A varying-coefficient partially linear transformation model for length-biased data with an application to HIV vaccine studies

Abstract

Prevalent cohort studies in medical research often give rise to length-biased survival data that require special treatments. The recently proposed varying-coefficient partially linear transformation (VCPLT) model has the virtue of providing a more dynamic content of the effects of the covariates on survival times than the well-known partially linear transformation (PLT) model by allowing flexible interactions between the covariates. However, no existing analysis of the VCPLT model has considered length-biased sampling. In this paper, we consider the VCPLT model when the data are length-biased and right censored, thereby extending the reach of this flexible and powerful tool. We develop a martingale estimating function-based approach to the estimation of this model, provide theoretical underpinnings, evaluate finite sample performance via simulations, and showcase its practical appeal via an empirical application using data from two HIV vaccine clinical trials conducted by the U.S. National Institute of Allergy and Infectious Diseases.

1. Introduction

Length-biased data frequently arise in prevalent cohort survival studies where only the individuals alive at the recruitment time are eligible for inclusion, and as such, the observed time intervals from initiation to failure for the cohort are longer than those for the general population. Length-biased data are typically also right-censored as it is not uncommon for some individuals to withdraw from the study before it ends.

When the data are length-biased, the probability of sampling an individual from the target population is directly proportional to the individual’s survival time. Ignoring the length-biased characteristic of the data has the adverse effect of overestimating the survival time for the population and drawing incorrect inference. A number of alternative methods for estimating the unbiased survival distribution from length-biased data have been proposed. These methods are mostly likelihood-based. Studies on methods that are conditional on the observed truncation times include Turnbull (1976), Lagakos et al. (1988) and Wang (1991). Studies by the unconditional approach all assume that the initiation times follow a stationary Poisson process (see Vardi, 1982, 1985; Gill et al., 1988; Asgharian et al., 2002; Asgharian and Wolfson, 2005; Luo and Tsai, 2009, among others). The unconditional approach is more difficult to implement, but it generally yields superior estimators to those obtained by the conditional approach.

The more recent strand of the length-biased sampling literature has moved beyond the estimation of survival distributions to address the relationship between survival times and the covariates. The dominant approaches to modeling survival times under length biased sampling have been the use of the Cox’s proportional hazards (PH) and the accelerated failure time (AFT) models (see, for example, Wang, 1996; Tsai, 2009; Ning et al., 2014). Shen et al. (2009) and Cheng and Huang (2014) considered the semiparametric linear transformation (SLT) model (Cheng et al., 1995) under length-biased sampling. The SLT model is a flexible formulation that includes the PH and AFT models as special cases. More recently, Wei et al. (2018) considered the partially linear transformation (PLT) model (Lu and Zhang, 2010) when the data are length-biased. The PLT model has an advantage over the SLT model in that it can account for non-linear covariate effects on the survival times. Wei et al. (2018) developed an estimating equation approach for estimating the covariate effects and proved that the proposed method has an optimal asymptotic property. In a recent article, Qiu and Zhou (2015) considered the varying-coefficient PLT (VCPLT) model, which extends the PLT model through the inclusion of varying-coefficients to allow for interaction effects between the covariates. Qiu and Zhou (2015) developed a martingale-based estimation method along the lines of Chen et al. (2002) and established an asymptotic theory of the method. However, Qiu and Zhou (2015) did not consider length-biased sampling in their work.

The object of this paper is to extend the reach of the VCPLT model to the case where the data are length-biased and right-censored. Compared to the PLT model in Wei et al. (2018), the VCPLT model provides a more dynamic content of the covariate effects on the survival times via the interactions between the covariates. In clinical trials and biomedical studies, the treatment effect often varies with other covariates. Our analysis is motivated by the HIV Vaccine Trials Network (HVTN) data shown in Section 6. One question of particular interest is whether one’s antibody response to Adenovirus Serotype 5 (Ad5), a common respiratory infection, depends on one’s past exposure to infectious disease of the herpes simplex virus type. We use the VCPLT model to investigate, among other things, how the level of the Ad5 titer modifies the association of herpes simplex viral infection with the age at HIV infection.

We refine the martingale-based estimation procedure of Wei et al. (2018) to account for the varying-coefficients, develop an iterative algorithm for computing the estimates and establish the asymptotic properties of the resultant estimators. In addition, we apply the proposed method to data from the HIV Vaccine Trials Network (HVTN)/Merck Inc. 502 ‘Step’ and HVTN 503 ‘Phambili’ studies conducted by the U.S. National Institute of Allergy and Infectious Diseases (NIAID) and Merck, Inc. Despite the fact that length-biased sampling is a naturally occurring phenomenon in many medical studies, publicly available data with length-biased characteristics are scarce. To the best of our knowledge, the dementia data set from the Canadian Study of Health and Aging (CSHA) is the only medical data set that has been used to examine the relationship between survival times and covariates when data are length-biased. Examples include Shen et al. (2009), Ning et al. (2011, 2014) and Wang and Wang (2014). These applications typically treat survival time as a function of diagnostic classification. There have been other studies based on non-medical data sets (Wang, 1996; Tsai, 2009; Chen et al., 2014; Wang and Wang, 2014; Wei et al., 2018), but otherwise studies on covariate effects on survival times under length-biased sampling with medical applications have been limited to the CSHA dementia data set. We consider our application to the HVTN data a noteworthy aspect of our work.

The reminder of this article is organised as follows. Section 2 describes the model framework. Section 3 develops the estimation methodology and an iterative algorithm for computing the estimates. Section 4 is devoted to an analysis of the estimators’ asymptotic properties and the development of a resampling process for estimating the asymptotic variance. Section 5 reports results of a simulation study that investigates the finite sample performance of the proposed estimator. In Section 6, we apply the proposed methodology to the HVTN 502 ’Step’ and 503 ’Phambili’ data sets. Some concluding remarks are contained in Section 7. Proofs of technical results are relegated to the Appendix.

2. Length-biased data and model framework

Let $\tilde{T}$ be the failure time of interest measured from the time of initiation to the time of failure, A be the truncation time measured from the initiation time to the time of enrolment, and V be the residual survival time measured from the time of enrollment to the time of failure. Under prevalent sampling, only subjects who are alive at the time of sampling are recruited, hence one can only observe T = A + V, the length-biased version of $\tilde{T}$, within the subset of $\tilde{T}>A$. Due to loss of follow-up, which is a commonplace with clinical trial studies, V is often right-censored. We let C be the censoring variable measured from the time of enrolment to censoring. Usually, C is assumed to be independent of A and V. The total censoring time is thus A + C. Asgharian and Wolfson (2005) showed that except for trivial situations, cov(A + V, A + C) = cov(A, V) + Var(A) > 0. Thus, T and A+C may be dependent as they share a common component A, and the length-biased data are informatively censored. This poses a challenge for the analysis because common methods for handling right-censored data typically do not take this feature into account.

Our modeling framework is the following varying-coefficient partially linear transformation (VCPLT) model (Qiu and Zhou, 2015):

$$H(\tilde{T})=-{\mathit{\beta }}^T\mathit{X}-{\mathit{f}}^T(Z)\mathit{W}+ϵ,$$ (1)

where H(⋅) is a monotonic increasing but otherwise unknown transformation function, X is a p × 1 dimensional time-independent covariate, β is a p × 1 vector of unknown coefficients, W is a q × 1 dimensional covariate that may interact with some exposure covariate Z, f(⋅) is an unspecified q × 1 dimensional smooth vector function with f(0) = 0, and ϵ is an error term with a completely specified distribution and independent of X, Z and W. We use λ_ϵ(t) and Λ_ϵ(t) to denote the hazard function and the cumulative hazard function of ϵ respectively. The inclusion of W in the VCPLT model allows the interaction effects between the covariates to be modeled and assessed. When W = 1, the VCPLT model reduces to the partially linear transformation (PLT) model (Lu and Zhang, 2010; Wei et al., 2018), which does not take into account the possibility of interactions between the covariates.

The observed data set {(A_i, Y_i, δ_i, X_i, Z_i, W_i), i = 1, …, n} consists of n independently and identically distributed (i.i.d) realisations from the population (A, Y, δ, X, Z, W), where Y = min(T,A + C), δ = I(V ≤ C), and I(⋅) is an indicator function. We assume that C and (A, V) are independent given the covariates X, Z and W. We denote the survival function of C as S_C(⋅), which may depend on the covariates.

3. Estimation methodology and a computational algorithm

3.1. Estimation methodology

The purpose of this section is to develop a strategy for estimating β, H(⋅) and f(⋅) in (1), under the assumption of length-biased and right-censored data. Denote the conditional density function and survival function of $\tilde{T}$ given the covariates X, Z and W as $f_{\tilde{T}}(t\mid \mathit{X},Z,\mathit{W})$ and $S_{\tilde{T}}(t\mid \mathit{X},Z,\mathit{W})$ respectively. Under the stationarity assumption of length-biased data, the conditional density function of T (Shen et al., 2009) is

$$f_{LB}(t\mid \mathit{X},Z,\mathit{W})=\frac{t{f}_{\tilde{T}}(t\mid \mathit{X},Z,\mathit{W})}{u(\mathit{X},Z,\mathit{W})},$$

where $u(\mathit{X},Z,\mathit{W})={\int }_0^{\infty }t{f}_{\tilde{T}}(t\mid \mathit{X},Z,\mathit{W})dt<\infty$ is a normalising constant. In the absence of right censoring, Asgharian and Wolfson (2005) showed that the conditional density function of A given V, X, Z and W is

$$f_{A\mid V}(A=a\mid V=v,\mathit{X},Z,\mathit{W})=\frac{f_{\tilde{T}}(a+v\mid \mathit{X},Z,\mathit{W})}{S_{\tilde{T}}(v\mid \mathit{X},Z,\mathit{W})}.$$ (2)

Under right censoring, denote $\tilde{V}=\min (V,C)$ as the observed residual lifetime. It follows from Huang and Qin (2012) that the conditional density function of A = a given δ = 1, $\tilde{V}=v$, X, Z and W is

$$P(A=a\mid \delta =1,\tilde{V}=v,\mathit{X},Z,\mathit{W})=\frac{f_{\tilde{T}}(a+v\mid \mathit{X},Z,\mathit{W})}{S_{\tilde{T}}(v\mid \mathit{X},Z,\mathit{W})}.$$ (3)

Comparing the density functions in (2) and (3), it is seen that when δ = 1, the conditional density function of A given $\tilde{V}$ is same as the conditional density function of A given V in the absence of right censoring. This exchangeability property has the potential of improving estimation efficiency. Wei et al. (2018) utilised this property and proposed a combined set of unbiased estimating equations for length-biased and right-censored data based on the PLT model. We generalise Wei et al.’s (2018) work from the PLT model to the VCPLT model.

Let us define $N_i(t)=I(Y_i\le t){\delta }_i,R_i^1(t)=I(A_i\le t\le Y_i)$, and $R_i^2(t)={\delta }_{i}I({\tilde{V}}_i\le t\le Y_i),i=1,\dots ,n$. We consider the risk function $R_i(t)=w_1{R}_i^1(t)+w_2{R}_i^2(t)$ that recognises auxiliary information, where w₁ and w₂ are composite weight that satisfy w₁ + w₂ = 1. Write

$$M_i(t)=N_i(t)-{\int }_0^t{R}_i(t)d{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\},$$ (4)

where ${(H_0,{\mathit{\beta }}_0^T,{\mathit{f}}_0^T)}^T$ consists of the true values of (H, β^T, f^T)^T. From the theory of counting process, it is clear that M_i(t) is a mean zero martingale process. Based on the mean zero property of M_i(t), assuming that f(⋅) is fixed, we can construct the following global estimating equations for β and H(⋅):

$$\sum _{i=1}^n[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{H(t)+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{f}}^T(Z_i){\mathit{W}}_i\}]=0,$$ (5)

and

$$\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_i[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{H(t)+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{f}}^T(Z_i){\mathit{W}}_i\}]=0,$$ (6)

where τ = inf{t : Pr(Y > t) = 0}, and H(⋅) is a nondecreasing function that satisfies H(0) = −∞ and has positive jumps only at the K uncensored observations 0 < t₁ < ⋯ < t_K. We assume that f(⋅) is sufficiently smooth for it to be estimated by local polynomial methods. Specifically, we assume that, for a given point u,

$$\mathit{f}(u)\approx \mathit{f}(z)+\dot{\mathit{f}}(z)(u-z)\equiv {\mathit{\alpha }}_0(z)+{\mathit{\alpha }}_1(u-z),$$

where z is in the neighbourhood of u and $\dot{\mathit{f}}(z)=d\mathit{f}(z)/dz$. Substituting the above into (5) and (6) results in the following local estimating equations that form the basis for the estimation of f(⋅):

$$\begin{gathered} \sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z)\left(\begin{array}{c}{\mathit{W}}_i\\ (Z_i-z){\mathit{W}}_i\end{array}\right)[d{N}_i(t)-R_i(t) \\ \times d{\text{Λ}}_{ϵ}\{H(t)+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{\alpha }}_0^T(z)Z_i+{\mathit{\alpha }}_1^T(z){\mathit{W}}_i(Z_i-z)\}]=0, \end{gathered}$$ (7)

where K(⋅) is a kernel function, K_h(⋅) = K(⋅/h)/h, and h denotes the bandwidth. The estimators of β, H(⋅) and f(⋅) may be obtained by solving the estimating equations (5), (6) and (7) iteratively. Our iterative algorithm, to be presented in the following subsection, is similar to those used in Carroll et al. (1997), Qiu and Zhou (2015) and Wei et al. (2018).

3.2. Computational algorithm

Our algorithm for computing β, H(⋅) and f(⋅) is as follows:

Step 0: Determine w₁ and w₂ that can achieve the highest efficiency. Let $\widetilde{{\mathit{f}}^{(0)}}(\cdot )$ be an initial value of f(⋅). Substitute $\widetilde{{\mathit{f}}^{(0)}}(\cdot )$ for f(⋅) in (5) and (6) and solve for H(⋅) and β by applying the algorithm of Chen et al. (2002). Denote the solutions as $\tilde{H}(\cdot )$ and $\tilde{\mathit{\beta }}$. Readers may refer to Remark 4 for the method for selecting w₁ and w₂.

Step 1: Given $\tilde{H}(\cdot )$ and $\tilde{\mathit{\beta }}$, obtain a solution for each of α₀(z) and α₁(z) by solving (7) at the observed points z = Z_i, i = 1, …, n. Denote the solutions as ${\tilde{\mathit{\alpha }}}_0(Z_i)$ and ${\tilde{\mathit{\alpha }}}_1(Z_i)$, and let $\tilde{\mathit{f}}(Z_i)={\tilde{\mathit{\alpha }}}_0(Z_i)$ be the estimator of f(Z_i), i = 1, …, n.

Step 2: Obtain a new set of $\tilde{H}(\cdot )$ and $\tilde{\mathit{\beta }}$ by solving (5) and (6) with f(Z_i) replaced by $\tilde{\mathit{f}}(Z_i)$, i = 1, …, n.

Step 3: Repeat Steps 1 and 2 until the solutions of β and H(⋅) converge. Denote the final estimators of β and H(⋅) as $\widehat{\mathit{\beta }}$ and $\widehat{H}(\cdot )$ respectively.

Step 4: Repeat Step 1 with $\widehat{H}(\cdot )$ and $\widehat{\mathit{\beta }}$ replacing $\tilde{H}(\cdot )$ and $\tilde{\mathit{\beta }}$ respectively. Let $\widehat{\mathit{f}}(z)$, the solution to α₀(z), be the final estimator of f(⋅) at the selected grid points, z = z_i, i = 1, …, s.

Remark 1. The algorithm entails the selection of an initial estimator ${\tilde{\mathit{f}}}^{(0)}$ for f(⋅). Similar to the work of Lu and Zhang (2010), Cai et al. (2007) and Qiu and Zhou (2015), we propose choosing ${\tilde{\mathit{f}}}^{(0)}$ by solving the following estimating equations:

$$\sum _{i=1}^n{K}_h(Z_i-z)[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{H(t)+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{\alpha }}_0^T(z){\mathit{W}}_i+{\mathit{\alpha }}_1^T(z){\mathit{W}}_i(Z_i-z)\}]=0$$ (8)

and

$$\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z)\left(\begin{array}{c}{\mathit{X}}_i\\ {\mathit{W}}_i(Z_i-z)\\ {\mathit{W}}_i\end{array}\right)[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{H(t)+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{\alpha }}_0^T(z){\mathit{W}}_i+{\mathit{\alpha }}_1^T(z){\mathit{W}}_i(Z_i-z)\}]=0.$$ (9)

Let ${\tilde{H}}_z(\cdot )$, $\tilde{\mathit{\beta }}(z)$, ${\tilde{\mathit{\alpha }}}_0(z)$ and ${\tilde{\mathit{\alpha }}}_1(z)$ be the solutions of (8) and (9). The initial estimator ${\tilde{\mathit{f}}}^{(0)}$ of f(⋅) is ${\tilde{\mathit{\alpha }}}_0(z)$.

Remark 2. The convergence criterion of our algorithm is based on the following l₂-norm:

$${\text{Δ}}^{(m)}={\left\{\sum _{j=1}^p{\left({\tilde{\mathit{\beta }}}_j^{(m)}-{\tilde{\mathit{\beta }}}_j^{(m-1)}\right)}^2+\sum _{j=1}^K{\left({\tilde{H}}^{(m)}(t_j)-{\tilde{H}}^{(m-1)}(t_j)\right)}^2\right\}}^{\frac{1}{2}},$$

where ${\tilde{\mathit{\beta }}}^{(m)}$ and ${\tilde{H}}^{(m)}(\cdot )$ are the solutions to β and H(⋅) respectively at the m^th iteration. The iterations end when Δ^(m) is less than a prescribed threshold value. We set the maximum iteration number to be 30.

Remark 3. Our estimation procedure involves selecting bandwidth parameters. Following Qiu and Zhou (2015) and Wei et al. (2018), we use the optimal bandwidth $h_1=C_0{n}^{-\frac{1}{5}}$ for estimating f(⋅), where C₀ is a constant that can be obtained by methods such as cross-validation, the approach of Cai et al. (2007), or rule-of-thumb. For the estimation of β and H(⋅), following Carroll et al. (1997) and Cai et al. (2007), we use the following ad-hoc bandwidth: $h_2=h_1\times n^{\frac{1}{5}}\times n^{-\frac{1}{3}}=C_0{n}^{-\frac{1}{3}}$. In our numerical analysis, we let C₀ be the standard deviation of Z.

Remark 4. Our estimation procedure involves selecting the weight w₁. Let {w_l : w_l ∈ (0, 1), l = 1, …, L} be the weight set. We select the optimal weight w₁ = w_l*, where $l^{\ast }={\text{argmin}}_{l=1,\dots ,L}{\sum }_{j=1}^P\mathit{SD}({\mathit{\beta }}_j)$, and SD is the standard deviation of estimates β_j, j = 1, …, P.

4. Asymptotic properties and estimation of asymptotic variance

4.1. Asymptotic properties

The purpose of this section is to investigate the asymptotic properties of the estimators $\widehat{\mathit{\beta }}$, $\widehat{H}(\cdot )$, ${\widehat{\mathit{\alpha }}}_0$ and ${\widehat{\mathit{\alpha }}}_1$. Let us define $k_2=\int u^{2}K(u)du,{\nu }_2=\int u^2{K}^2(u)du$,

$$B_1(t)=E[R(t){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}],$$

$$B_2(t)=E[R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}],$$

$$B(s,t)=\exp \left\{{\int }_s^t\frac{B_1(u)}{B_2(u)}d{H}_0(u)\right\},$$

$${\mathit{B}}_1^X(t)=E[\mathit{X}R(t){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}],$$

$${\mathit{B}}_2^X(t)=E[\mathit{X}R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}],$$

$$\mathit{\mu }(t)=\frac{1}{B_2(t)}\left\{{\mathit{B}}_2^X(t)+{\int }_t^{\tau }\left[{\mathit{B}}_1^X(s)-\frac{{\mathit{B}}_2^X(s)B_1(s)}{B_2(s)}\right]B(t,s)d{H}_0(s)\right\},$$

$${\lambda }_{ϵ}^{\ast }\{H_0(t)\}=B(t,0),\text{and}$$

$${\text{Λ}}_{ϵ}^{\ast }(x)={\int }_{-\infty }^x{\lambda }_{ϵ}^{\ast }(u)du,\text{for}x\in (-\infty ,\infty ),$$

for t ∈ (0, τ], where ${\dot{\lambda }}_{ϵ}(t)$ is the first-order derivative of λ_ϵ(t).

Furthermore, let

$${\mathit{A}}_1={\int }_0^{\tau }E[\{\mathit{X}-\mathit{\mu }(t)\}{\mathit{X}}^T{\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}R(t)]d{H}_0(t),\text{and}$$

$${\mathit{A}}_2={\int }_0^{\tau }E[\{\mathit{X}-\mathit{m}(t)\}\frac{{\mathit{e}}_1^T(\mathit{Z})}{{\mathit{e}}_{31}(\mathit{Z})}{\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}R(t)]d{H}_0(t),$$

where $\mathit{m}(t)=\mathit{\alpha }(t){\lambda }_{ϵ}^{\ast }\{H_0(t)\}/B_2(t)$, α(t) is a solution to the following Fredholm integral equation of the second kind (see, for example, Press et al., 1992, pp.782-785):

$$\mathit{\alpha }(t)-{\int }_0^{\tau }{\mathit{D}}_1(s,t)\mathit{\alpha }(s)d{H}_0(s)={\mathit{D}}_2(t),t\in [0,\tau ],$$

and D₁(⋅, ⋅), D₂(⋅), e₁(⋅) and e₃₁(⋅) are defined in the Appendix. Also, define

$$\text{Σ}=E{\left({\int }_0^{\tau }\{\mathit{X}-\mathit{m}(t)-({\mathit{X}}^{\ast }-{\mathit{m}}^{\ast })\}dM(t)\right)}^{\otimes 2},$$

where b^⊗2 = bb^T, and for l = 1, …, n,

$${\mathit{X}}_l^{\ast }={\int }_0^{\tau }E[\mathit{X}R(t){\mathit{W}}^T{\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}\mid Z=Z_l]{\mathit{e}}_{31}^{-1}(Z_l){\mathit{W}}_{l}g(Z_l)d{H}_0(t)$$

and

$${\mathit{m}}_l^{\ast }={\int }_0^{\tau }\mathit{m}(t)E[R(t){\mathit{W}}^T{\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z){\mathit{W}}^T\}\mid Z=Z_l]{\mathit{e}}_{31}^{-1}(Z_l){\mathit{W}}_{l}g(Z_l)d{H}_0(t).$$

Our derivation of the asymptotic properties of estimators require the following conditions:

• C1The covariates X, Z and W have compact support; the density g(⋅) of covariate Z is bounded, and its second derivative exists.
• C2The parameter β₀ is in the interior of a known compact set 𝓑₀; f₀ has a continuous and positive second derivative.
• C3λ_ϵ(⋅) is positive and has a continuous derivative; $\psi (\cdot )={\dot{\lambda }}_{ϵ}(\cdot )/{\lambda }_{ϵ}(\cdot )$ is continuous, and lim_t→∞ λ_ϵ(t) = 0 = lim_t→∞ ψ(t).
• C4τ is finite with $P(\tilde{T}>\tau )>0$.
• C5nh² / log(1/h) → ∞ and nh⁴ → 0, as n → ∞.
• C6There exist positive constants c₀ and c₁, such that sup_t∈[0,τ] B₂(t) > C₀, and sup_t∈[0,τ] [B₂(t) + |B₁(t)|] ≤ C₁,
• C7${\sup }_{t\in [0,\tau ]}{\int }_0^{\tau }|{\mathit{D}}_1(s,t)|d{H}_0(s)<\infty$,
• C8The matrices Σ and A = A₁ − A₂ are nondegenerate.

Theorem 1. Assume that Conditions (C1)-(C8) hold, and there exists a unique consistent estimator $\widehat{\beta }$ within a small neighborhood of β₀. Then as n → ∞, we have $\widehat{\mathit{\beta }}{\to }_p{\mathit{\beta }}_0$, and

$$\sqrt{n}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0){\to }_{d}N(0,{\mathit{A}}^{-1}\mathbf{\Sigma }{({\mathbf{A}}^{-1})}^T),$$

where ”→_p” and ”→_d” denote convergence in probability and convergence in distribution respectively.

Theorem 2. Assume that Conditions (C1)-(C8) hold. Then for t ∈ (0, τ], we have

$$\sqrt{n}(\widehat{H}(t)-H_0(t))=\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{{𝒦}_i(t)}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}+o_p(1),$$

where 𝒦_i(t)’s (defined in the Appendix) are independent processes with mean zero. In other words, $\sqrt{n}(\widehat{H}(t)-H_0(t))$ converges weakly to a mean zero Gaussian process.

Theorem 3. Assume that Conditions (C1)-(C8) hold and nh⁵ is bounded. Then as n → ∞,

$$\sqrt{nh}\left(\left[\begin{array}{c}{\widehat{\mathit{\alpha }}}_0(z)-{\mathit{f}}_0(z)\\ h({\widehat{\mathit{\alpha }}}_1(z)-{\dot{\mathit{f}}}_0(z))\end{array}\right]-{\mathit{b}}_n(z)\right){\to }_{d}N(0,{\mathbf{\Gamma }}_1^{-1}(z){\mathbf{\Sigma }}_2(z){\mathbf{\Gamma }}_1^{-1}(z)),$$

where b_n(z), Γ₁(z) and Σ₂(z) are defined in the Appendix.

The results given in the above theorems facilitate the construction of confidence intervals for β, H(⋅) and f(⋅).

4.2. Variance estimation

The asymptotic variances of the estimators given in the above theorems have complicated expressions; in particular, the evaluation of Σ involves solving integral equations which can be cumbersome. Here, we introduce an alternative bootstrap method that may be used to estimate the variances of the estimators. Let Φ_n be the empirical distribution with probability 1/n for each observation of (A_i, Y_i, δ_i, X_i, Z_i, W_i), i = 1, …, n. Generate a bootstrap sample of n i.i.d observations ($A_i^{\ast },Y_i^{\ast },{\delta }_i^{\ast },{\mathit{X}}_i^{\ast },Z_i^{\ast },{\mathit{W}}_i^{\ast }$), i = 1, …, n, from Φ_n. Replace ($A_i^{\ast },Y_i^{\ast },{\delta }_i^{\ast },{\mathit{X}}_i^{\ast },Z_i^{\ast },{\mathit{W}}_i^{\ast }$) for ($A_i,Y_i,{\delta }_i,{\mathit{X}}_i,Z_i,{\mathit{W}}_i$), i = 1, …, n, everywhere in (5), (6) and (7). Solve these estimating equations using the algorithm in Section 3.2 and obtain the estimators β*, H*(⋅) and f*(⋅). Repeat the above process B times and obtain a sequence of ${\mathit{\beta }}_j^{\ast }$’s, ${\mathit{f}}_j^{\ast }$(⋅)’s and $H_j^{\ast }$(⋅)’s, j = 1, …, B. The asymptotic variances of $\widehat{\mathit{\beta }}$, $\widehat{\mathit{f}}(\cdot )$ and $\widehat{H}(\cdot )$ can be estimated by the average of the empirical sample variances of ${\mathit{\beta }}_j^{\ast }$’s, ${\mathit{f}}_j^{\ast }$(⋅) and $H_j^{\ast }$(⋅)’s, j = 1, …, B, respectively. Gross and Lai (1996) established the asymptotic properties of this bootstrap method.

5. A simulation study

In this section, we report the results of a simulation study on the finite sample properties of the proposed method. Our experimental design is as follows. We generate the unbiased data $\tilde{T}$ from the VCPLT model (1) assuming the following hazard function of $ϵ:{\lambda }_{ϵ}(t)=\frac{\exp (t)}{1+r\cdot \exp (t)}$ (Dabrowska and Doksum, 1988), where r = 0 and r = 1 correspond to the PLPH and PLPO models respectively. We assume that the linear component of the model contains two covariates, X₁ and X₂, distributed as N(0,1) and Bernoulli(0.5) respectively, and independent of each other, and the nonlinear function contains the covariate Z from the U(0, 2) distribution and independent of X₁ and X₂. The covariate W that interacts with Z is assumed to have a N(0, 1) distribution. We let the nonparametric function be f(z) = 2z − z², and the transformation function be H(t) = 2 log(t) and H(t) = log(exp(t) − 1) when r = 0 and r = 1 respectively. The value of β₀ is set to (1, −1)^T.

Our generation of the length-biased data T follows the procedure of Shen et al. (2009). Specifically, we generate the truncation variable A from the U(0, τ) distribution independently of $\tilde{T}$, where τ is larger than the largest value of $\tilde{T}$ in order to fulfill the stationarity assumption. In our simulations, we set τ = 100, and select only the pairs ($A,\tilde{T}$) that satisfy $A<\tilde{T}$. With respect to the censoring mechanism, we examine both covariate-independent and covariate-dependent censoring. For the former, we generate the residual censoring variable C from the U(0, c₁) distribution; for the latter, we generate C from −2X₁ − X₂ + Exp(c₂), where c₁ and c₂ are chosen so that the censoring percentage (CR) is 20%, 40% or 80%. We set n = 100, B, the number of bootstrap replications for estimating the asymptotic variance of $\widehat{\mathit{\beta }}$, to 100, and the maximum number of iteration rounds for the computational algorithm described in Subsection 3.2 to 30. The convergence criterion of the algorithm is described in Remark 2. Our estimation is based on the Gaussian kernel. We set the bandwidth to h₁ = C₀n^−1/3 = Std(Z)n^−1/3 ≈ 0.6n^−1/3 for estimating β, and h₂ = C₀n^−1/5 = Std(Z)n^−1/5 ≈ 0.6n^−1/5 for estimating H(⋅) and f(⋅) (see Remark 3). We choose the weights w₁ and w₂ in R_i(t) in (4) using the method described in Remark 4. In addition, we compare our estimator of β with the estimator obtained by Qiu and Zhou’s (2015) method, and a naive estimator, $\tilde{\mathit{\beta }}$, which corresponds to Wei et al.’s (2018) estimator under the PLT model, and is obtained by setting the equal weight w₁ = w₂ = 0.5 within our framework. The reported sampling properties of estimators relating to the estimation objectives are based on 300 replications, and include results on the magnitude of bias (BIAS), standard deviation of estimates computed using the bootstrap method of Subsection 4.2 (SD), standard errors of estimates from the replicated samples (SE), mean squared error (MSE), and proximity of confidence interval coverage probabilities to the nominal target level of 0.95 (CP).

Tables 1 and 2 report results on the estimation of β under independent censoring and dependent censoring respectively. In all cases, the proposed estimator yields SDs and SEs that are close to each other, indicating that the proposed resampling method works well. The magnitude of BIAS is generally small, ranging between 0.48% and 8.62% of the true value. The proposed method also yields reasonably accurate proximity to the target confidence interval coverage of 0.95. The estimator’s performance in respect of CP also shows no obvious deterioration as CR increases, although the BIAS, SDs, SEs and MSEs generally increase with the censoring percentage. Generally speaking, a change from covariate independent censoring to dependent censoring has the effect of worsening the performance of the estimator. The proposed estimator of β is less biased but yields larger SDs and SEs when r = 1 than when r = 0. In the majority of cases, the proposed estimator delivers smaller SDs and SEs compared with the naive estimator, but at the expense of a larger bias. With few exceptions, the MSE of our proposed estimator is smaller than the corresponding MSE of the naive estimator. In all cases, Qiu and Zhou’s (2015) approach result in larger biases, SDs and SEs, and less accurate CPs. This indicates that ignoring the length-biasedness of data is not recommended.

Table 1:

Simulation results for estimating β under independent censoring

			proposed estimator		naive estimator		Qiu’s estimator
CR(%)	model		β1 = 1	β2 = −1	β1 = 1	β2 = −1	β1 = 1	β2 = −1
20%	r = 0	BIAS	0.0379	−0.0387	0.0302	−0.0212	0.0923	−0.1447
	(PLPH)	SD	0.1503	0.2201	0.1589	0.2221	0.5374	0.9811
		SE	0.1496	0.2027	0.1542	0.2123	0.5147	0.9891
		CP (95%)	96.00	97.67	93.67	96.56	98.01	87.21
		MSE	0.0240	0.0499	0.0262	0.0499	0.2973	0.9835
	r = 1	BIAS	−0.0329	−0.0504	−0.0123	0.0201	0.0304	−0.1044
	(PLPO)	SD	0.3162	0.5909	0.3223	0.5978	0.4556	0.9798
		SE	0.3195	0.5969	0.3128	0.6011	0.5088	0.9299
		CP (95%)	96.00	95.67	95.32	95.21	98.60	87.33
		MSE	0.1011	0.3517	0.1040	0.3578	0.2085	0.9709
40%	r = 0	BIAS	0.0684	−0.0812	0.0862	−0.0990	0.0436	−0.2210
	(PLPH)	SD	0.2128	0.2705	0.2254	0.3098	0.5333	0.9254
		SE	0.2311	0.2787	0.2098	0.2788	0.5984	0.9712
		CP (95%)	94.67	96.67	95.00	96.67	90.03	78.99
		MSE	0.0500	0.0798	0.0582	0.1042	0.2863	0.9052
	r = 1	BIAS	0.0466	−0.0048	0.0087	0.0076	0.0783	0.2134
	(PLPO)	SD	0.3460	0.5957	0.3464	0.6245	0.6012	1.0123
		SE	0.3474	0.5736	0.3493	0.6189	0.6779	1.2965
		CP (95%)	95.33	96.70	94.79	95.87	88.97	79.32
		MSE	0.1219	0.3549	0.1200	0.3901	0.3688	1.0703
80%	r = 0	BIAS	0.0824	−0.0913	0.0877	−0.0932	0.1132	−0.2109
	(PLPH)	SD	0.3465	0.4783	0.3687	0.4897	0.6078	0.9567
		SE	0.3621	0.4556	0.3889	0.4761	0.6568	1.0439
		CP (95%)	95.73	97.21	93.43	97.04	78.98	69.31
		MSE	0.1269	0.2371	0.1436	0.2485	0.3822	0.9598
	r = 1	BIAS	0.0734	0.0882	0.0638	0.0729	−0.1098	0.2276
	(PLPO)	SD	0.4021	0.6520	0.4193	0.6842	0.6109	1.0290
		SE	0.4376	0.6732	0.4426	0.7098	0.6894	1.3098
		CP (95%)	93.21	96.83	93.79	97.98	79.02	69.21
		MSE	0.1671	0.4330	0.1799	0.4735	0.3852	1.1106

Table 2:

Simulation results for estimating β under dependent censoring

			proposed estimator		naive estimator		Qiu’s estimator
CR(%)	model		β1 = 1	β2 = −1	β1 = 1	β2 = −1	β1 = 1	β2 = −1
20%	r = 0	BIAS	0.0754	−0.0679	0.0673	−0.0767	0.0923	−0.1898
	(PLPH)	SD	0.1648	0.2362	0.1785	0.2687	0.6732	0.9984
		SE	0.1707	0.2688	0.1766	0.2680	0.6145	1.2098
		CP (95%)	95.00	96.00	96.00	94.33	98.72	83.54
		MSE	0.0328	0.0604	0.0364	0.0781	0.4617	1.0328
	r = 1	BIAS	0.0058	−0.0088	0.0050	−0.0558	0.0672	0.1693
	(PLPO)	SD	0.3185	0.6176	0.3523	0.6089	0.6987	0.9876
		SE	0.3148	0.6141	0.3238	0.6155	0.7687	0.9542
		CP (95%)	94.33	96.00	95.89	95.67	97.73	84.02
		MSE	0.1015	0.3815	0.1241	0.3739	0.4927	1.0040
40%	r = 0	BIAS	0.0862	−0.0736	0.0895	0.0987	0.0798	−0.3454
	(PLPH)	SD	0.1805	0.2771	0.2292	0.3199	0.6983	1.0324
		SE	0.2054	0.3054	0.2080	0.3032	0.6278	1.2909
		CP (95%)	95.00	94.67	95.03	95.09	89.43	79.89
		MSE	0.0400	0.0822	0.0605	0.1121	0.3469	1.1852
	r = 1	BIAS	−0.0114	−0.0833	0.0017	−0.0563	0.0897	0.3176
	(PLPO)	SD	0.4092	0.6308	0.4333	0.6787	0.7187	1.2367
		SE	0.4189	0.6692	0.4226	0.6875	0.8898	1.3465
		CP (95%)	95.00	95.67	94.65	96.32	89.67	77.35
		MSE	0.1676	0.4048	0.1878	0.4638	0.5246	1.6300
80%	r = 0	BIAS	0.0931	−0.0987	0.0962	0.0872	−0.1134	−0.3665
	(PLPH)	SD	0.3564	0.5321	0.3787	0.5881	0.7321	1.1654
		SE	0.3789	0.5762	0.3846	0.5534	0.7843	1.2108
		CP (95%)	96.81	97.32	96.22	96.89	79.45	74.43
		MSE	0.1357	0.2929	0.1527	0.3535	0.5488	1.4924
	r = 1	BIAS	−0.0532	0.0673	0.0543	−0.0765	0.1354	0.3097
	(PLPO)	SD	0.4592	0.6609	0.4698	0.7102	0.7676	1.3319
		SE	0.4835	0.6988	0.4309	0.7765	0.7982	1.3607
		CP (95%)	97.21	96.98	94.65	96.32	76.71	73.62
		MSE	0.2137	0.4413	0.2237	0.5102	0.6078	1.8699

Tables 3 and 4 report the performance of the proposed estimator for the nonlinear function f(z) for r = 0 and r = 1 respectively. The remarks regarding the proposed estimator’s performance when estimating β generally carries over to the estimation of f(z). Specifically, the estimator never produces a very substantial bias, and the resampling method delivers SDs that are very close to the corresponding SEs. As is seen from Tables 3 and 4, in all but a few cases, $\widehat{f}(z)$ produces values that closely track the actual values of f(z). A plot of $\widehat{\mathit{f}}(z)$ alongside the true f(z) for r = 0 is given in Figure 1. We also plot $\widehat{H}(t)$ and H(t) in Figure 2. It can be seen that except when t is near 0, the curves of $\widehat{H}(t)$ and H(t) are very close. In all cases, the algorithm for computing the estimates developed in Subsection 3.2 converges in a small number of steps.

Table 3:

Simulation results for estimating f(⋅) when r = 0 (PLPH case)

Censoring mechanism	CR(%)	w 0	f(w0)	$\widehat{f}(w_0)$	BIAS	SD	SE
Independent censoring	20%	0.3	0.5100	0.5194	0.0094	0.2746	0.2590
		0.6	0.8400	0.8691	0.0291	0.2340	0.2275
		1.2	0.9600	0.9988	0.0388	0.2274	0.2257
		1.5	0.7500	0.7774	0.0274	0.2316	0.2177
		1.8	0.3600	0.3874	0.0274	0.3356	0.3089
	40%	0.3	0.5100	0.5176	0.0076	0.3105	0.3068
		0.6	0.8400	0.8505	0.0105	0.2572	0.2037
		1.2	0.9600	0.9877	0.0277	0.2609	0.2285
		1.5	0.7500	0.7518	0.0018	0.2605	0.2377
		1.8	0.3600	0.3491	−0.0109	0.3929	0.3832
	80%	0.3	0.5100	0.5461	0.0361	0.7944	0.9283
		0.6	0.8400	0.8448	0.0048	0.6894	0.7178
		1.2	0.9600	0.9586	−0.0014	0.6808	0.6860
		1.5	0.7500	0.7778	0.0278	0.7524	0.7561
		1.8	0.3600	0.3535	−0.0065	1.0253	1.2658
Dependent censoring	20%	0.3	0.5100	0.5171	0.0071	0.2556	0.283
		0.6	0.8400	0.8426	0.0026	0.2102	0.1929
		1.2	0.9600	0.9589	−0.0011	0.2099	0.1963
		1.5	0.7500	0.7362	−0.0138	0.2092	0.1877
		1.8	0.3600	0.3696	0.0096	0.3003	0.2462
	40%	0.3	0.5100	0.5221	0.0121	0.2940	0.2846
		0.6	0.8400	0.8492	0.0092	0.2453	0.2165
		1.2	0.9600	0.9906	0.0306	0.2468	0.2226
		1.5	0.7500	0.7786	0.0286	0.2424	0.2137
		1.8	0.3600	0.4027	0.0427	0.3416	0.3204
	80%	0.3	0.5100	0.5087	−0.0013	0.7951	1.0476
		0.6	0.8400	0.8636	0.0236	0.6871	0.8066
		1.2	0.9600	1.0069	0.0469	0.6902	0.7454
		1.5	0.7500	0.8158	0.0658	0.7887	0.8250
		1.8	0.3600	0.2236	−0.1364	1.1646	1.2160

Table 4:

Simulation results for estimating f(⋅) when r = 1 (PLPO case)

Censoring mechanism	CR(%)	w 0	f(w0)	$\widehat{f}(w_0)$	Bias	SD	SE
Independent censoring	20%	0.3	0.5100	0.5159	0.0059	0.5522	0.5441
		0.6	0.8400	0.8270	−0.0130	0.4555	0.4528
		1.2	0.9600	0.8780	−0.0820	0.4524	0.4030
		1.5	0.7500	0.6337	−0.1163	0.4735	0.4211
		1.8	0.3600	0.3092	−0.0508	0.6462	0.5838
	40%	0.3	0.5100	0.5195	0.0095	0.6411	0.6505
		0.6	0.8400	0.7954	−0.0446	0.5311	0.5104
		1.2	0.9600	0.9143	−0.0457	0.5074	0.4959
		1.5	0.7500	0.7002	−0.0498	0.5574	0.5668
		1.8	0.3600	0.3188	−0.0412	0.7996	0.8105
	80%	0.3	0.5100	0.5237	0.0137	0.8210	0.7998
		0.6	0.8400	0.8254	−0.0146	0.6782	0.6994
		1.2	0.9600	0.9798	0.0198	0.6635	0.6578
		1.5	0.7500	0.7887	0.0387	0.6802	0.6945
		1.8	0.3600	0.3203	−0.0397	0.8209	0.8197
Dependent censoring	20%	0.3	0.5100	0.5193	0.0093	0.6026	0.5720
		0.6	0.8400	0.8389	−0.0011	0.5030	0.4663
		1.2	0.9600	0.9112	−0.0488	0.5046	0.4620
		1.5	0.7500	0.6819	−0.0681	0.5150	0.4814
		1.8	0.3600	0.3486	−0.0115	0.7537	0.7173
	40%	0.3	0.5100	0.5213	0.0113	0.6713	0.6718
		0.6	0.8400	0.8425	0.0025	0.5447	0.4696
		1.2	0.9600	0.9588	−0.0012	0.5252	0.4868
		1.5	0.7500	0.7401	−0.0099	0.5606	0.5127
		1.8	0.3600	0.3590	−0.0010	0.8370	0.7779
	80%	0.3	0.5100	0.5324	0.0224	0.9014	0.9425
		0.6	0.8400	0.7876	−0.0524	0.6792	0.7003
		1.2	0.9600	0.9288	−0.0312	0.6659	0.6243
		1.5	0.7500	0.7765	0.0235	0.7021	0.7129
		1.8	0.3600	0.3773	0.0173	0.9867	0.9459

Figure 1:

The true and estimated curves of f(w) under r = 0 (PLPH case)

1(a): covariate-independent censoring with censoring rate = 20%, 1(b): covariate-independent censoring with censoring rate = 40%, 1(c): covariate-independent censoring with censoring rate = 80%, 1(d): covariate-dependent censoring with censoring rate = 20%, 1(e): covariate-dependent censoring with censoring rate = 40%, 1(f): covariate-dependent censoring with censoring rate = 80% In each of the four subfigures, the solid curve is the true curve, and the dashed curve is the estimator by the proposed method.

Figure 2:

The true and estimated curves of H(t) under r = 0 (PLPH case)

2(a): covariate-independent censoring with censoring rate = 20%, 2(b): covariate-independent censoring with censoring rate = 40%, 2(c): covariate-independent censoring with censoring rate = 80%, 2(d): covariate-dependent censoring with censoring rate = 20%, 2(e): covariate-dependent censoring with censoring rate = 40%, 2(f): covariate-dependent censoring with censoring rate = 80%, In each of the four subfigures, the solid curve is the true curve, and the dashed curve is the estimator by the proposed method.

Our next simulation example focuses on the two-dimensional W case, with two covariates, W₁ and W₂ that interact with Z. We consider the same X₁, X₂ and Z as in the last experiment, and generate W₁ and W₂ from i.i.d. N(0, 1) distributions. For this experiment, we let n = 500. Table 5 reports results on the performance of the estimators of β under independent censoring. In general, the results are very similar to those under the one-dimensional case with a smaller sample size presented in Table 1.

Table 5:

Simulation results for estimating β under independent censoring (n = 500)

			proposed estimator		naive estimator		Qiu’s estimator
CR(%)	model		β1 = 1	β2 = −1	β1 = 1	β2 = −1	β1 = 1	β2 = −1
20%	r = 0	BIAS	0.0237	−0.0366	0.0290	−0.0267	0.0873	−0.1453
	(PLPH)	SD	0.1309	0.2008	0.1487	0.2198	0.5309	0.9663
		SE	0.1412	0.2021	0.1519	0.2147	0.5226	0.9679
		CP (95%)	95.43	94.97	94.63	95.46	97.65	87.39
		MSE	0.0177	0.0417	0.0230	0.0490	0.2895	0.9548
	r = 1	BIAS	−0.0341	−0.0432	−0.0398	−0.0199	0.0699	−0.1193
	(PLPO)	SD	0.2987	0.5403	0.3213	0.5787	0.4532	0.9662
		SE	0.2876	0.5531	0.3167	0.5642	0.4763	0.9187
		CP (95%)	95.03	95.66	94.41	94.66	98.98	87.35
		MSE	0.0851	0.2938	0.1048	0.3353	0.1943	0.9478
40%	r = 0	BIAS	0.0543	−0.0383	0.0437	−0.0552	0.0306	−0.2029
	(PLPH)	SD	0.2029	0.2663	0.2108	0.2876	0.5337	0.9187
		SE	0.2034	0.2654	0.2077	0.2794	0.5692	0.9678
		CP (95%)	94.21	94.30	95.34	95.87	93.31	80.14
		MSE	0.0437	0.0724	0.0463	0.0858	0.2858	0.8852
	r = 1	BIAS	0.0498	−0.0135	0.0129	0.0043	0.0653	0.2308
	(PLPO)	SD	0.3232	0.5439	0.3432	0.5884	0.5998	0.9678
		SE	0.3356	0.5676	0.3467	0.5790	0.6372	1.0390
		CP (95%)	95.56	95.55	94.69	95.78	94.01	84.03
		MSE	0.2960	0.3549	0.1180	0.3462	0.3640	0.9899
80%	r = 0	BIAS	0.0836	−0.0673	0.0637	−0.0993	0.1189	−0.2093
	(PLPH)	SD	0.3378	0.4635	0.3638	0.4938	0.5990	1.0938
		SE	0.3392	0.4589	0.3573	0.4766	0.6493	1.0389
		CP (95%)	93.37	97.19	93.33	97.36	77.22	70.39
		MSE	0.1211	0.2194	0.1364	0.2537	0.3729	1.2402
	r = 1	BIAS	0.0704	0.0798	0.0604	0.0729	−0.1209	0.2092
	(PLPO)	SD	0.3935	0.6431	0.4094	0.6678	0.6035	1.2456
		SE	0.3987	0.6398	0.4236	0.6799	0.6547	1.2209
		CP (95%)	93.23	96.87	93.39	97.93	78.39	72.93
		MSE	0.1598	0.4199	0.1713	0.4513	0.3788	1.5953

6. An empirical example

The Human Immunodeficiency Virus (HIV) targets the immune system and weakens people’s defenses against infections and some types of cancer, and continues to cause great morbidity and mortality globally. Preventive HIV vaccine efficacy trials randomise HIV uninfected healthy volunteers to receive a candidate HIV vaccine or a placebo, and follow the study participants for occurrence of the study endpoint acquisition of HIV-1 infection. Understanding the distribution of age at the time of HIV-1 acquisition in placebo recipients aids the design and analysis of preventive HIV-1 efficacy trials, by informing age-related recruitment strategies to help assure sufficiently high predicted HIV-1 incidence, and informing sample size/power calculations for assessing vaccine efficacy. In addition, it is of interest to study whether and how the distribution of age at HIV-1 infection in placebo recipients differs by gender, whether a blood test indicates previous infection with herpes simplex virus type 2 (HSV-2), and the quantitative level of antibodies in blood to Adenovirus Serotype 5 (which we refer to as “Ad5 titer”). We address these questions from data collected in two efficacy trials – the HVTN 502 ‘Step’ trial with data from 2970 men and women enrolled in the U.S., Australia, Brazil, Peru, and the Caribbean, and the HVTN 503 ‘Phambili’ trial with data from 800 men and women in South Africa. These trials tested the same Adenovirus Serotype 5 vector Gag-Pol-Nef candidate HIV vaccine versus placebo. The Ad5 titer variable is of interest because Ad5 is a common upper respiratory infection and we seek understanding of whether antibody response to Ad5 – which may be a marker of immune system health or alternatively of frailty to acquire many previous Ad5 infections – associates with the age at HIV-1 infection. The variable HSV-2 status is of interest as a known marker of sexual exposure and we seek refined understanding of how it associates with HIV-1 infection in an age-dependent manner. We analysed the data sets of placebo recipients that had data on all of the three baseline covariates (only HSV-2 status was sometimes missing), yielding 919 placebo recipients in HVTN 502 with a right-censoring rate of 92.3%, and 399 placebo recipients in HVTN 503 with a right-censoring rate of 86.5%. The HVTN data are left-truncated as the data include only adults aged 18 years or above.

Our objective is to assess whether and how the time from age 18 until HIV-1 infection diagnosis differs between men and women, between individuals with or without HSV-2 infection at enrollment, and whether and how Ad5 titer modifies the association of HSV-2 infection with the time from age 18 until HIV-1 infection diagnosis. In addition, by comparing results between HVTN 502 and 503 that took place in the Americas/Australia and South Africa, respectively, we assessed whether these associations differed by global geographic region. In the notation of our method, the failure time of interest, $\tilde{T}$, is the time (in years) from age 18 until diagnosis of HIV-1 infection. Participants were eligible to enroll if they tested HIV negative and were between 18 and 50 years old. The fact that most enrollees were greater than 18 years old and only HIV-1 uninfected volunteers were eligible created the left-truncation of $\tilde{T}$.

A test of the stationary assumption using the method developed by Addona and Wolfson (2005) confirms the survival data in both data sets are evidently stationary. We consider three covariates, X₁: gender (1=male, 0=female), X₂: HSV-2 (1=HSV-2 positive at enrollment, 0=HSV-2 negative at enrollment), and Z: Ad5 titer (scaled to lie between 0 and 1), and the following specification of the transformation function:

$$H(\tilde{T})={\beta }_1{X}_1+{\beta }_2{X}_2+f(Z)X_2+ϵ,$$ (10)

where the hazard function of ϵ is same as in the numerical studies: ${\lambda }_{ϵ}(t)=\frac{\exp (t)}{1+r\cdot \exp (t)}$, and r = 0 and r = 1 correspond to PLPH and PLPO cases, respectively. We use the Gaussian Kernel, set the bandwidth h₂ = Std(Z)n^−1/3 for the estimation of H(⋅) and β, and h₁ = Std(Z)n^−1/5 for estimation of f(⋅).

Table 6 reports the estimates of β₁ and β₂, the estimated standard deviations (SD), and the two-sided p-values corresponding to the tests of the coefficients being different from zero, by the proposed, naive and Qiu and Zhou’s (2015) estimators. Results from the HVTN 502 and 503 trials, and under r = 0 (PLPH) and r = 1 (PLPO) are reported. Figure 3 plots the estimated curve $\widehat{f}(z)$ based on the proposed estimator versus z with a 95% pointwise confidence band.

Table 6:

Results of the empirical example

			r=0 (PLPH)		r=1 (PLPO)
Estimator	Dataset		β 1	β 2	β 1	β 2
Proposed estimator	HVTN 502	Est	0.1240	−0.2909	0.2035	−0.4798
		SD	0.0658	0.1610	0.1258	0.2966
		p-value	0.0595	0.0708	0.1057	0.0856
	HVTN 503	Est	0.8336	0.0644	1.5438	−0.1474
		SD	0.2433	0.2856	0.3214	0.4060
		p-value	0.0006	0.8216	0.0130	0.7166
Naive estimator	HVTN 502	Est	0.1201	−0.2994	0.1989	−0.4993
		SD	0.0842	0.1900	0.1347	0.3593
		p-value	0.1538	0.1151	0.1477	0.1646
	HVTN 503	Est	1.1954	0.0457	1.5019	−0.0567
		SD	0.3368	0.2784	0.3824	0.5047
		p-value	0.0004	0.8696	0.0009	0.9106
Qiu’s estimator	HVTN 502	Est	0.2156	−0.5770	1.4860	0.4455
		SD	0.1184	0.2771	0.5736	0.2392
		p-value	0.0686	0.0373	0.0010	0.0625
	HVTN 503	Est	1.3108	−0.2392	1.1531	−0.4209
		SD	0.3632	0.2939	0.6982	0.3287
		p-value	0.0003	0.4157	0.0986	0.2004

Figure 3:

Estimated curves of f(w) for the empirical example

3(a): r=0 for HVTN 502, 3(b): r=1 for HVTN 502, 3(c): r=0 for HVTN 503, 3(d): r=1 for HVTN 503. In each sub-figure, the solid-curve is the estimated curve of f(w) and the dashed curves are the 95% confidence bands

It can be seen from the top panel of Table 6 that by our method, the estimates of β₁ are always positive, although for HVTN 502, the evidence of β₁ being significantly different from zero is not as convincing as for HVTN 503. In other words, while the result indicates that women tend to be infected with HIV earlier in life than men, the evidence in favour of this result is strong in South Africa, but less so in the Americas/Australia. The results for estimating β₂ suggest that in the case of South Africa (HVTN 503), there is no apparent linkage between HSV-2 status and time of acquiring HIV infection, but in the Americas/Australia (HVTN 502), there is moderate evidence that those having previously infected with HSV-2 tend to be infected with HIV earlier in life. Figure 3 shows that for the Americas/Australia (HVTN 502), Ad5 titer is associated with an older age of infection for those with a positive HSV-2 status, but for South Africa (HVTN 503), the association is negligible. There is no remarkable difference between results obtained under r = 0 and r = 1. The results in Table 6 also show that estimates obtained from our and the naive methods exhibit the same signs and statistical significance of the unknown parameters. However, our proposed method invariably produces smaller standard errors of the estimates. Qiu and Zhou’s (2015) estimator yields results that differ markedly from those obtained by the other two estimators, as is expected because Qiu and Zhou’s (2015) method ignores the length-biasedness of the data.

7. Concluding remarks

This paper focuses on the VCPLT model and takes proper account of length-biased sampling, which is a commonplace in prevalent cohort survival studies. We have developed a methodology and an iterative algorithm for computing the estimates of the model, established an asymptotic theory for the estimator, and examined its finite sample performance.

Our empirical study focusing on the analysis of the HVTN 502 and 503 vaccine efficacy trials provides insights that females become HIV-1 infected at younger ages than males in South Africa (HVTN 503), but apparently not, or much less so, in the Americas/Australia (HVTN 502), and that being previously infected with HSV-2 was associated with a younger age of infection in the Americas/Australia, but not in South Africa. In addition, we found evidence that a continuous marker (Ad5 titer) was associated with an older age of infection only within one level of a binary covariate (previously infected with HSV-2) in HVTN 502. One implication of these results is that an efficacy trial recruitment strategy in South Africa that enriches enrollment of younger women and older men may increase the overall HIV risk level of participants and thus make the trial design more efficient due to a higher endpoint rate in placebo recipients, whereas there is only a nonsignificant trend that this recruitment strategy could aid efficient trial design in the Americas/Australia. The new methodology enables such analyses by accommodating the fact that the number of years from adulthood until HIV-1 infection diagnosis is a left-truncated failure time.

Appendix

Let ‖⋅‖ denote the ℒ² norm, and β₀ and f₀ be the true values of β and f. For any ϵ > 0, define $D_{ϵ}^1=\{\mathit{\beta }:\Vert \mathit{\beta }-{\mathit{\beta }}_0\Vert \le ϵ\}$, $D_{ϵ}^2=\{\mathit{f}:{\sup }_z|\mathit{f}(z)-{\mathit{f}}_0(z)|\le ϵ\}$, $D_{ϵ}^3=\{\dot{\mathit{f}}:\underset{z}{\sup }|\dot{\mathit{f}}(z)-{\dot{\mathit{f}}}_0(z)|\le ϵ\}$, and $D_{ϵ}=\{{({\mathit{\beta }}^T,{\mathit{\alpha }}_0^T,{\mathit{\alpha }}_1^T)}^T:\Vert ({\mathit{\beta }}^T,{\mathit{\alpha }}_0^T(z),{{\mathit{\alpha }}_1^T(z))}^T-{({\mathit{\beta }}_0^T,{\mathit{f}}_0^T(z),{\dot{\mathit{f}}}_0^T(z))}^T\Vert \le ϵ\}$.

Our proof of Theorem 1 requires the following lemma:

Lemma 1. Assume that Conditions (C1)-(C6) hold, and h → 0 and nh → 0 as n → ∞. Also, assume that the matrix A_z (defined in the proof) is finite and non-degenerate for any z. Then the one-step estimators $\tilde{\mathit{\beta }}(z)$, ${\tilde{\mathit{\alpha }}}_1(z)$, and ${\tilde{\mathit{\alpha }}}_0(z)$ are locally consistent.

Proof of Lemma 1. Let ${\tilde{H}}_z(t;\mathit{\beta },{\mathit{\alpha }}_0,{\mathit{\alpha }}_1)$ be the solution of (8) for fixed β, α₀, and α₁. Note that the left-hand side of (8) is monotone in H. By similar arguments to Chen et al. (2002), it can be shown that (8) has a unique solution ${\tilde{H}}_z(t;\mathit{\beta },{\mathit{f}}_0,{\dot{\mathit{f}}}_0)$, and for any t ∈ (0, τ],

$${\tilde{H}}_z(t;\mathit{\beta },{\mathit{f}}_0,{\dot{\mathit{f}}}_0){\to }_{a.s.}{H}_0(t),$$

where →_a.s. denotes the convergence almost surely. Let

$$\begin{gathered} {\tilde{\mathit{U}}}_z(\mathit{\beta },{\mathit{\alpha }}_1,{\mathit{\alpha }}_0)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z)\left(\begin{array}{c}{\mathit{X}}_i\\ {\mathit{W}}_i(Z_i-z)\\ {\mathit{W}}_i\end{array}\right)[d{N}_i(t)-R_i(t) \\ \times d{\text{Λ}}_{ϵ}\left\{{\tilde{H}}_z(t;\mathit{\beta },{\mathit{\alpha }}_0,{\mathit{\alpha }}_1)+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{\alpha }}_0^T(z){\mathit{W}}_i+{\mathit{\alpha }}_1^T(z){\mathit{W}}_i(Z_i-z)\right\}]. \end{gathered}$$

Then ${\tilde{H}}_z(t;\mathit{\beta },{\mathit{\alpha }}_0,{\mathit{\alpha }}_1)$ may be considered as a random mapping from D_ϵ to another open connected set B_n in 𝓡^p+2q.

We divide our proof into three steps. First, by the law of large numbers and Lemma 1 in Cai et al. (2007), we have

$$\begin{gathered} {\tilde{\mathit{U}}}_z(\mathit{\beta },{\dot{\mathit{f}}}_0,{\mathit{f}}_0)={\int }_0^{\tau }E(K_h(Z-z)\left(\begin{array}{c}\mathit{X}\\ (Z-z)\mathit{W}\\ \mathit{W}\end{array}\right)[dN(t)-R(t) \\ \times d{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}+{\dot{\mathit{f}}}_0^T(z)\mathit{W}(Z-z)\}])+o_p(1) \\ ={\int }_0^{\tau }E\left(\left(\begin{array}{c}\mathit{X}\\ 0\\ \mathit{W}\end{array}\right)[dN(t)-R(t)d{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}]\mid Z=z\right)g(z)+o_p(1) \\ =o_p(1), \end{gathered}$$

where g(z) is the density function of Z. Thus, B_n contains 0 with probability approaching 1.

Second, let

$$\begin{gathered} {\mathit{A}}_{11,z}=E({\mathit{X}}^{\otimes 2}R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}\mid Z=z)g(z), \\ {\mathit{A}}_{22,z}=E({\mathit{W}}^{\otimes 2}R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}\mid Z=z)g(z)k_2, \\ {\mathit{A}}_{33,z}=E({\mathit{W}}^{\otimes 2}R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}\mid Z=z)g(z)\text{and} \\ {\mathit{A}}_{13,z}=E(\mathit{X}{\mathit{W}}^{T}R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}\mid Z=z)g(z), \end{gathered}$$

where $k_2=\int x^{2}k(x)dx$. By techniques similar to those used above,

$${\underset{n\to \infty }{\lim }\frac{\partial {\tilde{\mathit{U}}}_z(\mathit{\beta },{\mathit{\alpha }}_1,{\mathit{\alpha }}_0)}{\partial ({\mathit{\beta }}^T,{\mathit{\alpha }}_1^T,{\mathit{\alpha }}_0^T)}|}_{\mathit{\beta }={\mathit{\beta }}_0,{\mathit{\alpha }}_0={\mathit{f}}_0,{\mathit{\alpha }}_1={\dot{\mathit{f}}}_0}=-\left(\begin{array}{ccc}{\mathit{A}}_{11,z}& 0& {\mathit{A}}_{13,z}\\ 0& {\mathit{A}}_{22,w}& 0\\ {\mathit{A}}_{13,z}^T& 0& {\mathit{A}}_{33,z}\end{array}\right)\triangleq {\mathit{A}}_z$$

in probability. Using methods similar to Lu and Zhang (2010) and the uniform law of large numbers, we can obtain,

$$\underset{\mathit{\beta }\in D_{ϵ}^1,{\mathit{\alpha }}_0\in D_{{ϵ}_z}^2,{\alpha }_1\in D_{{ϵ}_z}^3}{\sup }\Vert \frac{\partial {\tilde{\mathit{U}}}_z(\mathit{\beta },{\mathit{\alpha }}_1,{\mathit{\alpha }}_0)}{\partial ({\mathit{\beta }}^T,{\mathit{\alpha }}_1^T,{\mathit{\alpha }}_0^T)}-{\mathit{A}}_z\Vert {\to }_{a.s.}0,$$

as n → ∞, ϵ → ∞ and ϵ_z → ∞.

Third, as A_z is finite and non-degenerate by assumption, the map ${\tilde{\mathit{U}}}_z(\mathit{\beta },{\mathit{\alpha }}_1,{\mathit{\alpha }}_0)$ is a homeomorphism from D_ϵ to B_n. However, ${\tilde{\mathit{U}}}_z(\tilde{\mathit{\beta }},{\tilde{\mathit{\alpha }}}_1,{\tilde{\mathit{\alpha }}}_0)=0$. Thus, by letting ϵ → 0, it can be shown that $\tilde{\mathit{\beta }}(z)$, ${\tilde{\mathit{\alpha }}}_1(z)$ and ${\tilde{\mathit{\alpha }}}_0(z)$ are locally consistent.

Proof of Theorem 1. We need to prove the local consistency of estimators $\widehat{\mathit{\beta }}$, $\widehat{H}(\cdot )$, ${\widehat{\mathit{\alpha }}}_0(z)$ and ${\widehat{\mathit{\alpha }}}_1(z)$. Given that $\tilde{\mathit{\beta }}(z)$, ${\tilde{\mathit{\alpha }}}_1(z)$ and ${\tilde{\mathit{\alpha }}}_0(z)$ are locally consistent, then so are $\widehat{\mathit{\beta }}$, ${\widehat{\mathit{\alpha }}}_0(z)$ and ${\widehat{\mathit{\alpha }}}_1(z)$ from results of Carroll et al. (1997). Thus, we only have to prove the consistency of $\widehat{H}(\cdot )$. Let $\breve{H}$ be a limit of $\widehat{H}(t,{\mathit{\beta }}_0,{\mathit{f}}_0)$. By equation (5) and arguments similar to those used by Kim et al. (2013), we have

$$E[N(t)]={\int }_0^{t}E\left[R(s){\lambda }_{ϵ}\{\breve{H}(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}\right]d\breve{H}(s).$$ (11)

Differentiating (11) with respect to t, we have

$$\frac{d\breve{H}(t)}{dt}=\frac{dE[N(t)]}{dt}{\left(E[R(t){\lambda }_{ϵ}\{\breve{H}(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}]\right)}^{-1}.$$ (12)

As (12) is a Cauchy problem, it has a unique solution. As well, it is readily seen from equation (4) that H₀, the the true transformation function, satisfies (12). Hence, by Helly’s Lemma, $H_0=\breve{H}$ and we have $\widehat{H}(t,{\mathit{\beta }}_0,{\mathit{f}}_0){\to }_{a.s.}{H}_0$. Note that $\frac{\partial \widehat{H}(t,\mathit{\beta },{\mathit{\alpha }}_0)}{\partial \mathit{\beta }}$ and $\frac{\partial \widehat{H}(t,\mathit{\beta },{\mathit{\alpha }}_0)}{\partial {\mathit{\alpha }}_0}$ are bounded on [0, τ] if $\mathit{\beta }\in D_{{ϵ}_1}^1$ and ${\mathit{\alpha }}_0\in D_{{ϵ}_2}^2$ for some ϵ₁ and ϵ₂. Then by using the Taylor-series expansion, we have $\widehat{H}(t){\to }_{\text{a}.\text{s}.}{H}_0(t)$.

Next, we prove the asymptotic normality of $\widehat{\beta }$. Our proof focuses on the following asymptotic representation of $\widehat{\mathit{\beta }}$:

$$({\mathit{A}}_1-{\mathit{A}}_2)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }[({\mathit{X}}_i-\mathit{m}(t))-({\mathit{X}}_i^{\ast }-{\mathit{m}}_i^{\ast })]d{M}_i(t)+o_p(n^{-1/2}).$$ (13)

If it can be proven that if (13) is true, then it is clear that Theorem 1 is also true by the martingale central limit theorem. In order to prove (13), we first give the representation of ${\text{Λ}}_{ϵ}^{\ast }\{\widehat{H}(t;{\mathit{\beta }}_0,{\mathit{f}}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}$ and $\frac{\partial }{\partial \mathit{\beta }}\widehat{H}(t,{\mathit{\beta }}_0,{\mathit{f}}_0)$ that will be used to approximate the rate of Taylor-series expansion. Next, we give the representation of $\frac{1}{n}\sum _{i=1}^n{M}_i(t)$ and $\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t)$. Finally, we combine the results in the first two steps to obtain (13).

From equations (4) and (5), we have

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{M}_i(t)=\frac{1}{n}\sum _{i=1}^n{N}_i(t)-\frac{1}{n}\sum _{i=1}^n{\int }_0^t{R}_i(s)d{\text{Λ}}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^t{R}_i(s)[d{\text{Λ}}_{ϵ}\{\widehat{H}(s;{\mathit{\beta }}_0,{\mathit{f}}_0)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ -d{\text{Λ}}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}]. \end{gathered}$$

By some tedious calculations and results of the empirical process theory for Z-estimator and the definition of ${\lambda }_{ϵ}^{\ast }\{H_0(t)\}$, we obtain

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{M}_i(t)=\frac{1}{n}\sum _{i=1}^n{\int }_0^t{R}_i(s)d\left[\frac{{\lambda }_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}}{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}\left({\text{Λ}}_{ϵ}^{\ast }\{\widehat{H}(s;{\mathit{\beta }}_0,{\mathit{f}}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\}\right)\right] \\ +o_p(n^{-1/2}) \\ ={\int }_0^t\frac{B_2(s)}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}d\left[{\text{Λ}}_{ϵ}^{\ast }\{\widehat{H}(s;{\mathit{\beta }}_0,{\mathit{f}}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\}\right]+o_p(n^{-1/2}). \end{gathered}$$ (14)

Then it follows, for t ∈ (0, τ], that

$${\text{Λ}}_{ϵ}^{\ast }\{\widehat{H}(t;{\mathit{\beta }}_0,{\mathit{f}}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}=\frac{1}{n}\sum _{i=1}^n{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{M}_i(t)+o_p(n^{-1/2}).$$ (15)

Note that for any t ∈ (0, τ],

$$\sum _{i=1}^n\left[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{\widehat{H}(t;\mathit{\beta },\mathit{f})+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{f}}^T(Z_i){\mathit{W}}_i\}\right]=0$$ (16)

for the parameter β. Differentiating (16) with respect to β on both sides yields

$$\sum _{i=1}^n{R}_i(t)d\left[{\lambda }_{ϵ}\{\widehat{H}(t;\mathit{\beta },\mathit{f})+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{f}}^T(Z_i){\mathit{W}}_i\}\{{\mathit{X}}_i+\frac{\partial }{\partial \mathit{\beta }}\widehat{H}(t;\mathit{\beta },\mathit{f})\}\right]=0.$$

Following arguments similar to Chen et al. (2002) and Lu and Zhang (2010), it can be shown that

$$\begin{gathered} {\frac{\partial }{\partial \mathit{\beta }}\widehat{H}(t;\mathit{\beta },\mathit{f})|}_{\mathit{\beta }={\mathit{\beta }}_0,\mathit{f}={\mathit{f}}_0} \\ =-{\int }_0^t\frac{B(s,t)}{B_2(s)}E\left[\mathit{X}{\dot{\lambda }}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}R(s)\right]d{H}_0(s)+o_p(1) \\ =-{\int }_0^t\frac{B(s,t)}{B_2(s)}{\mathit{B}}_1^X(s)d{H}_0(s)+o_p(1)\triangleq -\mathit{a}(t)+o_p(1). \end{gathered}$$ (17)

Next we give the representation of $\frac{1}{n}\sum _{i=1}^n{M}_i(t)$ and $\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t)$.

Denote

$${\mathit{U}}_1(\mathit{\beta },\mathit{f})=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_i\{d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{\widehat{H}(t;\mathit{\beta },\mathit{f})+{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{f}}^T(Z_i){\mathit{W}}_i\}.$$

Taking the derivative of U₁(β, f) with respect to β, setting β = β₀ and f = f₀, and using the law of large numbers and (17), we obtain

$$\begin{gathered} {\frac{\partial {\mathit{U}}_1(\mathit{\beta },\mathit{f})}{\partial \mathit{\beta }}|}_{\mathit{\beta }={\mathit{\beta }}_0,\mathit{f}={\mathit{f}}_0} \\ =-\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }\{{\mathit{X}}_i-\mathit{\mu }(t)\}{\mathit{X}}_i^T{R}_i(t){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T({\mathit{Z}}_i){\mathit{W}}_i\}d{H}_0(t)+o_p(1) \\ =-{\int }_0^{\tau }E\left[\{\mathit{X}-\mathit{\mu }(t)\}{\mathit{X}}^{T}R(t){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}\right]d{H}_0(t)+o_p(1) \\ =-{\mathit{A}}_1+o_p(1). \end{gathered}$$ (18)

Let (${\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1$) be the solution of (7), and for t ∈ [0, τ], define ${\widehat{H}}_0(t;\mathit{\beta })=\widehat{H}(t;\mathit{\beta },{\widehat{\mathit{\alpha }}}_0)$. For any z, let

$$\begin{gathered} {\mathit{U}}_{21}({\mathit{\alpha }}_0,{\mathit{\alpha }}_1,H,\mathit{\beta })(z)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z){\mathit{W}}_i[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{H(t) \\ +{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{\alpha }}_0^T(z){\mathit{W}}_i+{\mathit{\alpha }}_1^T(z){\mathit{W}}_i(Z_i-z)\}], \end{gathered}$$

$$\begin{gathered} {\mathit{U}}_{22}({\mathit{\alpha }}_0,{\mathit{\alpha }}_1,H,\mathit{\beta })(z)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z){\mathit{W}}_i\frac{Z_i-z}{h}[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{H(t) \\ +{\mathit{\beta }}^T{\mathit{X}}_i+{\mathit{\alpha }}_0^T(z){\mathit{W}}_i+{\mathit{\alpha }}_1^T(z){\mathit{W}}_i(Z_i-z)\}],\text{and} \end{gathered}$$

$${\mathit{U}}_2({\mathit{\alpha }}_0,{\mathit{\alpha }}_1,H,\mathit{\beta })(z)=({\mathit{U}}_{21}^T({\mathit{\alpha }}_0,{\mathit{\alpha }}_1,H,\mathit{\beta })(z),{\mathit{U}}_{22}^T({\mathit{\alpha }}_0,{\mathit{\alpha }}_1,H,\mathit{\beta })(z)).$$

Then we have

$${\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,{\widehat{H}}_0(t;\widehat{\mathit{\beta }}),\widehat{\mathit{\beta }})(z)=0,$$

where (${\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1$) is the solution of (7) at convergence, and $(\widehat{\mathit{\beta }},{\widehat{H}}_0(t;\widehat{\mathit{\beta }}))$ is the solution of (5) and (6) at convergence. By the Taylor-series expansion and the law of large numbers, we obtain

$$\begin{gathered} 0={\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,{\widehat{H}}_0(\cdot ;\widehat{\mathit{\beta }}),\widehat{\mathit{\beta }})(z) \\ ={\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,{\widehat{H}}_0(\cdot ;{\mathit{\beta }}_0),{\mathit{\beta }}_0)(z)-{\mathit{E}}_1(z)+o_p(n^{-1/2}), \end{gathered}$$ (19)

where

$$\begin{gathered} {\mathit{E}}_1(z) \\ =\frac{1}{n}\sum _{i=1}^n{K}_h(Z_i-z)R_i(t)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right) \\ \times {\lambda }_{ϵ}\{{\widehat{H}}_0(t,{\widehat{\mathit{\beta }}}_0)+{\mathit{X}}_i^T{\widehat{\mathit{\beta }}}_0+{\widehat{\mathit{\alpha }}}_0^T(z){\mathit{W}}_i+{\widehat{\mathit{\alpha }}}_1^T(z){\mathit{W}}_i(Z_i-z)\}{\left({\mathit{X}}_i+{\frac{\partial {\widehat{H}}_0(t;\mathit{\beta })}{\partial \mathit{\beta }}|}_{\mathit{\beta }={\mathit{\beta }}_0}\right)}^T(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0). \end{gathered}$$

However, as ${\widehat{H}}_0(t;\mathit{\beta })=\widehat{H}(t;\mathit{\beta },{\widehat{\mathit{\alpha }}}_0)$, we have

$$\frac{1}{n}\sum _{i=1}^n\left[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{{\widehat{H}}_0(t;\mathit{\beta })+{\mathit{\beta }}^T{\mathit{X}}_i+{\widehat{\mathit{\alpha }}}_0^T(Z_i){\mathit{W}}_i\}\right]=0.$$ (20)

Taking derivative with respect to β on both sides of (20), applying the law of large numbers and using arguments similar to those in Step 2, we obtain

$$\begin{gathered} \left({\frac{\partial {\widehat{H}}_0(s;\mathit{\beta })}{\partial \mathit{\beta }}|}_{\mathit{\beta }={\mathit{\beta }}_0}\right)d{B}_1(s)+B_2(s)d\left({\frac{\partial \widehat{H}(s;\mathit{\beta })}{\partial \mathit{\beta }}|}_{\mathit{\beta }={\mathit{\beta }}_0}\right) \\ =-E[R(s)\mathit{X}{\dot{\lambda }}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}]d{H}_0(s)+o_p(1). \end{gathered}$$ (21)

Multiplying ${\lambda }_{ϵ}^{\ast }\{H_0(s)\}$ to both sides of (21) and performing integration with respect to s over the range (0, t) yields

$${\frac{\partial {\widehat{H}}_0(t;\mathit{\beta })}{\partial \mathit{\beta }}|}_{\mathit{\beta }={\mathit{\beta }}_0}=-\mathit{a}(t)+o_p(1).$$ (22)

Substituting (22) into E₁(z) leads to

$$\begin{gathered} {\mathit{E}}_1(z)=\frac{1}{n}\sum _{i=1}^n{K}_h(Z_i-z)R_i(t)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right){\lambda }_{ϵ}\{{\widehat{H}}_0(t,{\widehat{\mathit{\beta }}}_0)+{\mathit{X}}_i^T{\widehat{\mathit{\beta }}}_0+{\widehat{\mathit{\alpha }}}_0^T(z){\mathit{W}}_i \\ +{\widehat{\mathit{\alpha }}}_1^T(z){\mathit{W}}_i(Z_i-z)\}{({\mathit{X}}_i-\mathit{a}(t))}^T{\left(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0\right)}^T+o_p(n^{-1/2}) \\ =\left(\begin{array}{c}{\mathit{e}}_1^T(z)\\ 0^T\end{array}\right)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+o_p(n^{-1/2}), \end{gathered}$$ (23)

where ${\mathit{e}}_1^T(z)=g(z)E[R(t)\mathit{W}{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}{(\mathit{X}-\mathit{a}(t))}^T\mid Z=z]$. On the other hand,

$$\begin{gathered} {\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,{\widehat{H}}_0(\cdot ;{\mathit{\beta }}_0),{\mathit{\beta }}_0)(z) \\ ={\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,H_0,{\mathit{\beta }}_0)(z)-\frac{1}{n}\sum _{i=1}^n{K}_h(Z_i-z)R_i(t)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right) \\ \times {\lambda }_{ϵ}\{{\widehat{H}}_0(t)+{\mathit{X}}_i^T{\mathit{\beta }}_0+{\widehat{\mathit{\alpha }}}_0^T(z){\mathit{W}}_i+{\widehat{\mathit{\alpha }}}_1^T(z){\mathit{W}}_i(Z_i-z)\}({\widehat{H}}_0(t;{\mathit{\beta }}_0)-{\widehat{H}}_0(t))+o_p(n^{-1/2}) \\ ={\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,H_0,{\mathit{\beta }}_0)(z)-{\int }_0^{\tau }g(z)\left(E\left[\begin{array}{c}\frac{R(t)\mathit{W}{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}\\ 0\end{array}\mid Z=z\right]\right) \\ \times d\left({\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}\right)+o_p(n^{-1/2}) \\ ={\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,H_0,{\mathit{\beta }}_0)(z)-{\int }_0^{\tau }\left(\begin{array}{c}{\mathit{e}}_2(z,t)\\ 0\end{array}\right)d\left({\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}\right) \\ +o_p(n^{-1/2}), \end{gathered}$$ (24)

where

$${\mathit{e}}_2(z,t)=g(z)E\left[\frac{R(t)\mathit{W}{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}\mid Z=z\right].$$

Moreover,

$$\begin{gathered} {\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,H_0,{\mathit{\beta }}_0)(z) \\ ={\mathit{U}}_2({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)--\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z)R_i(t)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right) \\ \times [{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{X}}_i^T{\mathit{\beta }}_0+{\widehat{\mathit{\alpha }}}_0^T(z){\mathit{W}}_i+{\widehat{\mathit{\alpha }}}_1^T(z){\mathit{W}}_i(Z_i-z)\} \\ -{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{X}}_i^T{\mathit{\beta }}_0+{\mathit{f}}_0^T(z){\mathit{W}}_i+{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i({\mathit{Z}}_i-z)\}] \\ ={\mathit{U}}_2({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)-\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z)R_i(t)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right) \\ \times \left({\mathit{W}}_i^T,{\mathit{W}}_i^T\left(\frac{Z_i-z}{h}\right)\right){\lambda }_{ϵ}\{H_0(t)+{\mathit{X}}_i^T{\mathit{\beta }}_0+{\mathit{f}}_0^T(z){\mathit{W}}_i+{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i(Z_i-z)\}\times \left(\begin{array}{c}{\widehat{\mathit{\alpha }}}_0(z){-}_0(z)\\ h({\widehat{\mathit{\alpha }}}_1(z)-{\dot{\mathit{f}}}_0(z))\end{array}\right)+o_p(n^{-1/2}). \end{gathered}$$

Thus, if we define

$${\mathit{e}}_{31}(z)=g(z)E[R(t)\mathit{W}{\mathit{W}}^T{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}\mid Z=z]$$

and

$${\mathit{e}}_3(z)=\left(\begin{array}{cc}{\mathit{e}}_{31}(z)& 0\\ 0& k_2{\mathit{e}}_{31}(z)\end{array}\right),$$

then we have

$${\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,H_0,{\mathit{\beta }}_0)(w)={\mathit{U}}_2({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)-{\mathit{e}}_3(z)\left(\begin{array}{c}{\widehat{\mathit{\alpha }}}_0(z)-{\mathit{f}}_0(z)\\ h({\widehat{\mathit{\alpha }}}_1(z)-{\dot{\mathit{f}}}_0(z))\end{array}\right)+o_p(n^{-1/2})$$ (25)

Hence, combining (19), (23)–(25), we get

$$\begin{gathered} {\mathit{e}}_3(z)\left(\begin{array}{c}{\widehat{\mathit{\alpha }}}_0(z)-{\mathit{f}}_0(z)\\ h({\widehat{\mathit{\alpha }}}_1(z)-{\dot{\mathit{f}}}_0(z))\end{array}\right) \\ ={\mathit{U}}_2({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)-{\int }_0^{\tau }\left(\begin{array}{c}{\mathit{e}}_2(z,t)\\ 0\end{array}\right)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}] \\ -\left(\begin{array}{c}{\mathit{e}}_1^T(z)\\ 0\end{array}\right)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+o_p(n^{-1/2}), \end{gathered}$$

which yields

$$\begin{gathered} {\widehat{\mathit{\alpha }}}_0(z)-{\mathit{f}}_0(z)={\mathit{e}}_{31}^{-1}(z){\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)-{\mathit{e}}_{31}^{-1}(z){\mathit{e}}_1^T(z)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0) \\ -{\int }_0^{\tau }{\mathit{e}}_{31}^{-1}(z){\mathit{e}}_2(z,t)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]+o_p(n^{-1/2}) \end{gathered}$$ (26)

and

$$h({\widehat{\mathit{\alpha }}}_1(z)-{\dot{\mathit{f}}}_0(z))=\frac{1}{k_2}{\mathit{e}}_{31}^{-1}(z){\mathit{U}}_{22}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)+o_p(n^{-1/2}).$$ (27)

By the definition of ${\widehat{H}}_0(t;\mathit{\beta })$,

$$\sum _{i=1}^n\left[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\widehat{\mathit{\alpha }}}_0^T(Z_i){\mathit{W}}_i\}\right]=0.$$

By the Taylor-series expansion and derivations analogous to Step 1,

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{M}_i(t) \\ =\frac{1}{n}\sum _{i=1}^n[N_i(t)-{\int }_0^t{R}_i(s)d{\text{Λ}}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\widehat{\mathit{\alpha }}}_0^T(Z_i){\mathit{X}}_i\} \\ -R_i(s)d{\text{Λ}}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}] \\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^t{R}_i(s)({\widehat{\mathit{\alpha }}}_0^T(Z_i)-{\mathit{f}}_0^T(Z_i)){\mathit{W}}_{i}d{\lambda }_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(n^{-1/2}) \\ +{\int }_0^t\frac{B_2(s)}{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(s;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\}]+o_p(n^{-1/2}). \end{gathered}$$ (28)

Let

$${\mathit{U}}_3(\mathit{\beta },{\widehat{H}}_0(t;\mathit{\beta }),{\widehat{\mathit{\alpha }}}_0)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_i[d{N}_i(t)-R_i(t)d{\text{Λ}}_{ϵ}\{{\widehat{H}}_0(t;\mathit{\beta })+{\mathit{\beta }}^T{\mathit{X}}_i+{\widehat{\mathit{\alpha }}}_0^T(Z_i){\mathit{W}}_i\}].$$

By arguments similar to those used in Steps 2 and 3, we have

$$\begin{gathered} 0={\mathit{U}}_3(\widehat{\mathit{\beta }},{\widehat{H}}_0(t;\widehat{\mathit{\beta }}),{\widehat{\mathit{\alpha }}}_0) \\ ={\mathit{U}}_3({\mathit{\beta }}_0,{\widehat{H}}_0(t;{\mathit{\beta }}_0),{\mathit{f}}_0)-{\mathit{A}}_1(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)-\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(t){\mathit{X}}_i{\mathit{W}}_i^T({\widehat{\mathit{\alpha }}}_0(Z_i)-{\mathit{f}}_0(Z_i)) \\ \times d{\lambda }_{ϵ}\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(n^{-1/2}). \end{gathered}$$

However, because

$$\begin{gathered} {\mathit{U}}_3({\mathit{\beta }}_0,{\widehat{H}}_0(t;{\mathit{\beta }}_0),{\mathit{f}}_0) \\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(t){\mathit{X}}_{i}d[{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ -{\text{Λ}}_{ϵ}\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}]+\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t) \\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t) \\ -\frac{1}{n}\sum _{i=1}^n{R}_i(t){\mathit{X}}_i\frac{{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]+o_p(n^{-1/2}), \end{gathered}$$

we obtain

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t) \\ ={\mathit{A}}_1(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(t){\mathit{X}}_i{\mathit{W}}_i^T({\widehat{\mathit{\alpha }}}_0(Z_i)-{\mathit{f}}_0(Z_i)) \\ \times d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ +\frac{1}{n}\sum _{i=1}^n{R}_i(t){\mathit{X}}_i\frac{{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}} \\ \times [{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]+o_p(n^{-1/2}). \end{gathered}$$ (29)

Substituting (26) into (28) yields

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{M}_i(t)={\int }_0^t\frac{B_2(s)}{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(s;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\}] \\ +\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(s)({\mathit{e}}_{31}^{-1}(z){\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(z)-{\mathit{e}}_{31}^{-1}(z){\mathit{e}}_1^T(z)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0) \\ {-{\int }_0^{\tau }{\mathit{e}}_{31}^{-1}(z){\mathit{e}}_2(z,t)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}])}^T \\ \times {\mathit{W}}_{i}d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(n^{-1/2}). \end{gathered}$$

Define

$$\begin{gathered} d{\mathit{J}}_1(t)=E[R(t){\mathit{W}}^T{\mathit{e}}_{31}^{-1}(Z){\mathit{e}}_1^T(Z)d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^{T}X+{\mathit{f}}_0^T(Z)\mathit{W}\}],\text{and} \\ {J}_2(s,t)=E[R(t){\mathit{e}}_2^T(Z,s){\mathit{e}}_{31}^{-1}(Z)\mathit{W}{\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}]. \end{gathered}$$

Then

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{M}_i(t)-\frac{1}{n}\sum _{i=1}^n{\int }_0^t{R}_i(s){\mathit{U}}_{21}^T({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i){\mathit{e}}_{31}^{-1}(Z_i){\mathit{W}}_{i}d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ ={\int }_0^t\frac{B_2(s)}{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(s;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\}]-{\int }_0^{t}d{\mathit{J}}_1(s)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0) \\ -{\int }_0^t{\int }_0^{\tau }{J}_2(s,u)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(u;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(u)\}]d{H}_0(s)+o_p(n^{-1/2}). \end{gathered}$$ (30)

Let ${\mathit{A}}_{22}={\int }_0^{\tau }\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}d{\mathit{J}}_1(t)$. Then we have

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}d{M}_i(t)-\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}{R}_i(t) \\ \times {\mathit{U}}_{21}^T({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i){\mathit{e}}_{31}^{-1}(Z_i){\mathit{W}}_{i}d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ ={\int }_0^{\tau }\mathit{\alpha }(t)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]-\left({\int }_0^{\tau }\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}d{\mathit{J}}_1(t)\right)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0) \\ -{\int }_0^{\tau }\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}{\int }_0^{\tau }{J}_2(s,t)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(s;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\}]d{H}_0(t)+o_p(n^{-1/2}) \\ ={\int }_0^{\tau }\left[\mathit{\alpha }(t)-{\int }_0^{\tau }\mathit{\alpha }(s)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}{J}_2(s,t)d{H}_0(s)\right]d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}] \\ -{\mathit{A}}_{22}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+o_p(n^{-1/2}). \end{gathered}$$ (31)

On the other hand, substituting (26) into (29), we have

$$\begin{gathered} \frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t) \\ ={\mathit{A}}_1(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+\frac{1}{n}\sum _{i=1}^n{R}_i(t){\mathit{X}}_i\frac{{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}] \\ +\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(t){\mathit{X}}_i{\mathit{W}}_i^T({\mathit{e}}_{31}^{-1}(Z_i){\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i)-{\mathit{e}}_{31}^{-1}(Z_i){\mathit{e}}_1^T(Z_i)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0) \\ -{\int }_0^{\tau }{\mathit{e}}_{31}^{-1}(Z_i){\mathit{e}}_2(Z_i,t)d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]) \\ \times d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(n^{-1/2}). \end{gathered}$$

Let

$$\begin{gathered} {\mathit{J}}_3(t)=E\left[R(t)\mathit{X}\frac{{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}\right], \\ {\mathit{J}}_4(t)={\int }_0^{\tau }E\left[\mathit{X}R(s){\mathit{W}}^T{\mathit{e}}_{31}^{-1}(Z){\mathit{e}}_2(Z,t){\dot{\lambda }}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}\right]d{H}_0(s),\text{and} \\ {\mathit{A}}_{21}={\int }_0^{\tau }E\left[\mathit{X}R(t){\mathit{W}}^T{\mathit{e}}_{31}^{-1}(Z){\mathit{e}}_1^T(Z){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}\right]d{H}_0(t). \end{gathered}$$

Then by the law of large numbers, we obtain

$$\begin{gathered} ({\mathit{A}}_1-{\mathit{A}}_{21})(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+{\int }_0^{\tau }[{\mathit{J}}_3(t)-{\mathit{J}}_4(t)]d[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]-\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_{i}d{M}_i(t) \\ =-\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(t){\mathit{X}}_i{\mathit{W}}_i^T{\mathit{e}}_{31}^{-1}(Z_i){\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i) \\ \times d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(n^{-1/2}). \end{gathered}$$ (32)

By the assumption of Theorem 1, and from (31) and (32),

$$({\mathit{A}}_1-{\mathit{A}}_2)(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }[{\mathit{X}}_i-\mathit{m}(t)]d{M}_i(t)-({\mathit{G}}_1-{\mathit{G}}_2)+o_p(n^{-1/2}),$$ (33)

where

$$\begin{gathered} \mathit{m}(t)=\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}, \\ {\mathit{A}}_2={\mathit{A}}_{21}-{\mathit{A}}_{22}, \\ {\mathit{G}}_1=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{R}_i(t){\mathit{X}}_i{\mathit{W}}_i^T{\mathit{e}}_{31}^{-1}(Z_i){\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i)d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\},\text{and} \\ {\mathit{G}}_2=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }\mathit{\alpha }(t)\frac{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}{B_2(t)}{R}_i(t){\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i) \\ {\mathit{e}}_{31}^{-1}(Z_i){\mathit{W}}_{i}d{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}. \end{gathered}$$

On the other hand, by the Taylor-series expansion and the definition of ${\mathit{U}}_{21}({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)$, we have

$${\mathit{G}}_1=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{X}}_i^{\ast }d{M}_i(t)+o_p(n^{-1/2}),$$ (34)

and

$${\mathit{G}}_2=\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{\mathit{m}}_i^{\ast }d{M}_i(t)+o_p(n^{-1/2}).$$ (35)

Substituting (34) and (35) into (33) leads to (13). This completes the proof of Theorem 1.

To prove Theorem 2, we need the following lemma:

Lemma 2. Assume that conditions (C1) to (C8) hold. Then ${\widehat{\mu }}_n(t)=\sqrt{n}[{\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;{\mathit{\beta }}_0)\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}]$ satisfies the following integral equation asymptotically:

$${\widehat{\mu }}_n(t)-{\int }_0^{\tau }a(s,t)d{\widehat{\mu }}_n(s)=W_n(t),t\in (0,\tau ],$$ (36)

where $a(s,t)={\int }_0^{\tau }\frac{{\lambda }_{ϵ}^{\ast }\{H_0(u)\}}{B_2(u)}{\mathit{J}}_2(s,u)d{H}_0(u)$, and $W_n(t)=n^{-1/2}\sum _{i=1}^n{w}_i(t)$ is a sum of independent mean zero functions.

Remark 5. By using integration by parts, we can write (36) as a Fredholm integral equation of the second kind with the kernel $\frac{\partial a(t,s)}{\partial s}$ (see, for example, Press et al., 1992, pp.782-785), i.e.,

$${\widehat{\mu }}_n(t)+{\int }_0^{\tau }{\widehat{\mu }}_n(s)\frac{\partial a(s,t)}{\partial s}ds=W_n(t)+{a(s,t){\widehat{\mu }}_n(s)|}_{s=0}^{\tau }.$$

In order for (36) to yield a unique solution, we assume that

$$\underset{t\in (0,\tau ]}{\sup }{\int }_0^{\tau }|\frac{\partial a(s,t)}{\partial s}|ds<\infty .$$ (37)

By arguments analogous argument to Lu and Zhang (2010), we can construct the following solution to (36):

$${\widehat{\mu }}_n(t)=W_n(t)+{\int }_0^{\tau }b(s,t)d{W}_n(s),$$ (38)

where b(t, s) is the unique solution to

$$b(s,t)-{\int }_0^{\tau }a(s,t)\frac{\partial b(s,u)}{\partial u}du=a(s,t),s,t\in (0,\tau ].$$

Thus, given condition (37), ${\widehat{\mu }}_n(t)$ as defined in (38) is the unique solution to the integral equation (36).

Proof of Lemma 2. From (30), we have

$$\begin{gathered} {\widehat{\mu }}_n(t)-{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{\mathit{J}}_1(s)\sqrt{n}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)-{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(u)\}}{B_2(u)}{\int }_0^{\tau }{\mathit{J}}_2(s,u)d\{{\widehat{\mu }}_n(s)\}d{H}_0(u) \\ =\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{M}_i(s)-\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}{R}_i(s){\mathit{U}}_{21}^T({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i) \\ \times {\mathit{e}}_{31}^{-1}(Z_i){\mathit{W}}_{i}d{\lambda }_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(1). \end{gathered}$$ (39)

Let $a(s,t)={\int }_0^{\tau }\frac{{\lambda }_{ϵ}^{\ast }\{H_0(u)\}}{B_2(u)}{J}_2(s,u)d{H}_0(u)$. Then (39) becomes

$$\begin{gathered} {\widehat{\mu }}_n(t)-{\int }_0^{\tau }a(s,t)d{\widehat{\mu }}_n(s) \\ ={\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{\mathit{J}}_1(s)\sqrt{n}(\widehat{\mathit{\beta }}-{\mathit{\beta }}_0)+\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{M}_i(s) \\ -\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}{R}_i(s){\mathit{U}}_{21}^T({\mathit{f}}_0,{\dot{\mathit{f}}}_0,H_0,{\mathit{\beta }}_0)(Z_i){\mathit{e}}_{31}^{-1}(Z_i){\mathit{W}}_i \\ \times d{\lambda }_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}+o_p(1). \end{gathered}$$

By (13) and arguments similar to (25), we have

$$\begin{gathered} {\widehat{\mu }}_n(t)-{\int }_0^{\tau }a(s,t)d\{{\widehat{\mu }}_n(s)\} \\ =\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{M}_i(s)-\frac{1}{\sqrt{n}}\sum _{i=1}^n{\int }_0^t{\tilde{\mathit{m}}}_i(s)d{M}_i(s) \\ +\frac{1}{\sqrt{n}}\sum _{i=1}^n\left({\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}d{\mathit{J}}_1(s)\right){\mathit{A}}^{-1}{\int }_0^{\tau }[({\mathit{X}}_i-\mathit{m}(t))-({\mathit{X}}_i^{\ast }-{\mathit{m}}_i^{\ast })]d{M}_i(t)+o_p(1) \\ \triangleq \frac{1}{\sqrt{n}}\sum _{i=1}^n{\mathit{W}}_i(t)+o_p(1), \end{gathered}$$

where

$${\tilde{\mathit{m}}}_l(t)={\int }_0^t\frac{{\lambda }_{ϵ}^{\ast }\{H_0(s)\}}{B_2(s)}E[R(s){\mathit{W}}^T{\dot{\lambda }}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}\mid Z=Z_l]{\mathit{e}}_{31}^{-1}(Z_l){\mathit{W}}_{l}g(Z_l)d{H}_0(s),l=1,\dots ,n.$$

This completes the proof.

Proof of Theorem 2. By the Taylor-series expansion, Lemma 2 and (13), we have

$$\begin{gathered} \sqrt{n}({\text{Λ}}_{ϵ}^{\ast }\{{\widehat{H}}_0(t;\widehat{\mathit{\beta }})\}-{\text{Λ}}_{ϵ}^{\ast }\{H_0(t)\}) \\ =-{\int }_0^t\frac{E[R(s){\mathit{X}}^T{\dot{\lambda }}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}]}{B_2(s)}d{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\} \\ \frac{1}{\sqrt{n}}\sum _{i=1}^n{\mathit{A}}^{-1}{\int }_0^{\tau }[({\mathit{X}}_i-\mathit{m}(t))-({\mathit{X}}_i^{\ast }-{\mathit{m}}_i^{\ast })]d{M}_i(t) \\ +W_n(t)+{\int }_0^{\tau }b(s,t)d{W}_n(s)+o_p(1) \\ =\frac{1}{\sqrt{n}}\sum _{i=1}^n{𝒦}_i(t)+o_p(1). \end{gathered}$$

where

$$\begin{gathered} {𝒦}_i(t)={\int }_0^{\tau }\frac{E[R(s){\mathit{X}}^T{\dot{\lambda }}_{ϵ}\{H_0(s)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(Z)\mathit{W}\}]}{B_2(s)}d{\text{Λ}}_{ϵ}^{\ast }\{H_0(s)\} \\ \times {\mathit{A}}^{-1}{\int }_0^{\tau }[({\mathit{X}}_i-m(t))-({\mathit{X}}_i^{\ast }-{\mathit{m}}_i^{\ast })]d{M}_i(t)+w_i(t) \\ +{\int }_0^{\tau }b(s,t)d{w}_i(s). \end{gathered}$$

This yields

$$\sqrt{n}[\widehat{H}(t)-H_0(t)]=\frac{1}{\sqrt{n}}\sum _{i=1}^n\frac{{𝒦}_i(t)}{{\lambda }_{ϵ}^{\ast }\{H_0(t)\}}+o_p(1),$$

which can be shown to converge weakly to a mean zero Gaussian Process by the functional central limit theorem. This completes the proof.

Proof of Theorem 3. By Theorems 1 and 2, and applying the Taylor-series expansion, we have

$$\begin{gathered} {\sup }_{t\in (0,\tau ]}\Vert \frac{1}{n}\sum _{i=1}^n{K}_h(Z_i-z)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right)[{\Lambda }_{ϵ}\{\widehat{H}(t)+{\widehat{\mathit{\beta }}}^T{\mathit{X}}_i+{\widehat{\mathit{\alpha }}}_0^T(Z_i){\mathit{W}}_i+{\widehat{\mathit{\alpha }}}_1^T(z){\mathit{W}}_i(Z_i-z)\} \\ +{\Lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\widehat{\mathit{\alpha }}}_0^T(Z_i){\mathit{W}}_i+{\widehat{\mathit{\alpha }}}_1^T(z){\mathit{W}}_i(Z_i-z)\}]\Vert =O_p(n^{-1/2}). \end{gathered}$$ (40)

However, ${\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,\widehat{H},\widehat{\mathit{\beta }})=0$, hence ${\mathit{U}}_2({\widehat{\mathit{\alpha }}}_0,{\widehat{\mathit{\alpha }}}_1,H_0,{\mathit{\beta }}_0)=O_p(n^{-1/2})=o_p(1/\sqrt{nh})$. Let $\tilde{\mathit{\alpha }}(z)={({\mathit{\alpha }}_0^T(z),h{\mathit{\alpha }}_1^T(z))}^T$, ${\tilde{\mathit{\alpha }}}_0(z)={({\mathit{f}}_0^T(z),h{\dot{\mathit{f}}}_0^T(z))}^T$, and $\widehat{\tilde{\mathit{\alpha }}}(z)={({\widehat{\mathit{\alpha }}}_0^T(z),h{\widehat{\mathit{\alpha }}}_1^T(z))}^T$. Then by the Taylor-series expansion, we obtain

$${\mathit{U}}_2(\widehat{\tilde{\mathit{\alpha }}},H_0,{\mathit{\beta }}_0)={\mathit{U}}_2({\tilde{\mathit{\alpha }}}_0,H_0,{\mathit{\beta }}_0)+\frac{\partial }{\partial \tilde{\mathit{\alpha }}}{\mathit{U}}_2({\tilde{\mathit{\alpha }}}^{\ast },H_0,{\mathit{\beta }}_0)\{\widehat{\tilde{\mathit{\alpha }}}(z)-{\tilde{\mathit{\alpha }}}_0(z)\},$$ (41)

where ${\tilde{\mathit{\alpha }}}^{\ast }$ lies between $\widehat{\tilde{\mathit{\alpha }}}(z)$ and ${\tilde{\mathit{\alpha }}}_0(z)$, and thus ${\tilde{\mathit{\alpha }}}^{\ast }{\to }_p{\tilde{\mathit{\alpha }}}_0$. Hence by the law of large numbers, we have

$$-\frac{\partial {\mathit{U}}_2({\tilde{\mathit{\alpha }}}^{\ast },H_0,{\mathit{\beta }}_0)}{\partial \tilde{\mathit{\alpha }}}{\to }_{a.s.}{\Gamma }_1(z),$$ (42)

where

$${\mathrm{\Gamma }}_1(w)=E\left[\left(\begin{array}{cc}\mathit{W}{\mathit{W}}^T& 0\\ 0& \mathit{W}{\mathit{W}}^T{k}_2\end{array}\right)R(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}|Z=z\right].$$

Substituting (4) into ${\mathit{U}}_2(\tilde{\mathit{\alpha }},H_0,{\mathit{\beta }}_0)$, we get

$$\begin{gathered} {\mathit{U}}_2({\tilde{\mathit{\alpha }}}_0,H_0,{\mathit{\beta }}_0) \\ =\frac{1}{n}\sum _{i=1}^n{\int }_0^{\tau }{K}_h(Z_i-z)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right)d{M}_i(t)+\frac{1}{n}\sum _{i=1}^n{K}_h(Z_i-z)\left(\begin{array}{c}{\mathit{W}}_i\\ {\mathit{W}}_i\left(\frac{Z_i-z}{h}\right)\end{array}\right)R_i(t) \\ \times [{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T{Z}_i){\mathit{W}}_i\}-{\text{Λ}}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(z){\mathit{W}}_i+{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i(Z_i-z)\}] \\ \triangleq {\mathit{I}}_1+{\mathit{I}}_2. \end{gathered}$$

By the martingale central limit theorem, we have

$$\sqrt{nh}{\mathit{I}}_1{\to }_{d}N(0,{\Sigma }_2(z)),$$

where

$${\mathbf{\Sigma }}_2(z)=hE{\left[{\int }_0^{\tau }{K}_h(Z-z)\left(\begin{array}{c}\mathit{W}\\ \mathit{W}\left(\frac{Z-z}{h}\right)\end{array}\right)dM(t)\right]}^{\otimes 2}.$$ (43)

Let

$$\begin{gathered} {\widetilde{\text{Λ}}}_{ϵ}(Z_i)=R_i(t)[{\Lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\} \\ -{\Lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(z){\mathit{W}}_i+{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i(Z_i-z)\}]. \end{gathered}$$

Expanding ${\widetilde{\text{Λ}}}_{ϵ}(Z_i)$ at z by the Taylor-series expansion yields

$${\widetilde{\text{Λ}}}_{ϵ}(Z_i)={\widetilde{\text{Λ}}}_{ϵ}(z)+{\widetilde{\text{Λ}}}_{ϵ}^{\prime }(z)(Z_i-z)+\frac{{\widetilde{\text{Λ}}}_{ϵ}^{″}(z)}{2}{(Z_i-z)}^2+o_p({(Z_i-z)}^2),$$ (44)

where ${\widetilde{\text{Λ}}}_{ϵ}^{\prime }(z)$ and ${\widetilde{\text{Λ}}}_{ϵ}^{″}(z)$ denote the first and second derivatives of function ${\widetilde{\text{Λ}}}_{ϵ}(z)$ respectively. It follows from the definition of ${\widetilde{\text{Λ}}}_{ϵ}(Zi)$ that

$$\begin{gathered} {\widetilde{\text{Λ}}}_{ϵ}^{\prime }(Z_i)=R_i(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}{\dot{\mathit{f}}}_0^T(Z_i){\mathit{W}}_i \\ -R_i(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(z){\mathit{W}}_i+{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i(Z_i-z)\}{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i,\text{and} \\ {\widetilde{\text{Λ}}}_{ϵ}^{″}(Z_i)=R_i(t){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}{({\dot{\mathit{f}}}_0^T(Z_i){\mathit{W}}_i)}^2 \\ +R_i(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(Z_i){\mathit{W}}_i\}{\ddot{\mathit{f}}}_0^T(Z_i){\mathit{W}}_i \\ -R_i(t){\dot{\lambda }}_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(z){\mathit{W}}_i+{\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i(Z_i-z)\}{({\dot{\mathit{f}}}_0^T(z){\mathit{W}}_i)}^2. \end{gathered}$$

Substituting these two equations into (44), we get

$${\widetilde{\text{Λ}}}_{ϵ}(Z_i)=\frac{1}{2}{R}_i(t){\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T{\mathit{X}}_i+{\mathit{f}}_0^T(z){\mathit{W}}_i\}{\ddot{\mathit{f}}}_0^T{\mathit{W}}_i{(Z_i-z)}^2+o_p({(Z_i-z)}^2).$$ (45)

Plugging (45) into the definition of I₂, and by nonparametric techniques, we can obtain

$$\begin{gathered} {\mathit{I}}_2=E\left[\left(\begin{array}{c}{\mathit{W}}^{\otimes 2}{k}_2\\ 0\end{array}\right)\frac{h^2}{2}{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{X}+{\mathit{f}}_0^T(z)\mathit{W}\}|Z=z\right]g(z){\ddot{\mathit{f}}}_0(z)+o_p(h^2) \\ ={\Gamma }_1(z){\mathit{b}}_n(z)+o_p(h^2), \end{gathered}$$ (46)

where

$${\mathit{b}}_n(z)={\mathbf{\Gamma }}_1^{-1}(z)g(z){\ddot{\mathit{f}}}_0(z)E\left[\left(\begin{array}{c}{\mathit{W}}^{\otimes 2}{k}_2\\ 0\end{array}\right)\frac{h^2}{2}{\lambda }_{ϵ}\{H_0(t)+{\mathit{\beta }}_0^T\mathit{W}+{\mathit{f}}_0^T(z)\mathit{W}\}|Z=z\right].$$

The proof of Theorem 3 is complete by combining the results of (42), (43) and (46), and applying the Slutsky Theorem.