (1 − Δ_lik)Y_lik(t)is bounded and monotone function in t. This implies $n^{-1}{\sum }_{i=1}^{n_l}(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)$ converges to 0 in probability uniformly in t and consequently, α̂_lk(t) and α̃_l converges to the same limit uniformly in t. This ensures α_* (t) also converges to the same limit as α̃_l. Combining these results, it follows from Slutsky’s theorem and Condition (H) that

$$\begin{array}{l}{n}^{1/2}\left\{{\widehat{\alpha }}_{lk}{(t)}^{-1}-{\stackrel{\sim }{\alpha }}_l^{-1}\right\}=\frac{1}{{\stackrel{\sim }{\alpha }}_l{p}_l{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)\}}{n}^{-1/2}\left\{\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)\right\}\\ +\left[\frac{{\stackrel{\sim }{\alpha }}_l}{{\alpha }_{\ast }{(t)}^2}·\frac{n}{n_l}·\frac{n_l}{{\sum }_{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)}-\frac{1}{{\stackrel{\sim }{\alpha }}_l{p}_l{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)\}}\right]n^{-1/2}\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)\\ =\frac{1}{{\stackrel{\sim }{\alpha }}_l{p}_l{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)\}}{n}^{-1/2}\left\{\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)\right\}+o_p(1).\end{array}$$

(ii) can be shown via similar arguments.

Now, we state the asymptotic behavior of the regression parameter estimator β̃_II in the following theorem:

Theorem 3

Under Conditions (A) – (H), β̃_II solving Ũ_II (β) = 0 is a consistent estimator of β₀. Also, n^1/2(β̃_II − β₀) is asymptotically normally distributed with mean zero and with variance matrix Σ_II (β₀) of the following form

$$A{({\beta }_0)}^{-1}\sum _{l=1}^L{p}_l\left\{Q_l({\beta }_0)+\frac{1-{\alpha }_l}{{\alpha }_l}{V}_l^{II,(1)}({\beta }_0)+(1-{\alpha }_l)\sum _{k=1}^{K}pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right)V_{lk}^{II,(2)}({\beta }_0)\right\}A{({\beta }_0)}^{-1}$$

where

$$\begin{array}{l}{V}_l^{II,(1)}(\beta )=Va{r}_l\left(\sum _{k=1}^K(1-{\mathrm{\Delta }}_{l1k}){\int }_0^{\tau }\left[R_{l1k}(\beta ,u)-\frac{Y_{l1k}(u)E_l\{(1-{\mathrm{\Delta }}_{l1k})R_{l1k}(\beta ,u)\}}{E_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}\right]d{\mathrm{\Lambda }}_{0k}(u)\right)\mathit{and}\\ V_{lk}^{II,(2)}(\beta )=Va{r}_l\left[{\mathit{M}}_{\stackrel{\sim }{z},l1k}(\beta ,\tau )-{\int }_0^{\tau }{Y}_{l1k}(u)\frac{E_l\{{d\mathit{M}}_{\stackrel{\sim }{z},l1k}(\beta ,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{E_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}|{\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right].\end{array}$$

PROOF

The proof for the consistency of β̃_II is based on the application of the Inverse Function Theorem in Foutz (1977). One can show β̃_II to be consistent for β₀ provided:

1. $\frac{\partial \left\{n^{-1}{\stackrel{\sim }{U}}_{II}(\beta )\right\}}{\partial {\beta }^T}$ exists and is continuous in an open neighborhood of β₀,

2. $\frac{\partial \left\{n^{-1}{\stackrel{\sim }{U}}_{II}({\beta }_0)\right\}}{\partial {\beta }^T}$ is negative definite with probability going to one as n → ∞,

3. $\frac{\partial \left\{n^{-1}{\stackrel{\sim }{U}}_{II}(\beta )\right\}}{\partial \beta }$ converges to A(β₀) in probability uniformly for β in an open neighborhood about β₀,

4. n⁻¹Ũ_II (β) → 0 in probability.

One can write

$$\begin{array}{c}\frac{\partial \left\{n^{-1}{\stackrel{\sim }{U}}_{II}(\beta )\right\}}{\partial {\beta }^T}=-n^{-1}\sum _{k=1}^K{\int }_0^{\tau }{\stackrel{\sim }{V}}_k(\beta ,u)d\left\{\sum _{l=1}^L\sum _{i=1}^{n_l}{\omega }_{\mathit{lik}}^{II}(u)N_{\mathit{lik}}(u)\right\}\text{where}\\ {\stackrel{\sim }{V}}_k(\beta ,t)=\frac{{\stackrel{\sim }{S}}_{II,k}^{(2)}(\beta ,t){\stackrel{\sim }{S}}_{II,k}^{(0)}(\beta ,t)-{\stackrel{\sim }{S}}_{II,k}^{(1)}{(\beta ,t)}^{\otimes 2}}{{\stackrel{\sim }{S}}_{II,k}^{(0)}{(\beta ,t)}^2}\end{array}$$ (7)

Then, (i) is clearly satisfied on the basis of (7) and Condition (F). Now, following Andersen & Gill (1982),

$$\begin{array}{l}\Vert \left[-\frac{\partial \{n^{-1}{\stackrel{\sim }{U}}^{II}(\beta )\}}{\partial {\beta }^T}\right]-A(\beta )\Vert \le \Vert \sum _{k=1}^K{\int }_0^{\tau }\{{\stackrel{\sim }{V}}_k(\beta ,u)-v_k(\beta ,u)\}{n}^{-1}d\left\{\sum _{l=1}^L\sum _{i=1}^{n_l}{N}_{\mathit{lik}}(u)\right\}\Vert \\ +\Vert \sum _{k=1}^K{\int }_0^{\tau }\{{\stackrel{\sim }{V}}_k(\beta ,u)-v_k(\beta ,u)\}d\left[n^{-1}\sum _{l=1}^L\sum _{i=1}^{n_l}\{{\omega }_{\mathit{lik}}^{II}(u)-1\}{N}_{\mathit{lik}}(t)\right]\Vert \\ +\Vert \sum _{k=1}^K{\int }_0^{\tau }{v}_k(\beta ,u)n^{-1}d\left\{\sum _{l=1}^L\sum _{i=1}^{n_l}{M}_{\mathit{lik}}(\beta ,u)\right\}\Vert +\Vert \sum _{k=1}^K{n}^{-1}\sum _{l=1}^L\sum _{i=1}^{n_l}\{{\omega }_{\mathit{lik}}^{II}(u)-1\}{\int }_0^{\tau }{v}_k(\beta ,u){dM}_{\mathit{lik}}(\beta ,u)\Vert \\ +\Vert \sum _{k=1}^K{\int }_0^{\tau }{v}_k(\beta ,u)\{{\stackrel{\sim }{S}}_{II,k}^{(0)}(\beta ,u)-s_k^{(0)}(\beta ,u)\}{\lambda }_{0k}(u)du\Vert \end{array}$$ (8)

Each of the terms on the right side of the above inequality can be shown to converge to zero, uniformly in β ∈ . To show the first term on the right side of (8), we will first show that

$$\underset{\begin{array}{c}t\in [0,\tau ]\\ \beta \in \mathcal{B}\end{array}}{sup}\Vert {\stackrel{\sim }{V}}_k(\beta ,t)-v_k(\beta ,t)\Vert \stackrel{p}{\to }0\text{as}n\to \infty \text{for}k=1,\dots ,K.$$

It suffices to show that ${sup}_{t\in [0,\tau ],\beta \in \mathcal{B}}\left|\right|{\stackrel{\sim }{S}}_{II,k}^{(d)}(\beta ,t)-S_k^{(d)}(\beta ,t)\left|\right|\stackrel{p}{\to }0$ as n → ∞ for d = 0, 1 and 2. One can write

$$\begin{array}{l}{n}^{1/2}\left\{{\stackrel{\sim }{S}}_{II,k}^{(d)}(\beta ,t)-S_k^{(d)}(\beta ,t)\right\}=n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_k}-1\right){\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})Y_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}\\ +\left\{{\widehat{\alpha }}_{lk}{(t)}^{-1}-{\stackrel{\sim }{\alpha }}_l^{-1}\right\}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}}){\xi }_{li}{Y}_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}\\ +\left\{{\widehat{q}}_{lk}{(t)}^{-1}-{\stackrel{\sim }{q}}_{lk}^{-1}\right\}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li}){\eta }_{\mathit{lik}}{Y}_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}.\end{array}$$ (9)

Then by lemma 3,

$$\begin{array}{l}(9)=n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})Y_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}-1\right){\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})Y_{\mathit{lik}}(t)Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}})\frac{Y_{\mathit{lik}}(t)}{p_l{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)\}}\\ \times \left\{n^{-1}\sum _{j=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{ljk}})\frac{{\xi }_{lj}}{{\stackrel{\sim }{\alpha }}_l}{Y}_{\mathit{ljk}}(t)Z_{\mathit{ljk}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{ljk}}(t)}\right\}\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(1-\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}\right)(1-{\xi }_{li}){\mathrm{\Delta }}_{\mathit{lik}}\frac{Y_{\mathit{lik}}(t)}{p_l{\text{E}}_l\{{\mathrm{\Delta }}_{l1k}{Y}_{l1k}(t)\}}\\ \times \left\{n^{-1}\sum _{j=1}^{n_l}{\mathrm{\Delta }}_{\mathit{ljk}}\frac{(1-{\xi }_{lj})}{(1-{\stackrel{\sim }{\alpha }}_l)}\frac{{\eta }_{\mathit{ljk}}}{{\stackrel{\sim }{q}}_{lk}}{Y}_{\mathit{ljk}}(t)Z_{\mathit{ljk}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{ljk}}(t)}\right\}+o_p(1)\end{array}$$ (10)

It follows from lemma 2 that, for d = 0, 1 and 2, $n^{-1}{\sum }_{j=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{ljk}})\frac{{\xi }_{lj}}{{\stackrel{\sim }{\alpha }}_l}{Y}_{\mathit{ljk}}(t)Z_{\mathit{ljk}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{ljk}}(t)}$ and $n^{-1}{\sum }_{j=1}^{n_l}{\mathrm{\Delta }}_{\mathit{ljk}}\frac{(1-{\xi }_{lj})}{(1-{\stackrel{\sim }{\alpha }}_l)}\frac{{\eta }_{\mathit{ljk}}}{{\stackrel{\sim }{q}}_{lk}}{Y}_{\mathit{ljk}}(t)Z_{\mathit{ljk}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{ljk}}(t)}$ converge to p_lE_l{(1 − Δ_l1k)Y_l1k(t)Z_l1k(t)^⊗de^βTZ_l1k(t)} and p_lpr(Δ_l1k = 1)E_l{Y_l1k(t)Z_l1k(t)^⊗de^βTZ_l1k(t) | Δ_l1k = 1, ξ_l1 = 0 in probability uniformly in t under Condition (G), respectively. Thus, from (10)

$$\begin{array}{l}{n}^{1/2}\left\{{\stackrel{\sim }{S}}_{II,k}^{(d)}(\beta ,t)-S_k^{(d)}(\beta ,t)\right\}\\ =n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}(1-{\mathrm{\Delta }}_{\mathit{lik}})\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right)Y_{\mathit{lik}}(t)\\ \times \left[Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}-\frac{{\text{E}}_l\left\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)Z_{l1k}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{l1k}(t)}\right\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(t)\}}\right]\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}(1-{\xi }_{li}){\mathrm{\Delta }}_{\mathit{lik}}\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}-1\right)Y_{\mathit{lik}}(t)[Z_{\mathit{lik}}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{\mathit{lik}}(t)}\\ -\frac{{\text{E}}_l\left\{Y_{l1k}(t)Z_{l1k}{(t)}^{\otimes d}{e}^{{\beta }^T{Z}_{l1k}(t)}\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}}{{\text{E}}_l\{Y_{l1k}(t)\mid {\mathrm{\Delta }}_{l1k}=1\}}]+o_p(1)\end{array}$$ (11)

It then follows from lemma 2 that, for d = 0, 1 and 2, $n^{1/2}\left\{{\stackrel{\sim }{S}}_{II,k}^{(d)}(\beta ,t)-S_k^{(d)}(\beta ,t)\right\}$ converges weakly to zero-mean Gaussian processes under Condition (G). Consequently, together with condition (F),

$$\underset{\begin{array}{c}t\in [0,\tau ]\\ \beta \in \mathcal{B}\end{array}}{sup}\Vert {\stackrel{\sim }{S}}_{II,k}^{(d)}(\beta ,t)-s_k^{(d)}(\beta ,t)\Vert \stackrel{p}{\to }0\text{as}n\to \infty \text{for}d=0,1,\text{and}2.$$ (12)

Since $s_k^{(0)}(\beta ,t)$ is bounded away from zero on × [0, τ] by condition (F), it follows from the above convergence results that for k = 1, …, K, Ũ_k(β, t) converges to v_k(β, t) in probability uniformly in t and β.

By combining these results with the Lenglart inequality for $n^{-1}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{N}_{\mathit{lik}}(\tau )$ (Andersen & Gill, 1982, p1115), it follows that the first term on the right side of (8) converges to zero in probability, uniformly in β ∈ , as n → ∞.

The second term and the fourth terms on the right side of (8) can be shown to converge to zero by applying lemma 2. The third term can be shown to converge to zero by the Lenglart inequality for ${\sum }_{k=1}^K{\int }_0^{\tau }{v}_k(\beta ,t)n^{-1}d{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{M}_{\mathit{lik}}(t)$ (Andersen & Gill, 1982, p1115).

Conditions (D), (E) and (F) ensure the boundedness of sup_t,β{v_k(β, t)}_jj′ and Λ_0k(τ) for k = 1, …, K and j, j^′ = 1, …, p. Thus, together with the uniform convergence of ${\stackrel{\sim }{S}}_{II,k}^{(0)}(\beta ,t)$ to $s_k^{(0)}(\beta ,t)$ in probability, the last term on the right side of (8) converges to zero in probability, uniformly in β ∈ as n → ∞. Hence,

$$-\frac{\partial \left\{n^{-1}{\stackrel{\sim }{U}}_{II}(\beta )\right\}}{\partial {\beta }^T}\stackrel{p}{\to }A(\beta )\text{as}n\to \infty \text{uniformly}\text{in}\beta \in \mathcal{B}$$

and, thus, (ii) and (iii) are satisfied.

For (iv), we will show that n^−1/2Ũ_II (β₀) is asymptotically equivalent to

$$\begin{array}{l}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )+n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K(1-{\mathrm{\Delta }}_{\mathit{lik}})\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right){\int }_0^{\tau }[R_{\mathit{lik}}({\beta }_0,u)-Y_{\mathit{lik}}(u)\\ \times \frac{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})R_{l1k}({\beta }_0,u)\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}]d{\mathrm{\Lambda }}_{0k}(u)+n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}-1\right)\\ \times \left[{\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )-{\int }_0^{\tau }{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{\text{d}{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}\right].\end{array}$$ (13)

Specifically, one can decompose n^−1/2Ũ^II (β₀) into the following four parts:

$$\begin{array}{l}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0,\tau )+n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\int }_0^{\tau }\left\{e_k(\beta ,u)-\frac{{\stackrel{\sim }{S}}_{II,k}^{(1)}({\beta }_0,u)}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\right\}{dM}_{\mathit{lik}}({\beta }_0,u)\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\int }_0^{\tau }\{{\omega }_{\mathit{lik}}^{II}(u)-1\}{\stackrel{\sim }{Z}}_{\mathit{lik}}({\beta }_0,u){dM}_{\mathit{lik}}(u)\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\int }_0^{\tau }\{{\omega }_{\mathit{lik}}^{II}(u)-1\}\left\{e_k(\beta ,u)-\frac{{\stackrel{\sim }{S}}_{II,k}^{(1)}({\beta }_0,u)}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\right\}{dM}_{\mathit{lik}}({\beta }_0,u)\end{array}$$ (14)

The second term on the right-hand side of (14) can be shown to converge to zero uniformly in t. Note that, for fixed t, $n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{M}_{\mathit{lik}}({\beta }_0,t)$ is a sum of i.i.d. zero-mean random variables. Based on Conditions (C) and (E), M_lik(β₀, t) is of bounded variation and therefore can be written as a difference of two monotone functions in t. It then follows from the example of 2.11.16 of van der Vaart & Wellner (1996, p215) that $n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{M}_{\mathit{lik}}({\beta }_0,t)$ converges weakly to a zero-mean Gaussian process, say (t). It can be shown that E[{ (t) − (s)}⁴] ≤ C{Λ_0k(t) − Λ_0k(s)}² for some constant C > 0. Specifically, E[{ (t) − (s)}⁴] = 3(E[{ (t) − (s)}²])² since (t) is a zero-mean normal random variable for a fixed t. Then E[{ (t) − (s)}²] = E{ (t)²}+E{ (s)²}−2E{ (t) (s)} = E{ (t)²}− E{ (s)²} for s ≤ t. Since $\{\text{E}{\mathcal{W}}_{Mk}{(t)}^2\}=\text{E}\{M_{\mathit{lik}}{({\beta }_0,t)}^2\}=\text{E}\{{\int }_0^t{Y}_{\mathit{lik}}(u)e^{{\beta }_0^T{Z}_{\mathit{lik}}(u)}{\lambda }_{0k}(u)du\},\text{E}[{\{{\mathcal{W}}_{Mk}(t)-{\mathcal{W}}_{Mk}(s)\}}^2]=\text{E}\{{\int }_s^t{Y}_{\mathit{lik}}(u)e^{{\beta }_0^T{Z}_{\mathit{lik}}(u)}{\lambda }_{0k}(u)du\}\le e^{C_z}\text{E}\{{\int }_s^t{\lambda }_{0k}(u)du\}={\stackrel{\sim }{C}}_z\{{\mathrm{\Lambda }}_{0k}(t)-{\mathrm{\Lambda }}_{0k}(s)\}$ by the boundedness condition (C). Since Λ_0k(·) is differentiable and λ₀(·) is bounded on [0, τ], there exists a constant M, such that Λ_0k(t) − Λ_0k(s) ≤ M (t − s) for s ≤ t. Therefore, $\text{E}[{\{{\mathcal{W}}_{Mk}(t)-{\mathcal{W}}_{Mk}(s)\}}^2]\le C_z^{\ast }(t-s)$ and $\text{E}[{\{{\mathcal{W}}_{Mk}(t)-{\mathcal{W}}_{Mk}(s)\}}^4]\le 3{(\text{E}[{\{{\mathcal{W}}_{Mk}(t)-{\mathcal{W}}_{Mk}(s)\}}^2])}^2\le {\stackrel{\sim }{C}}_z^{\ast }{(t-s)}^2$ for some constant $C_z^{\ast }$. Then, by the Kolmogorov-Centsov Theorem (Karatzas & Shereve, 1988, p53), (t) has continuous sample paths. In addition, since ${\stackrel{\sim }{S}}_{II,k}^{(1)}({\beta }_0,t)/{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,t)$ is of bounded variation based on (12) and Conditions (C) and (F), we can write ${\stackrel{\sim }{S}}_{II,k}^{(1)}({\beta }_0,t)/{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,t)=Z_{k1}^{\ast }(t)-Z_{k2}^{\ast }(t)$ where both $Z_{k1}^{\ast }(t)$ and $Z_{k2}^{\ast }(t)$ are nonnegative, monotone in t and bounded. Therefore, ${\stackrel{\sim }{S}}_{II,k}^{(1)}({\beta }_0,t)/{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,t)$ is a sum of two monotone functions. Hence, it follows from lemma 1 that the second term on the right-hand side of (14) converges to 0 uniformly in t.

By similar arguments, the fourth term on the right-hand side of (14) can be shown to converge to 0 uniformly in t.

The third term on the right-hand side of (14) can be further decomposed as

$$\begin{array}{l}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K(1-{\mathrm{\Delta }}_{\mathit{lik}})\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right){\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K(1-{\mathrm{\Delta }}_{\mathit{lik}}){\xi }_{li}{\int }_0^{\tau }\left\{{\widehat{\alpha }}_{lk}^{-1}(u)-{\stackrel{\sim }{\alpha }}_l^{-1}\right\}{\stackrel{\sim }{Z}}_{\mathit{lik}}({\beta }_0,u){dM}_{\mathit{lik}}({\beta }_0,u)\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_k}-1\right){\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li}){\eta }_{\mathit{lik}}{\int }_0^{\tau }\left\{{\widehat{q}}_{lk}^{-1}(u)-{\stackrel{\sim }{q}}_{lk}^{-1}\right\}{\stackrel{\sim }{Z}}_{\mathit{lik}}({\beta }_0,u){dM}_{\mathit{lik}}({\beta }_0,u)\end{array}$$ (15)

The second term on the right side of (15) is asymptotically equivalent to

$$n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K(1-{\mathrm{\Delta }}_{\mathit{lik}})\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right){\int }_0^{\tau }{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})R_{l1k}({\beta }_0,u)\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}d{\mathrm{\Lambda }}_{0k}(u)$$ (16)

by (i) in lemma 3 and applying lemma 2. Likewise, by (ii) in lemma 3 and applying lemma 2, the last term on the right side of (15) is asymptotically equivalent to

$$n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(1-\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}\right){\int }_0^{\tau }{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{\text{d}{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}$$ (17)

By combining (16) and (17), (15) is asymptotically equivalent to

$$\begin{array}{l}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K(1-{\mathrm{\Delta }}_{\mathit{lik}})\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right){\int }_0^{\tau }\left[R_{\mathit{lik}}({\beta }_0,u)-Y_{\mathit{lik}}(u)\frac{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})R_{l1k}({\beta }_0,u)\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}\right]d{\mathrm{\Lambda }}_{0k}(u)\\ +n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_i)\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}-1\right)\\ \times \left[{\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )-{\int }_0^{\tau }{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{\text{d}{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}\right]\end{array}$$ (18)

Combining the above results, we have shown that n^−1/2Ũ_II (β₀) is asymptotically equivalent to (13). Under the regularity conditions, the first term on the right-hand side of (13) is asymptotically zero-mean normal with covariance matrix ${\sum }_{l=1}^L{p}_l{Q}_l({\beta }_0)$ where $Q_l({\beta }_0)=\text{E}{\{{\sum }_{k=1}^K{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0,\tau )\}}^{\otimes 2}$ by Spiekerman & Lin (1998). The second and the third terms on the right-hand side of (13) can be shown to be asymptotically zero-mean normal with covariance matrix ${\sum }_{l=1}^L{p}_l\frac{1-{\alpha }_l}{{\alpha }_l}{V}_l^{II,(1)}({\beta }_0)$ and ${\sum }_{l=1}^L{p}_l(1-{\alpha }_l){\sum }_{k=1}^{K}pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right)V_{lk}^{II,(2)}({\beta }_0)$ by lemma 2, respectively. It follows from conditional expectation arguments that these three terms are mutually independent. Therefore, n^−1/2Ũ_II (β₀) is asymptotically normally distributed with mean zero and with finite variance

$$\sum _{l=1}^L{p}_l\left\{Q_l({\beta }_0)+\frac{1-{\alpha }_l}{{\alpha }_l}{V}_l^{II,(1)}({\beta }_0)+(1-{\alpha }_l)\sum _{k=1}^{K}pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right)V_{lk}^{II,(2)}({\beta }_0)\right\}.$$

Hence n⁻¹Ũ_II (β) converges to zero in probability. Thus, (iv) is satisfied.

By (i),(ii),(iii) and (iv), it follows that there exists a unique sequence β̃_II s.t. n⁻¹Ũ_II (β̃_II ) = 0 with probability converging to one as n → 0 and with β̃_II converging in probability to β₀ by Theorem 2 (Foutz, 1977).

The asymptotic normality of β̃_II follows from the consistency of β̃_II and a Taylor series expansion of Ũ_II (β). This completes the proof.

The asymptotic properties of ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}({\stackrel{\sim }{\beta }}_{II},t)$ (k = 1, …, K) are summarized in the following theorem.

Theorem 4

Under Conditions (A) – (H), for each k = 1, …, K, ${\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}({\stackrel{\sim }{\beta }}_{II},t)$ converges in probability to Λ_0k(t) uniformly in t ∈ [0, τ]. Also, $W^{II}(t)=n^{1/2}{[\{{\stackrel{\sim }{\mathrm{\Lambda }}}_{01}^{II}({\stackrel{\sim }{\beta }}_{II},t)-{\mathrm{\Lambda }}_{01}(t)\},\dots ,\{{\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}({\stackrel{\sim }{\beta }}_{II},t)-{\mathrm{\Lambda }}_{0K}(t)\}]}^T$ converges weakly to a zero-mean Gaussian process (t) in D[0, τ]^K where ${\mathcal{W}}^{II}(t)={\{{\mathcal{W}}_1^{II}(t),\dots ,{\mathcal{W}}_K^{II}(t)\}}^T$. The covariance function between ${\mathcal{W}}_j^{II}(t_1)$ and ${\mathcal{W}}_k^{II}(t_2)$ is

$$\begin{array}{l}{\phi }_{jk}^{II}(t_1,t_2)({\beta }_0)=\sum _{l=1}^L{p}_l(E_l\{{\nu }_{l1j}({\beta }_0,t_1){\nu }_{l1k}({\beta }_0,t_2)\}+\frac{1-{\alpha }_l}{{\alpha }_l}{E}_l\{{\psi }_{l1j}^{II}({\beta }_0,t_1){\psi }_{l1k}^{II}({\beta }_0,t_2)\}+(1-{\alpha }_l)\\ \times [I(j=k)pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right){\mathit{Cov}}_l\left\{{\zeta }_{l1k}^{(1)}({\beta }_0,t_1),{\zeta }_{l1k}^{(1)}({\beta }_0,t_2)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}\\ +pr({\mathrm{\Delta }}_{l1j}=1)\left(\frac{1-q_{lj}}{q_{lj}}\right){\mathit{Cov}}_l\left\{{\zeta }_{l1j}^{(1)}({\beta }_0,t_1),r_k{({\beta }_0,t_2)}^{T}A{({\beta }_0)}^{-1}{\zeta }_{l1j}^{(2)}({\beta }_0,t_2)\mid {\mathrm{\Delta }}_{l1j}=1,{\xi }_{l1}=0\right\}\\ +pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right){\mathit{Cov}}_l\left\{{\zeta }_{l1k}^{(1)}({\beta }_0,t_2),r_j{({\beta }_0,t_1)}^{T}A{({\beta }_0)}^{-1}{\zeta }_{l1k}^{(2)}({\beta }_0,t_1)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}\\ +\sum _{m=1}^{K}pr({\mathrm{\Delta }}_{l1m}=1)\left(\frac{1-q_{lm}}{q_{lm}}\right)\\ \times r_j{({\beta }_0,t_1)}^{T}A{({\beta }_0)}^{-1}{\mathit{Cov}}_l\left\{{\zeta }_{l1m}^{(2)}({\beta }_0,t_1),{\zeta }_{l1m}^{(2)}({\beta }_0,t_2)\mid {\mathrm{\Delta }}_{l1m}=1,{\xi }_{l1}=0\right\}A{({\beta }_0)}^{-1}{r}_k({\beta }_0,t_2)]).\end{array}$$

where

$$\begin{array}{l}{\zeta }_{\mathit{lik}}^{(1)}(\beta ,t)={\int }_0^t\frac{1}{s_k^{(0)}(\beta ,u)}\left[{dM}_{\mathit{lik}}(\beta ,u)-Y_{\mathit{lik}}(u)\frac{{\text{E}}_l\{{dM}_{l1k}(\beta ,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}\right]\text{and}\\ {\zeta }_{\mathit{lik}}^{(2)}(\beta ,t)={\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}(\beta ,\tau )-{\int }_0^t{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{{d\mathbf{M}}_{\stackrel{\sim }{z},l1k}(\beta ,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}.\end{array}$$

PROOF

One can make the following decomposition

$$\begin{array}{l}{n}^{1/2}\{{\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}({\stackrel{\sim }{\beta }}_{II},t)-{\mathrm{\Lambda }}_{0k}(t)\}\\ =n^{1/2}{\int }_0^t\left\{\frac{1}{n{\stackrel{\sim }{S}}_{II,k}^{(0)}({\stackrel{\sim }{\beta }}_{II},u)}-\frac{1}{n{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\right\}d\left\{\sum _{l=1}^L\sum _{i=1}^{n_l}{M}_{\mathit{lik}}(\beta ,u)\right\}\\ +n^{1/2}{\int }_0^t\left\{\frac{1}{n{\stackrel{\sim }{S}}_{II,k}^{(0)}({\stackrel{\sim }{\beta }}_{II},u)}-\frac{1}{n{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\right\}d\left[\sum _{l=1}^L\sum _{i=1}^{n_l}\left\{{\omega }_{\mathit{lik}}^{II}(u)-1\right\}{M}_{\mathit{lik}}(\beta ,u)\right]\\ +n^{1/2}{\int }_0^t\left\{\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\stackrel{\sim }{\beta }}_{II},u)}-\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\right\}{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)d{\mathrm{\Lambda }}_{0k}(u)+{\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}d\left\{n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}{M}_{\mathit{lik}}(\beta ,u)\right\}\\ +{\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}d[n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left\{{\omega }_{\mathit{lik}}^{II}(u)-1\right\}{M}_{\mathit{lik}}(\beta ,u)\}\end{array}$$ (19)

By the Taylor expansion of ${\stackrel{\sim }{S}}_{II,k}^{(0)}{({\stackrel{\sim }{\beta }}_{II},u)}^{-1}$ around β₀, the first term on the right-hand side of (19) is equivalent to

$${\int }_0^t\left\{-\frac{{\stackrel{\sim }{S}}_{II,k}^{(1)}{({\beta }^{\ast },u)}^T}{{\stackrel{\sim }{S}}_{II,k}^{(0)}{({\beta }^{\ast },u)}^2}\right\}({\stackrel{\sim }{\beta }}_{II}-{\beta }_0)d\left\{n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}{M}_{\mathit{lik}}({\beta }_0,u)\right\}$$ (20)

where β^* is on the line segment between β̃_II and β₀. Then, as n → ∞, (20) converges to 0 uniformly in t in probability by lemma 1 since ${\stackrel{\sim }{S}}_{II,k}^{(1)}(\beta ,u)/{\stackrel{\sim }{S}}_{II,k}^{(0)}(\beta ,u)$ is of bounded variation, β̃_II is consistent for β₀, and $n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{M}_{\mathit{lik}}({\beta }_0,u)$ converges weakly to a zero-mean Gaussian process with continuous sample path. The second term can be shown to converge to 0 uniformly in t in probability by similar arguments.

Again, it follows from the Taylor expansion of ${\stackrel{\sim }{S}}_{II,k}^{(0)}{({\stackrel{\sim }{\beta }}_{II},u)}^{-1}$ around β₀, the uniform convergence of ${\stackrel{\sim }{S}}_{II,k}^{(1)}(\beta ,u)$ and ${\stackrel{\sim }{S}}_{II,k}^{(0)}(\beta ,u)$, the consistency of β̂_II for β₀ and the boundedness of Λ_0k(t) on [0, τ) that the third term is asymptotically equivalent to

$$n^{1/2}{r}_k{({\beta }_0,t)}^T\left({\stackrel{\sim }{\beta }}_{II}-{\beta }_0\right).$$

The fourth term can be shown to be asymptotically equivalent to

$${\int }_0^t\frac{1}{s_k^{(0)}({\beta }_0,u)}d\left\{n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}{M}_{\mathit{lik}}({\beta }_0,u)\right\}$$

by lemma 1 since ${\stackrel{\sim }{S}}_{II,k}^{(0)}(\beta ,u)$ converges to $s_k^{(0)}(\beta ,u)$ uniformly in t and $n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{M}_{\mathit{lik}}({\beta }_0,u)$ converges weakly to a zero-mean Gaussian process with continuous sample path. For the last term on the right-hand side of (19), it follows from lemma 3, and the uniform convergence of ${\stackrel{\sim }{S}}_{II,k}^{(0)}{({\beta }_0,t)}^{-1}$ to $s_k^{(0)}{({\beta }_0,t)}^{-1}$, where $s_k^{(0)}({\beta }_0,t)$ is bounded away from 0 that

$$\begin{array}{l}{\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}d\left[n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left\{{\omega }_{\mathit{lik}}^{II}(u)-1\right\}{M}_{\mathit{lik}}({\beta }_0,u)\right]\\ =n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}[\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}}){\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}{Y}_{\mathit{lik}}(u)e^{{\beta }_0^T{Z}_{\mathit{lik}}(u)}d{\mathrm{\Lambda }}_{0k}(u)\\ -(1-{\mathrm{\Delta }}_{\mathit{lik}}){\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\left\{{\widehat{\alpha }}_{lk}{(u)}^{-1}-{\stackrel{\sim }{\alpha }}_l^{-1}\right\}{\xi }_{li}{Y}_{\mathit{lik}}(u)e^{{\beta }_0^T{Z}_{\mathit{lik}}(u)}d{\mathrm{\Lambda }}_{0k}(u)\\ +{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_k}-1\right){\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}{dM}_{\mathit{lik}}({\beta }_0,u)\\ +{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li}){\int }_0^t\frac{1}{{\stackrel{\sim }{S}}_{II,k}^{(0)}({\beta }_0,u)}\left\{{\widehat{q}}_k{(u)}^{-1}-{\stackrel{\sim }{q}}_k^{-1}\right\}{\eta }_{\mathit{lik}}{dM}_{\mathit{lik}}({\beta }_0,u)]\\ =n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}(\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right)(1-{\mathrm{\Delta }}_{\mathit{lik}}){\int }_0^t{Y}_{\mathit{lik}}(u)\\ \times \left[e^{{\beta }_0^T{Z}_{\mathit{lik}}(u)}-\frac{{\text{E}}_l\left\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)e^{{\beta }_0^T{Z}_{l1k}(u)}\right\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}\right]\frac{d{\mathrm{\Lambda }}_{0k}(u)}{s_k^{(0)}({\beta }_0,u)}\\ -{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(1-\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}\right){\int }_0^t\frac{1}{s_k^{(0)}({\beta }_0,u)}\left[{dM}_{\mathit{lik}}(u)-Y_{\mathit{lik}}(u)\frac{{\text{E}}_l\{{dM}_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l(Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1)}\right]\end{array}$$ (21)

Now by combining the above results and using the asymptotic expansion of n^1/2(β̃_II − β₀) where

$$\begin{array}{l}{n}^{1/2}\left({\stackrel{\sim }{\beta }}_{II}-{\beta }_0\right)=A{({\beta }_0)}^{-1}{n}^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\sum _{k=1}^K({\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )\\ +(1-{\mathrm{\Delta }}_{\mathit{lik}})\left(\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}-1\right){\int }_0^{\tau }\left[R_{\mathit{lik}}({\beta }_0,u)-Y_{\mathit{lik}}(u)\frac{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})R_{l1k}({\beta }_0,u)\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}\right]d{\mathrm{\Lambda }}_{0k}(u)\\ +{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}-1\right)\\ \times \left[{\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}({\beta }_0,\tau )-{\int }_0^{\tau }{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{\text{d}{\mathbf{M}}_{\stackrel{\sim }{z},l1k}({\beta }_0,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}\right])+o_p(1),\end{array}$$

we have

$$n^{1/2}\{{\stackrel{\sim }{\mathrm{\Lambda }}}_{0k}^{II}({\stackrel{\sim }{\beta }}_{II},t)-{\mathrm{\Lambda }}_{0k}(t)\}=n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}{\nu }_{\mathit{lik}}({\beta }_0,t)+n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right){\psi }_{\mathit{lik}}^{II}({\beta }_0,t)+n^{-1/2}\sum _{l=1}^L\sum _{i=1}^{n_l}{\nu }_{\mathit{lik}}^{\ast }({\beta }_0,t)+o_p(1)$$

where

$$\begin{array}{l}{\nu }_{\mathit{lik}}(\beta ,t)=r_k{(\beta ,t)}^{T}A{(\beta )}^{-1}\sum _{m=1}^K{\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lim}}(\beta ,\tau )+{\int }_0^t{\{s_k^{(0)}(\beta ,u)\}}^{-1}{dM}_{\mathit{lik}}(\beta ,u),\\ {\psi }_{\mathit{lik}}^{II}(\beta ,t)=r_k{(\beta ,t)}^{T}A{(\beta )}^{-1}\sum _{m=1}^K(1-{\mathrm{\Delta }}_{\mathit{lim}}){\int }_0^T[R_{\mathit{lim}}(\beta ,u)\\ -\frac{Y_{\mathit{lim}}(u){\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1m})R_{l1m}(\beta ,u)\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1m})Y_{l1m}(u)\}}]d{\mathrm{\Lambda }}_{0m}(u)+(1-{\mathrm{\Delta }}_{\mathit{lik}}){\int }_0^t{Y}_{\mathit{lik}}(u)[e^{{\beta }^T{Z}_{\mathit{lik}}(u)}\\ -\frac{{\text{E}}_l\left\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)e^{{\beta }^T{Z}_{l1k}(u)}\right\}}{{\text{E}}_l\{(1-{\mathrm{\Delta }}_{l1k})Y_{l1k}(u)\}}]\frac{d{\mathrm{\Lambda }}_{0k}(u)}{s_k^{(0)}(\beta ,u)}\text{and}\\ {\nu }_{\mathit{lik}}^{\ast }(\beta ,t)=r_k{(\beta ,t)}^{T}A{(\beta )}^{-1}\sum _{m=1}^K{\mathrm{\Delta }}_{\mathit{lim}}(1-{\xi }_i)\left(\frac{{\eta }_{\mathit{lim}}}{{\stackrel{\sim }{q}}_{lm}}-1\right){\zeta }_{\mathit{lim}}^{(2)}(\beta ,t)+{\mathrm{\Delta }}_{\mathit{lik}}(1-{\xi }_{li})\left(\frac{{\eta }_{\mathit{lik}}}{{\stackrel{\sim }{q}}_{lk}}-1\right){\zeta }_{\mathit{lik}}^{(1)}(\beta ,t),\\ {\zeta }_{\mathit{lik}}^{(1)}(\beta ,t)={\int }_0^t\frac{1}{s_k^{(0)}(\beta ,u)}\left[{dM}_{\mathit{lik}}(\beta ,u)-Y_{\mathit{lik}}(u)\frac{{\text{E}}_l\{{dM}_{l1k}(\beta ,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}\right]\text{and}\\ {\zeta }_{\mathit{lik}}^{(2)}(\beta ,t)={\mathbf{M}}_{\stackrel{\sim }{z},\mathit{lik}}(\beta ,\tau )-{\int }_0^t{Y}_{\mathit{lik}}(u)\frac{{\text{E}}_l\{{d\mathbf{M}}_{\stackrel{\sim }{z},l1k}(\beta ,u)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\}}{{\text{E}}_l\{Y_{l1k}(u)\mid {\mathrm{\Delta }}_{l1k}=1\}}\end{array}$$

Now, let $W^{(1)}(t)={\{W_1^{(1)}(t),\dots ,W_K^{(1)}(t)\}}^T$ where $W_k^{(1)}(t)=n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{\nu }_{\mathit{lik}}({\beta }_0,t),W^{(2)}(t)={\{W_1^{(2)}(t),\dots ,W_K^{(2)}(t)\}}^T$ where $W_k^{(2)}(t)=n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}\left(1-\frac{{\xi }_{li}}{{\stackrel{\sim }{\alpha }}_l}\right){\psi }_{\mathit{lik}}^{II}({\beta }_0,t)$, and $W^{(3)}(t)={\{W_1^{(3)}(t),\dots ,W_K^{(3)}(t)\}}^T$ where $W_k^{(3)}(t)=n^{-1/2}{\sum }_{l=1}^L{\sum }_{i=1}^{n_l}{\nu }_{\mathit{lik}}^{\ast }({\beta }_0,t)$ for k = 1, …, K. Then, W ⁽¹⁾(t) converges weakly to a zero-mean Gaussian process ${\mathcal{W}}^{(1)}(t)={\{{\mathcal{W}}_1^{(1)}(t),\dots ,{\mathcal{W}}_K^{(1)}(t)\}}^T$ in D[0, τ]^K where the covariance function between ${\mathcal{W}}_j^{(1)}(t_1)$ and ${\mathcal{W}}_k^{(1)}(t_2)$ is ${\sum }_{l=1}^L{p}_l{\text{E}}_l\{{\nu }_{l1j}({\beta }_0,t_1){\nu }_{l1k}({\beta }_0,t_2)\}$ by Spiekerman & Lin (1998, Thm.2). W ⁽²⁾(t) also can be shown to converge weakly to a zero-mean Gaussian process ${\mathcal{W}}^{(2)}(t)={\{{\mathcal{W}}_1^{(2)}(t),\dots ,{\mathcal{W}}_K^{(2)}(t)\}}^T$. For any finite number of time points (t₁, …, t_D), the finite dimensional distribution of W ⁽²⁾(t) is asymptotically the same as those of (t) by lemma 2 and Cramer-Wold device. Since the space D[0, τ]^K is equipped with the uniform metric, it suffices to show the marginal tightness of $W_k^{(2)}(t)$ for each k. The marginal tightness follows directly by applying lemma 2 to $W_k^{(2)}(t)$. Thus, W ⁽²⁾(t) converges weakly to a zero-mean Gaussian process where the covariance function between ${\mathcal{W}}_j^{(2)}(t_1)$ and ${\mathcal{W}}_k^{(2)}(t_2)$ is ${\sum }_{l=1}^L{p}_l\frac{1-{\alpha }_l}{{\alpha }_l}{\text{E}}_l\{{\psi }_{l1j}^{II}({\beta }_0,t_1){\psi }_{l1k}^{II}({\beta }_0,t_2)\}$. The weak convergence W⁽³⁾(t) to a zero-mean Gaussian process (t) follows from the similar arguments with the covariance function between ${\mathcal{W}}_j^{(2)}(t_1)$ and ${\mathcal{W}}_k^{(3)}(t_3)$ being

$$\begin{array}{l}\sum _l^L{p}_l(1-{\alpha }_l)[I(j=k)pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right){\mathit{Cov}}_l\left\{{\zeta }_{l1k}^{(1)}({\beta }_0,t_1),{\zeta }_{l1k}^{(1)}({\beta }_0,t_2)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}\\ +pr({\mathrm{\Delta }}_{l1j}=1)\left(\frac{1-q_{lj}}{q_{lj}}\right){\mathit{Cov}}_l\left\{{\zeta }_{l1j}^{(1)}({\beta }_0,t_1),r_k{({\beta }_0,t_2)}^{T}A{({\beta }_0)}^{-1}{\zeta }_{l1j}^{(2)}({\beta }_0,t_2)\mid {\mathrm{\Delta }}_{l1j}=1,{\xi }_{l1}=0\right\}\\ +pr({\mathrm{\Delta }}_{l1k}=1)\left(\frac{1-q_{lk}}{q_{lk}}\right){\mathit{Cov}}_l\left\{{\zeta }_{l1k}^{(1)}({\beta }_0,t_2),r_j{({\beta }_0,t_1)}^{T}A{({\beta }_0)}^{-1}{\zeta }_{l1k}^{(2)}({\beta }_0,t_1)\mid {\mathrm{\Delta }}_{l1k}=1,{\xi }_{l1}=0\right\}\\ +\sum _{m=1}^{K}pr({\mathrm{\Delta }}_{l1m}=1)\left(\frac{1-q_{lm}}{q_{lm}}\right)\\ \times r_k{({\beta }_0,t_1)}^{T}A{({\beta }_0)}^{-1}{\mathit{Cov}}_l\left\{{\zeta }_{l1m}^{(2)}({\beta }_0,t_1),{\zeta }_{l1m}^{(2)}({\beta }_0,t_2)\mid {\mathrm{\Delta }}_{l1m}=1,{\xi }_{l1}=0\right\}A{({\beta }_0)}^{-1}{r}_j({\beta }_0,t_2)].\end{array}$$

It follows from the conditional expectation argument that these three terms are mutually independent. Therefore, W^II(t) = W ⁽¹⁾(t) + W ⁽²⁾(t) + W ⁽³⁾(t) converges weakly to a zero-mean Gaussian process (t) = (t) + (t) + (t) where the covariance function between ${\mathcal{W}}_j^{II}(t_1)$ and ${\mathcal{W}}_k^{II}(t_2)$ is ${\phi }_{jk}^{II}(t_1,t_2)({\beta }_0)$. This completes the proofs.

3.4 Proofs of Theorems 1 and 2

Proofs for Theorems 1 and 2 basically follow the same steps used for those for Theorems 3 and 4. However, the steps involving the asymptotic expansions of α̂_lk(t)⁻¹ and q̂_lk(t)⁻¹ around α̃_l and q̃_lk (lemma 3) can now be omitted. Specifically, the third and fourth terms in (9) and (10), and the second and fourth terms in (15) and (21) vanish.

4 Discussion

In this paper, we considered fitting marginal hazards models for failure time data with multiple disease outcomes from two types of stratified case-cohort study designs: the original and the generalized stratified case-cohort designs. In either design, subcohort members are sampled via stratified random sampling with possibly different sampling proportions within each stratum where the strata were constructed based on the information available for the entire cohort members. After the selection of the subcohort members, we sample all the remaining cases outside the subcohort under the original stratified case-cohort design whereas we are allowed to select a part of cases outside the subcohort via stratified random sampling under the generalized stratified case-cohort design. For estimation, we proposed weighted estimating equation approach for regression parameters and Breslow-Aalen type estimator for cumulative baseline hazards functions. We also provided a detailed proofs for deriving the asymptotic properties of the proposed estimators. The proposed estimators were shown to have desirable asymptotic properties such as consistency and asymptotic normality.

One modification to the generalized case-cohort study design in the current paper might be worth considering. Instead of sampling cases outside the sub-cohort within each stratum separately, one might want to sample cases outside the subcohort from the whole cohort regardless of their strata. Our proposed methods can be easily adapted to this design simply by redefining the strata, i.e., defining cases outside the subcohort as a separate single stratum.