Combined estimating equation approaches for the additive hazards model with left-truncated and interval-censored data

Abstract

Interval-censored failure time data arise commonly in various scientific studies where the failure time of interest is only known to lie in a certain time interval rather than observed exactly. In addition, left truncation on the failure event may occur and can greatly complicate the statistical analysis. In this paper, we investigate regression analysis of left-truncated and interval-censored data with the commonly used additive hazards model. Specifically, we propose a conditional estimating equation approach for the estimation, and further improve its estimation efficiency by combining the conditional estimating equation and the pairwise pseudo-score-based estimating equation that can eliminate the nuisance functions from the marginal likelihood of the truncation times. Asymptotic properties of the proposed estimators are discussed including the consistency and asymptotic normality. Extensive simulation studies are conducted to evaluate the empirical performance of the proposed methods, and suggest that the combined estimating equation approach is obviously more efficient than the conditional estimating equation approach. We then apply the proposed methods to a set of real data for illustration.

1 Introduction

Interval-censored failure time data arise frequently in many scientific studies including clinical trials, epidemiological surveys and cancer screening studies, and due to the periodical follow-ups in these situations, the failure time of interest cannot be monitored exactly but is only known to fall into a certain time interval. For example, in HIV/AIDS studies, urine samples are intermittently provided by the individuals at their clinical visits to examine the occurrence of the CMV shedding. Therefore, the CMV shedding time is only known to occur in a time interval formed by dates of the last negative urine test and the first positive urine test (Goggins and Finkelstein 2000). Another example of interval-censored data is given by Finkelstein (1986), where the failure event of interest is the occurrence of breast retraction among early breast cancer patients.

In practice, one may encounter several types of interval-censored data. One type of interval-censored data that has been discussed vastly in the literature is called case I interval-censored data or current status data (Sun 1999; Li et al. 2017, 2021; Ma et al. 2015). Such data arise when each individual under study is observed only once at one monitoring time and the failure event of interest is known only to occur before or after this monitoring time. In other words, the failure time of interest is either left- or right-censored rather than observed exactly, and the interval containing the failure time includes either zero or infinity. When there exist two or multiple observation times for each subject, the obtained data are usually referred to as case II or case K interval-censored data depending on the number of observations (Sun 2006). In this situation, the failure time of interest falls into a certain time interval, which may be given by the half-open interval or the interval with finite endpoints.

In many applications, the failure times of interest are often subject to left truncation under the cross-sectional sampling scheme, where only individuals that have not experienced the failure event at the study enrolment are included in the study. Therefore, the collected data suffer from the biased sampling since they are not representative for the whole population under study. For the analysis of left-truncated data, a large number of methods have been proposed under right censoring (Lai and Ying 1991; Wang et al. 1993; Gross and Lai 1996; Shen et al. 2009; Qin and Shen 2010; Huang and Qin 2013; Shen et al. 2017; Wu et al. 2018). For instance, Wang et al. (1993) proposed a partial likelihood-based estimation method for the proportional hazards model; Lai and Ying (1991) developed some rank-based estimators for the censored linear regression; Shen et al. (2009) assumed that the truncation times satisfy the stationary assumption and proposed some unbiased estimating equation approaches to estimate the transformation and accelerated failure time models; Huang and Qin (2013) and Wu et al. (2018) considered the general left-truncated data and proposed the pairwise likelihood-based methods for the additive and proportional hazards models, respectively.

Since interval-censored data are often subject to the left truncation in scientific fields, some methods have already been developed for their analysis (Pan and Chappell 1998, 2002; Kim 2003; Wang et al. 2015a, b; Gao and Chan 2019; Shen et al. 2019; Wang et al. 2021). In particular, for the left-truncated and case I interval-censored data, Kim (2003) and Wang et al. (2015b) proposed the conditional maximum likelihood estimation approaches for the proportional and additive hazards models, respectively. With respect to the left-truncated and general interval-censored data, Pan and Chappell (2002) proposed a marginal likelihood approach as well as a monotone maximum likelihood approach for the proportional hazards model; Wang et al. (2015a) proposed a conditional sieve maximum likelihood estimation method for the additive hazards model; Shen et al. (2019) developed a conditional likelihood estimation approach for the general transformation models. It is worth noting that the aforementioned methods were established without incorporating the marginal distribution information of the observed truncation times, which may lose some estimation efficiency as anticipated. Corresponding to this, Gao and Chan (2019) proposed a nonparametric maximum likelihood estimation approach for the proportional hazards model under the stationary assumption for the truncation times. Motivated by the work of Huang and Qin (2013); Wang et al. (2021) further proposed a pairwise likelihood estimation approach for the proportional hazards model.

This study aims at investigating regression techniques for the left-truncated and interval-censored data with the additive hazards model. The motivations of considering the additive hazards model are mainly threefold. Firstly, the additive hazards model serves as an essential alternative to the commonly adopted proportional hazards model in survival analysis. This is because that the additive hazards model provides a different perspective of covariate effect on the failure time compared with the proportional hazards model and may be more plausible than the latter in some practical applications (Aranda-Ordaz 1983; Buckley 1984; Lin and Ying 1994; Lin et al. 1998). Secondly, unlike the proportional hazards model, it is easy to derive a tractable estimating equation for the additive hazards model under interval censoring by using the counting process formulations (Lin et al. 1998; Wang et al. 2010). Therefore, one can easily adapt the established estimating equations to accommodate left truncation. Thirdly, under the additive hazards model, the interpretation of the estimated covariate effects can be converted into that of the ratio of the estimated survival probabilities between different groups at a given time, which does not involve the baseline hazard function. Therefore, even without estimating the baseline hazard function, the additive hazards model with only the estimated regression parameters still has appealing interpretations as commented in the real data analysis of Sect. 6.

A major difficulty in estimating the additive hazards model with left-truncated and interval-censored data is how to develop a reliable and easily implemented estimation procedure. The first contribution of this work is to develop a simple conditional estimating equation method through modifying the risk sets of the estimating equation proposed by Lin et al. (1998) for analyzing the left-truncated and case I interval-censored data. Since the conditional estimating equation method is found to be very inefficient, we further improve its estimation efficiency by developing a combined estimating equation approach with the use of the marginal likelihood of the truncation times. A desirable feature of the two proposed estimating equations is that they have simple and tractable form, and do not involve any nuisance functions including the baseline hazard function in the additive hazards model and the density function of the truncation time. Therefore, they can be readily solved by the simple and standard optimization procedure, such as the Newton–Raphson algorithm.

Since the left-truncated and general interval-censored data are also frequently encountered in many applications and have complicated structure, the second contribution of the work is to develop two easily implemented methods to handle such data. Specifically, we first derive a conditional estimating equation approach by utilizing the working independence strategy for interval-censored data (Betensky et al. 2001; Zhu et al. 2008). Then, as above, a combined estimating equation approach by allowing for the marginal likelihood of the truncation times is further proposed to improve the estimation efficiency of the former method. The numerical results obtained from extensive simulation studies imply that the proposed estimation methods can give reliable and stable performance. In particular, by utilizing the distribution information of the truncation times, the combined estimating equation approach can greatly improve the estimation efficiency of the conditional estimating equation method.

The rest of this article is arranged as follows. In Sect. 2, we introduce some notation, the assumed model as well as the data structure. In Sect. 3, we first consider the left-truncated and case I interval-censored data and develop the conditional and combined estimating equation approaches for the additive hazards model. In Sect. 4, we generalize the proposed estimation methods to the situation of left-truncated and general interval-censored data. In Sect. 5, we conduct extensive simulation studies to investigate the finite-sample performance of the proposed methods, followed by an application to a real dataset in Sect. 6. Some discussions and concluding remarks are given in Sect. 7.

2 Notation, model and data structure

Let $T^{*}$ denote the underlying failure time of interest (i.e. the time to the failure event of interest), and ${{\varvec{Z}}}^{*}$ be a p-dimensional vector of covariates in the target population. Let $A^{*}$ denote the underlying left truncation time (i.e. the time to the study enrolment) that is assumed to be independent of $T^{*}$. When $T^{*}$ suffers from left truncation by $A^{*}$, we know that only subjects with $T^{*} \ge A^{*}$ are collected or sampled in the study. Let T, A, and ${{\varvec{Z}}}$ be the failure time, truncation time and covariate vector of a sampled subject (i.e. satisfying $T \ge A$), respectively. Then $(T, A, {{\varvec{Z}}})$ has the same joint distribution as $(T^{*}, A^{*}, {{\varvec{Z}}}^{*})$ given that $T^{*} \ge A^{*}$. Here it should be noted that the aforementioned independent assumption between $T^*$ and $A^*$ does not imply the independence between T and A since subject with larger A would have larger T under the left truncation scheme (Wu et al. 2018).

To describe the effects of ${{\varvec{Z}}}^{*}$ on $T^{*}$, we assume that $T^{*}$ follows the additive hazards model with the conditional hazard function

$$\begin{aligned} \lambda (t \mid {{\varvec{Z}}}^{*})=\lambda (t)+\varvec{\beta }^{\top } {{\varvec{Z}}}^{*}, \end{aligned}$$

(1)

where $\lambda (t)$ is the unknown baseline hazard function and $\varvec{\beta }$ is a p-dimensional vector of regression parameters. Thus $\Lambda (t)= \int _0^t \lambda (s) ds$ is the cumulative baseline hazard function of $T^{*}$. Let f and S denote the density and survival functions of $T^{*}$, respectively, and let g be the density function of $A^{*}$. Then the joint density function (A, T) at (a, t) can be expressed as

$$\begin{aligned} h(a,t)=\frac{g(a)f(t)}{\int _0^\infty S(u)g(u)du}=\frac{f(t)}{S(a)} \times \ \frac{S(a)g(a)}{\int _0^\infty S(u)g(u)du} \qquad (0\le a\le t) \end{aligned}$$

where f(t)/S(a) is the conditional density of T given A and $S(a)g(a)/\int _0^\infty S(u)g(u)du$ is the marginal density of A (Huang and Qin 2013).

We first consider the situation of left-truncated and case I interval-censored data, in which each subject is examined only once for the occurrence of the event of interest at the observation or censoring time. Let C denote the observation or censoring time after the study enrolment, and assume that C is conditional independent of (A, T) given ${{\varvec{Z}}}$. Under case I interval censoring mechanism, T cannot be observed directly but is only known to be smaller than $A+C$ or larger than $A+C$. Define $\delta =I(A+C\le T)$, that is, T is right-censored if $\delta =1$ and left-censored otherwise. Then the observed data include n i.i.d. replicates of $\{C, \delta , A, {{\varvec{Z}}}\}$, namely, $\{C_i, \delta _i, A_i, {{\varvec{Z}}}_i; i = 1, \ldots , n\}$.

3 Estimation procedure

In this section, we first develop a conditional estimating equation approach by adapting the estimating equation proposed by Lin et al. (1998) to account for the left truncation, and then develop a combined estimating equation approach to improve the estimation efficiency of the former method.

3.1 Conditional estimating equation approach

For each subject i, let $\tilde{C}_i = A_i+C_i$ and define the counting process $N_i(t)=\delta _{i}I(\tilde{C}_i \le t)$, which jumps by one at time t when (i) subject i is monitored at time t, $\tilde{C}_i = t$, and (ii) subject i is still found to be failure-free at this time, that is, $T_i \ge t$. The process $N_i(t)$ can be regarded as right censoring at time t when subject i is monitored, $\tilde{C}_i = t$, and found to have experienced the failure at this time, $\delta _{i}=0$, since this process always takes the value of 0 under this situation. By following the arguments given in Lin et al. (1998), we can derive the intensity model of $N_i(t)$, which takes the form

$$\begin{aligned} dH_i(t)= R_i(t) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i t\}dH(t), \end{aligned}$$

(2)

where $R_i(t) =I(A_i \le t \le \tilde{C}_i)$ is the at-risk process under left truncation, $dH(t)=\exp \{-\Lambda (t)\}d\Lambda _c(t)$, and $\Lambda _c(t)$ is the cumulative hazard function of $\tilde{C}_i$.

The above expression (2) suggests that

$$\begin{aligned} M_i(t)=N_i(t)-\int _{0}^t R_i(s) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_i s\}dH(s) \quad (i=1, \ldots , n) \end{aligned}$$

are martingales with respect to the $\sigma$-filtration: $\sigma \{N_i(s), R_i(s), {{\varvec{Z}}}_i; s\le t,i=1, \ldots , n\}$. Thus, one can make inference about $\varvec{\beta }$ by applying the partial likelihood approach to (2). Specifically, the partial likelihood score function $\varvec{\beta }$ is

$$\begin{aligned} \Phi (\varvec{\beta })=\sum _{i=1}^n\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S^{(1)}(\varvec{\beta },t)}{S^{(0)}(\varvec{\beta },t)}\right\} dN_i(t), \end{aligned}$$

(3)

and the observed information matrix for $\varvec{\beta }$ is

$$\begin{aligned} I(\varvec{\beta })=\sum _{i=1}^n\int _{0}^{\tau } \left\{ \frac{S^{(2)}(\varvec{\beta },t)}{S^{(0)}(\varvec{\beta },t)}-\frac{S^{(1)}(\varvec{\beta },t)^{\otimes {2}}}{S^{(0)}(\varvec{\beta },t)^2}\right\} dN_i(t), \end{aligned}$$

(4)

where

$$\begin{aligned} S^{(k)}(\varvec{\beta },t)=\sum _{j=1}^n R_j(t) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_jt\}( {{\varvec{Z}}}_jt)^{\otimes {k}}, \end{aligned}$$

${{\varvec{a}}}^{\otimes {0}}=1, {{\varvec{a}}}^{\otimes {1}}={{\varvec{a}}}$ and ${{\varvec{a}}}^{\otimes {2}}={{\varvec{a}}}{{\varvec{a}}}^{\top }$ for a column vector ${{\varvec{a}}}$, and $\tau$ is a prespecified constant such that $P(A+C \ge \tau ) > 0$.

After normalising, the conditional estimating equation for $\varvec{\beta }$ is given by

$$\begin{aligned} U(\varvec{\beta })=\frac{1}{n}\Phi (\varvec{\beta }) = \frac{1}{n}\sum _{i=1}^n\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S^{(1)}(\varvec{\beta },t)}{S^{(0)}(\varvec{\beta },t)}\right\} dN_i(t) = 0. \end{aligned}$$

(5)

Solving $U(\varvec{\beta })=0$ yields the estimate of $\varvec{\beta }$, which is denoted by $\hat{\varvec{\beta }}_U$. Let $\varvec{\beta }_0$ be the true value of $\varvec{\beta }$. The following theorem shows the asymptotic property of $\hat{\varvec{\beta }}_U$, and the proof will be presented in the “Appendix A”.

Theorem 1

Under Conditions (C1)–(C3) given in the “Appendix A”, we have that, as $n\rightarrow \infty$, $\sqrt{n}(\hat{\varvec{\beta }}_U -\varvec{\beta }_0)$ converges in distribution to a zero-mean normal random vector with covariance matrix $\Sigma (\varvec{\beta }_0)^{-1}$, where $\Sigma (\varvec{\beta }_0)$ is the limit of $n^{-1}I(\varvec{\beta }_0)$ that can be consistently estimated by $n^{-1}I(\hat{\varvec{\beta }}_U)$.

Note that the conditional estimating function proposed above is anticipated to loss some estimation efficiency since it does not incorporate the distribution information of the truncation times. To improve the efficiency, in what follows, we will provide a combined estimating equation approach by combining the conditional estimating equation and the pseudo-score-based estimating equation obtained from the pairwise information of the truncation times.

3.2 Combined estimating equation approach

To allow for the information provided by the truncation times, we adopt the pairwise likelihood method given in Kalbfleisch (1978) and Liang and Qin (2000). As in Huang and Qin (2013); Wu et al. (2018) and others, assume that the truncation time A does not depend on ${{\varvec{Z}}}$ and is not degenerate. Note that all observed truncation times satisfy $A<\tau$ in practical situations since we often set $\tau$ to be the maximum of the observed observation times. Note that, for $i < j$, by conditioning on having observed $(A_i, A_j)$ but without knowing the order of $A_i$ and $A_j$, the pairwise pseudo-likelihood of the observed $(A_i, A_j)$ conditional on $A_i \le \tau$, $A_j \le \tau$ and $({{\varvec{Z}}}_i, {{\varvec{Z}}}_j)$ is

$$\begin{aligned} \frac{\frac{S(A_i \mid {{\varvec{Z}}}_i)g(A_i)}{\int _0^{\tau }S(u \mid {{\varvec{Z}}}_i)g(u)du} \times \frac{S(A_j \mid {{\varvec{Z}}}_j)g(A_j)}{\int _0^{\tau }S(u \mid {{\varvec{Z}}}_j)g(u)du}}{\frac{S(A_i \mid {{\varvec{Z}}}_i)g(A_i)}{\int _0^{\tau }S(u \mid {{\varvec{Z}}}_i)g(u)du} \times \frac{S(A_j \mid {{\varvec{Z}}}_j)g(A_j)}{\int _0^{\tau }S(u \mid {{\varvec{Z}}}_j)g(u)du} +\frac{S(A_i \mid {{\varvec{Z}}}_j)g(A_i)}{\int _0^{\tau }S(u \mid {{\varvec{Z}}}_j)g(u)du} \times \frac{S(A_j \mid {{\varvec{Z}}}_i)g(A_j)}{\int _0^{\tau }S(u \mid {{\varvec{Z}}}_i)g(u)du}}, \end{aligned}$$

which equals

$$\begin{aligned}{} & {} \frac{\exp (-\varvec{\beta }^{\top } {{\varvec{Z}}}_iA_i-\varvec{\beta }^{\top } {{\varvec{Z}}}_jA_j)}{\exp (-\varvec{\beta }^{\top } {{\varvec{Z}}}_iA_i-\varvec{\beta }^{\top } {{\varvec{Z}}}_jA_j)+\exp (-\varvec{\beta }^{\top } {{\varvec{Z}}}_jA_i-\varvec{\beta }^{\top } {{\varvec{Z}}}_iA_j)}\\{} & {} \quad =\frac{1}{1+\exp \{\varvec{\beta }^{\top } ({{\varvec{Z}}}_i-{{\varvec{Z}}}_j)(A_i-A_j)\}}. \end{aligned}$$

It is worth noting that the pairwise pseudo likelihood depends on the regression parameter $\varvec{\beta }$ of the assumed additive hazards model (1) and may render some additional information to improve the estimation efficiency. Moreover, as a by-product, the pairwise likelihood method used above can enable us to eliminate the nuisance functions in the marginal distribution of the truncation times, which can greatly facilitate the subsequent estimation procedure. Define $\rho _{ij}=\rho (A_i, {{\varvec{Z}}}_i, A_j, {{\varvec{Z}}}_j)=({{\varvec{Z}}}_i-{{\varvec{Z}}}_j)(A_i-A_j)$. Then the logarithm of the pairwise pseudo-likelihood and the normalised score function of $\varvec{\beta }$ are given by

$$\begin{aligned} \sum _{1 \le i<j \le n} - \log \left\{ 1+\exp (\varvec{\beta }^{\top } \rho _{ij})\right\} , \end{aligned}$$

and

$$\begin{aligned} \psi (\varvec{\beta }) =\frac{2}{n(n-1)}\sum _{1 \le i<j \le n}^{}\psi _{ij}(\varvec{\beta }), \end{aligned}$$

(6)

respectively, where

$$\begin{aligned} \psi _{ij}(\varvec{\beta })=\psi _{ij}(\varvec{\beta }; A_i, {{\varvec{Z}}}_i, A_j, {{\varvec{Z}}}_j)=\frac{-\rho _{ij}}{1+\exp (-\varvec{\beta }^{\top }\rho _{ij})}. \end{aligned}$$

Since $U(\varvec{\beta })$ and $\psi (\varvec{\beta })$ both can give consistent estimate of $\varvec{\beta }$, the combined estimating equation proposed is thus defined as

$$\begin{aligned} \epsilon (\varvec{\beta })=U(\varvec{\beta })+\psi (\varvec{\beta })=0. \end{aligned}$$

(7)

Denote by $\hat{\varvec{\beta }}_{\epsilon }$ the estimate of $\varvec{\beta }$ that solves the equation $\epsilon (\varvec{\beta })=0$. By following the lines of Huang and Qin (2013), we have the following theorem on the asymptotic property of $\hat{\varvec{\beta }}_{\epsilon }$, and relegate the proof to the “Appendix A”.

Theorem 2

Under Conditions (C1)–(C3) given in the “Appendix A”, we can conclude that, as $n\rightarrow \infty$, $\sqrt{n} (\hat{\varvec{\beta }}_{\epsilon }-\varvec{\beta }_0)$ converges to a normal distribution with mean zero and covariance matrix $[\Sigma (\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}[\Sigma (\varvec{\beta }_0)+V_1(\varvec{\beta }_0)][\Sigma (\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}$, where $\Sigma (\varvec{\beta }_0)$ is the limit of $n^{-1}I(\varvec{\beta }_0)$, $V_1(\varvec{\beta }_0)=4E\{\psi _{12}(\varvec{\beta }_0)\psi _{13}(\varvec{\beta }_0)\}$ and $V_2(\varvec{\beta }_0)=-E\{\partial \psi _{12}(\varvec{\beta }_0)/\partial \varvec{\beta }\}=E[\rho _{12}^{\otimes 2}\exp (-\varvec{\beta }^{\top }_0\rho _{12})/\{1+\exp (-\varvec{\beta }_0^{\top }\rho _{12})^2\}]$. In particular, $\Sigma (\varvec{\beta }_0)$ can be estimated by $n^{-1}I(\hat{\varvec{\beta }}_{\epsilon })$, and $V_1(\varvec{\beta }_0)$ and $V_2(\varvec{\beta }_0)$ can be estimated by

$$\begin{aligned} \frac{4}{n-1}\sum _{i=1}^{n}\left\{ \frac{1}{n-1}\sum _{j=1,j\ne i}^{n}\psi _{ij}(\hat{\varvec{\beta }}_{\epsilon })\right\} ^{\otimes {2}}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{n(n-1)}\sum _{i\ne j}\frac{\rho _{ij}^{\otimes {2}}\exp (-\hat{\varvec{\beta }}_{\epsilon }^{\top }\rho _{ij})}{\{1+\exp (-\hat{\varvec{\beta }}_{\epsilon }^{\top }\rho _{ij})\}^2}, \end{aligned}$$

respectively (Sen 1960).

4 Generations to the left-truncated and general interval-censored data

In this section, we extend the proposed methods above to the scenario of the left-truncated and general interval-censored data. Without loss of generality, assume that each subject is observed twice after the study enrolment, and the two monitoring times measured from the study enrolment are denoted by U and V, respectively, with $0<U< V <\infty$. In this situation, we have left-censored T if $T-A < U$, right-censored T if $T-A \ge V$, and interval-censored T if $T-A \ge U$ and $T-A< V$, where A is the truncation time as defined above. Define $\delta ^{(1)}=I(A+U \le T)$ and $\delta ^{(2)}=I(A+V\le T)$.

Let $\{U_i, V_i, \delta _{i}^{(1)},\delta _{i}^{(2)}, A_i, {{\varvec{Z}}}_i; i = 1, \ldots , n\}$ be n i.i.d. replicates of $\{U, V, \delta ^{(1)}, \delta ^{(2)}, A, {{\varvec{Z}}}\}$. Define $\tilde{U}_i = A_i+U_i$ and $\tilde{V}_i = A_i+V_i$. As in Lin et al. (1998), for each i, define the counting process $N^{(1)}_{i}(t)=\delta ^{(1)}_{i}I(\tilde{U}_i\le t)$ with the intensity model

$$\begin{aligned} dH^{(1)}_{i}(t)= R^{(1)}_{i}(t) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i t\}dH_{1}(t), \end{aligned}$$

where $R^{(1)}_{i}(t) =I(A_i \le t \le \tilde{U}_i)$, $dH_{1}(t)=\exp \{-\Lambda (t)\}d\Lambda _U(t)$, and $\Lambda _U(t)$ is the cumulative hazard function of $\tilde{U}_i$. The above intensity model suggests that

$$\begin{aligned} M^{(1)}_{i}(t) = N^{(1)}_{i}(t) - \int _0^t R^{(1)}_{i}(s) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i s\}dH_{1}(s) \quad (i=1, \ldots , n) \end{aligned}$$

are martingales with respect to the $\sigma$-filtration: $\sigma \{N^{(1)}_{i}(s), R^{(1)}_{i}(s), {{\varvec{Z}}}_i; s\le t,i=1, \ldots , n\}$.

Similarly, we can also define the counting process related to $\tilde{V}_i$ as $N^{(2)}_i(t)=\delta ^{(2)}_{i}I(\tilde{V}_i\le t)$, and its intensity model is given by

$$\begin{aligned} dH^{(2)}_{i}(t)= R^{(2)}_{i}(t) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i t\}dH_{2}(t), \end{aligned}$$

where $R^{(2)}_{i}(t) =I(\tilde{U}_i\le t \le \tilde{V}_i)$, $dH_{2}(t)=\exp \{-\Lambda (t)\}d\Lambda _V(t)$, and $\Lambda _V(t)$ is the cumulative hazard function of $\tilde{V}_i$. The above intensity model implies that

$$\begin{aligned} M^{(2)}_{i}(t) = N^{(2)}_{i}(t) - \int _0^t R^{(2)}_{i}(s) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i s\}dH_{2}(s) \quad (i=1, \ldots , n) \end{aligned}$$

are martingales with respect to the $\sigma$-filtration: $\sigma \{N^{(2)}_{i}(s), R^{(2)}_{i}(s), {{\varvec{Z}}}_i; s\le t,i=1, \ldots , n\}$.

In what follows, we derive a conditional estimating equation approach for $\varvec{\beta }$ under the conditional independent assumption between the examination and failure times given the covariates. The proposal below mainly utilizes the idea of working independence for interval-censored data (Betensky et al. 2001; Zhu et al. 2008), which treats the multiple examinations from a subject as the single examinations from different subjects. Therefore, the partial likelihood score function of $\varvec{\beta }$ can be expressed as

$$\begin{aligned} \Phi (\varvec{\beta }) = \sum _{i=1}^n\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_1^{(1)}(\varvec{\beta },t)}{S_1^{(0)}(\varvec{\beta },t)}\right\} dN^{(1)}_{i}(t)+ \sum _{i=1}^n\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_2^{(1)}(\varvec{\beta },t)}{S_2^{(0)}(\varvec{\beta },t)}\right\} dN^{(2)}_{i}(t), \nonumber \\ \end{aligned}$$

(8)

where

$$\begin{aligned} S_1^{(k)}(\varvec{\beta },t)=\sum _{j=1}^n R^{(1)}_{j}(t) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_jt\}( {{\varvec{Z}}}_jt)^{\otimes {k}}, \\ S_2^{(k)}(\varvec{\beta },t)=\sum _{j=1}^n R^{(2)}_{j}(t) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_jt\}( {{\varvec{Z}}}_jt)^{\otimes {k}}, \end{aligned}$$

${{\varvec{a}}}^{\otimes {0}}=1, {{\varvec{a}}}^{\otimes {1}}={{\varvec{a}}}$ and ${{\varvec{a}}}^{\otimes {2}}={{\varvec{a}}}{{\varvec{a}}}^{\top }$ for a column vector ${{\varvec{a}}}$, and here $\tau$ is a prespecified constant such that $P(A+U \ge \tau ) > 0$ and $P(A+V \ge \tau ) > 0$.

The observed information matrix for $\varvec{\beta }$ is

$$\begin{aligned} I(\varvec{\beta }) = \sum _{i=1}^n \sum _{k=1}^2 \int _{0}^{\tau }\left\{ \frac{S_k^{(2)}(\varvec{\beta },t)}{S_k^{(0)}(\varvec{\beta },t)}-\frac{S_k^{(1)}(\varvec{\beta },t)^{\otimes {2}}}{S_k^{(0)}(\varvec{\beta },t)^2}\right\} dN^{(k)}_{i}(t). \end{aligned}$$

(9)

For each i, define

$$\begin{aligned} a_i(\varvec{\beta }) =&\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_1^{(1)}(\varvec{\beta },t)}{S_1^{(0)}(\varvec{\beta },t)}\right\} dM^{(1)}_{i}(t)\\ =&\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_1^{(1)}(\varvec{\beta },t)}{S_1^{(0)}(\varvec{\beta },t)}\right\} dN^{(1)}_{i}(t)\\&- \int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_1^{(1)}(\varvec{\beta },t)}{S_1^{(0)}(\varvec{\beta },t)}\right\} \frac{R^{(1)}_{i}(t) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i t\}}{S_1^{(0)}(\varvec{\beta },t)} \left\{ \sum _{i=1}^n dN^{(1)}_{i}(t)\right\} , \end{aligned}$$

and

$$\begin{aligned} b_i(\varvec{\beta })=&\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_2^{(1)}(\varvec{\beta },t)}{S_2^{(0)}(\varvec{\beta },t)}\right\} dM^{(2)}_{i}(t)\\ =&\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_2^{(1)}(\varvec{\beta },t)}{S_2^{(0)}(\varvec{\beta },t)}\right\} dN^{(2)}_{i}(t)\\&- \int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_2^{(1)}(\varvec{\beta },t)}{S_2^{(0)}(\varvec{\beta },t)}\right\} \frac{R^{(2)}_{i}(t) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i t\}}{S_2^{(0)}(\varvec{\beta },t)} \left\{ \sum _{i=1}^n dN^{(2)}_{i}(t)\right\} . \end{aligned}$$

Let $\hat{\varvec{\beta }}_{U2}$ denote the estimate of $\varvec{\beta }$ obtained by solving the normalised conditional estimating equation

$$\begin{aligned} U(\varvec{\beta })=\frac{1}{n} \Phi (\varvec{\beta }) = 0. \end{aligned}$$

(10)

We establish the asymptotic property of $\hat{\varvec{\beta }}_{U2}$ in the following theorem and defer the detailed proof to the “Appendix B”.

Theorem 3

Under Conditions (C4)–(C6) given in the “Appendix B”, we can conclude that, as $n\rightarrow \infty$, $\sqrt{n}(\hat{\varvec{\beta }}_{U2}-\varvec{\beta }_0)$ converges in distribution to a zero-mean normal random vector with covariance matrix $\Sigma _B(\varvec{\beta }_0)^{-1} \Sigma _A(\varvec{\beta }_0) \Sigma _B(\varvec{\beta }_0)^{-1}$, where $\Sigma _A(\varvec{\beta }_0)$ is the limit of

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n [a_i(\varvec{\beta }_0) + b_i(\varvec{\beta }_0)] [ a_i(\varvec{\beta }_0) + b_i(\varvec{\beta }_0)]^{\top }, \end{aligned}$$

and $\Sigma _B(\varvec{\beta }_0)$ is the limit of $n^{-1}I(\varvec{\beta }_0)$. In particular, $\Sigma _B(\varvec{\beta }_0)$ can be estimated by $n^{-1}I(\hat{\varvec{\beta }}_{U2})$, and $\Sigma _A(\varvec{\beta }_0)$ can be estimated by

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n [a_i(\hat{\varvec{\beta }}_{U2}) + b_i(\hat{\varvec{\beta }}_{U2})] [ a_i(\hat{\varvec{\beta }}_{U2}) + b_i(\hat{\varvec{\beta }}_{U2})]^{\top }. \end{aligned}$$

Under the left-truncated and general interval-censored data, the combined estimating equation for the additive hazards model can be defined as

$$\begin{aligned} \epsilon (\varvec{\beta })= U(\varvec{\beta })+\psi (\varvec{\beta })=0, \end{aligned}$$

(11)

where $U(\varvec{\beta })$ and $\psi (\varvec{\beta })$ are given in (10) and (6), respectively. Let $\hat{\varvec{\beta }}_{\epsilon 2}$ be the estimate of $\varvec{\beta }$ obtained by solving the equation $\epsilon (\varvec{\beta })=0$ (11). By following the arguments of Huang and Qin (2013), we can derive the following theorem on the asymptotic property of $\hat{\varvec{\beta }}$. The proof will be presented in the “Appendix B”.

Theorem 4

Under Conditions (C4)–(C6) given in the “Appendix B”, we can conclude that, as $n\rightarrow \infty$, $\sqrt{n} (\hat{\varvec{\beta }}_{\epsilon 2}-\varvec{\beta }_0)$ converges to a normal distribution with mean zero and covariance matrix $[\Sigma _B(\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}[\Sigma _A(\varvec{\beta }_0)+V_1(\varvec{\beta }_0)][\Sigma _B(\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}$, where $\Sigma _A(\varvec{\beta }_0)$ is the limit of

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n [a_i(\varvec{\beta }_0) + b_i(\varvec{\beta }_0)] [ a_i(\varvec{\beta }_0) + b_i(\varvec{\beta }_0)]^{\top }, \end{aligned}$$

$\Sigma _B(\varvec{\beta }_0)$ is the limit of $n^{-1}I(\varvec{\beta }_0)$, $V_1(\varvec{\beta }_0)=4E\{\psi _{12}(\varvec{\beta }_0)\psi _{13}(\varvec{\beta }_0)\}$ and $V_2(\varvec{\beta }_0)=-E\{\partial \psi _{12}(\varvec{\beta }_0)/\partial \varvec{\beta }\}$ $=E[\rho _{12}^{\otimes 2}\exp (-\varvec{\beta }^{\top }_0\rho _{12})/\{1+\exp (-\varvec{\beta }_0^{\top }\rho _{12})^2\}]$.

In the above, $\Sigma _B(\varvec{\beta }_0)$ can be estimated by $n^{-1}I(\hat{\varvec{\beta }}_{\epsilon 2})$, $\Sigma _A(\varvec{\beta }_0)$ can be estimated by $\frac{1}{n} \sum _{i=1}^n [a_i(\hat{\varvec{\beta }}_{\epsilon 2}) + b_i(\hat{\varvec{\beta }}_{\epsilon 2})] [ a_i(\hat{\varvec{\beta }}_{\epsilon 2}) + b_i(\hat{\varvec{\beta }}_{\epsilon 2})]^{\top }$, and $V_1(\varvec{\beta }_0)$ and $V_2(\varvec{\beta }_0)$ can be estimated by

$$\begin{aligned} \frac{4}{n-1}\sum _{i=1}^{n}\left\{ \frac{1}{n-1}\sum _{j=1,j\ne i}^{n}\psi _{ij}(\hat{\varvec{\beta }}_{\epsilon 2})\right\} ^{\otimes {2}}, \end{aligned}$$

and

$$\begin{aligned} \frac{1}{n(n-1)}\sum _{i\ne j}\frac{\rho _{ij}^{\otimes {2}}\exp (-\hat{\varvec{\beta }}_{\epsilon 2}^{\top }\rho _{ij})}{\{1+\exp (-\hat{\varvec{\beta }}_{\epsilon 2}^{\top }\rho _{ij})\}^2}, \end{aligned}$$

respectively (Sen 1960).

5 Simulation studies

We conduct extensive simulation studies to evaluate the empirical performance of the proposed methods in finite samples. In the first study, we considered the left-truncated and case I interval-censored data, and assumed that there existed two covariates $Z_1$ and $Z_2$, where $Z_1$ follows the Bernoulli distribution with a success probability of 0.5 and $Z_2$ follows the uniform distribution over (0, 1). The underlying failure time of interest $T^*$ was then generated from the additive hazards model (1) with $\lambda (t) = 1$ and the true values of $(\beta _1, \beta _2)$ being (0.5, 0.5). The underlying truncation time $A^*$ was assumed to follow either the exponential distribution with mean 10 or the uniform distribution over (0, 100). Due to the presence of left truncation, only the pairs satisfying $T^* \ge A^*$ were kept in the simulated data, and thus $T=T^*$ and $A=A^*$ under this situation. The censoring time from the study enrolment C was generated from the exponential distribution with mean 2/3 and $\tilde{C} = C + A$, which yielded about 50% right censoring rates. We set n = 100, 200, 400 or 800 and used 1000 replications. To analyze the simulated data, we considered the following three competing methods: the combined estimating equation approach, the conditional estimating equation approach, and the naive method proposed by Lin et al. (1998) that ignores the existence of left truncation.

Tables 1 and 2 present the numerical results corresponding to the situations where the truncation time follows the exponential and uniform distributions, respectively. The tables include the estimation bias (Bias) given by the average of the estimates minus the true value, the sample standard errors (SSE) of the estimates, the average standard error estimates (SEE) and the 95% empirical coverage probabilities (CP). One can see from Tables 1 and 2 that the combined estimating equation approach can give unbiased estimates, the bias decreases when the sample size increases, the corresponding standard error estimates match the average standard error estimates well, and the empirical coverage probabilities are around the nominal value 95%, which implies that the normal approximation to the distribution of the estimates seems to be reasonable. As anticipated, the combined estimating equation approach proposed is much more efficient than the conditional estimating equation approach since the former utilized the distribution information of truncation times in the estimation procedure. In addition, it is worth noting that the naive method that ignores the existence of left truncation produced severely biased estimates. This finding can also be anticipated since the obtained left-truncated data are no longer representative for the whole population under study.

Table 1 Simulation results for the left-truncated and case I interval-censored data when the truncation time follows the exponential distribution, including the estimated bias (Bias), the sample standard errors (SSE) of the estimates, the average standard error estimates (SEE) and the $95\%$ empirical coverage probabilities (CP)

Table 2 Simulation results for the left-truncated and case I interval-censored data when the truncation time follows the uniform distribution, including the estimated bias (Bias), the sample standard errors (SSE) of the estimates, the average standard error estimates (SEE) and the $95\%$ empirical coverage probabilities (CP)

Table 3 Simulation results for the left-truncated and general interval-censored data when the truncation time follows the exponential distribution, including the estimated bias (Bias), the sample standard errors (SSE) of the estimates, the average standard error estimates (SEE) and the $95\%$ empirical coverage probabilities (CP)

Table 4 Simulation results for the left-truncated and general interval-censored data when the truncation time follows the uniform distribution, including the estimated bias (Bias), the sample standard errors (SSE) of the estimates, the average standard error estimates (SEE) and the $95\%$ empirical coverage probabilities (CP)

Table 5 Analysis results of the MHCPS data, including the estimated covariate effect (Est), the standard error estimate (Std) and the p-value

In the second study, we considered the scenario of the left-truncated and general interval-censored data. For each subject, we assumed that there existed two observation times $\tilde{U}$ and $\tilde{V}$, where we set $\tilde{U}=A+U$, $\tilde{V}=A+V$, U follows uniform distribution over (0, 0.5) and V follows uniform distribution over $(U+0.3, U+0.8)$. We kept the other simulation specifications being the same as above. Under this configuration, the left and right censoring rates are both approximately 33%. As in the first simulation study, we performed the analysis with the two proposed estimating equation approaches, and also considered the naive estimating equation method that ignores the existence of left truncation, which can be implemented with the proposed conditional estimating equation approach by setting $A=0$. The simulation results given in Tables 3 and 4 correspond to the situations where the truncation time follows the exponential and uniform distributions, respectively, from which one can find the similar conclusions as above.

In the simulation studies above, we also tried to apply the generalized method of moments (GMM) (Hansen 1982) to further enhance the estimation efficiency. Let $G(\varvec{\beta }) = (U(\varvec{\beta })^{\top }, \psi (\varvec{\beta })^{\top })^{\top }$ and W be a positive-definite weight function matrix. A consistent estimator of $\varvec{\beta }$ can be obtained through minimizing $G(\varvec{\beta })^{\top }W^{-1}G(\varvec{\beta })$, and the optimal matrix that gives the efficient estimation is $W=\textrm{var}\{G(\varvec{\beta })\}$. However, the numerical results (not shown here) indicate that the GMM method does not lead to higher efficiency compared with the proposed combined estimating equation approach. As commented by Huang and Qin (2013), this phenomenon arises partially because the optimal weight function involves estimating the second moments of $U(\varvec{\beta }_0)$ and $\psi (\varvec{\beta }_0)$, which usually needs larger sample size to obtain the superiority of an efficient GMM estimator.

6 Real data analysis

We now apply the proposed methods to a set of left-truncated and interval-censored data arising from the Massachusetts Health Care Panel Study (MHCPS) analyzed by Pan and Chappell (1998, 2002) and Gao and Chan (2019) and others. In 1975, the study enrolled individuals over the age of 65 in Massachusetts, and after that, three subsequent follow-ups were taken at the 1.25, 6, and 10 years after the enrolment to determine if individuals were still living actively, resulting in the interval-censored observation on the time to loss of active life. The main objective of the study was to evaluate the effect of gender (male or female) on the loss of active life for elderly people. For this, the failure time of interest is defined as the age at loss of active life. In addition, since only people who were active at the enrolment were included in the study, the time to the loss of active life was also subject to left truncation with the truncation time being the age at enrolment (Pan and Chappell 2002). Therefore, we had the left-truncated and interval-censored data on the time to loss of active life. After deleting a small proportion of unrealistic records of the raw data, a total of 1025 individuals with the age ranging from 65 to 97.3 were finally included in the current analysis. The right censoring rate is $45.8\%$.

In this MHCPS dataset, the data for the failure times of interest were given in the form of $\{[L_i, R_i); i = 1, \ldots , n\}$ and contained a mixture of interval-censored and right-censored observations. To implement the proposed methods, we followed the strategy given in Wang et al. (2010) and made an adjustment: for subject i with $[L_i, R_i)$, we let $\tilde{V}_i = L_i$ and $\tilde{U}_i$ to be the smallest observation time in the study if $R_i = \infty$; and we set $\tilde{U}_i = L_i$ and $\tilde{V}_i = R_i$ when $L_i > 0$ and $R_i < \infty$. Define $Z_i = 1$ if the ith subject was male and $Z_i = 0$ if this subject was female.

Table 5 presents the analysis results of the MHCPS data with the combined estimating equation approach, the conditional estimating equation approach, and the naive estimating equation method that ignores the existence of left truncation. They include the estimated covariate effect (Est), the standard error estimate (Std) and the p-value for testing the covariate effect being zero. One can find from Table 5 that the results obtained from the combined estimating equation approach suggest that males have significantly higher risk of losing active life than females, which is in accordance with the conclusion given by Gao and Chan (2019) under the proportional hazards model. In addition, one can find from Table 5 that the conditional and the naive estimating equation approaches give consistent conclusions, and it is apparent that they tend to underestimate the covariate effect compared with the combined estimating equation approach. The attenuated covariate effect estimates obtained by the conditional and the naive estimating equation approaches may arise because the two approaches either fail to utilize the distribution information of the truncation times or ignore the existence of left truncation. In general, they both lead to larger estimation bias compared with the combined estimating equation approach as shown in the simulation studies.

As discussed before, under the assumed additive hazards model (1), the interpretation of the estimated covariate effect can be converted into that of the ratio of the estimated survival probabilities between different groups at a given time. For example, the estimated ratio of the probabilities of having the active life after 80 years old between male and female under the combined estimating equation approach is $S(80 \mid Z_i = 1)/S(80 \mid Z_i = 0) = \exp \{-\Lambda (80)-0.00165 \times 80\}/\exp \{-\Lambda (80)\} \approx 0.876$. In contrast, the estimated ratios of the probabilities of having the active life after 80 years old between male and female are about 0.955 and 0.961 under the conditional and the naive estimating equation approaches, respectively.

7 Concluding remarks

Additive hazards model is an essential alternative to the commonly used proportional hazards model in failure time data analysis, and assumes an additive covariate effect on the hazard function of the failure time of interest. In this paper, we studied the additive hazards regression analysis of the left-truncated and interval-censored failure time data including the case I and general interval censoring schemes. By utilizing the order information of the pairwise truncation times, we developed the combined estimating equation approach for the estimation, which yielded more efficient estimators than the conditional estimating equation approach. The tractable estimating equations derived were very easy to solve with the routine root-finding method, such as Newton–Raphson algorithm. The asymptotic properties of the proposed estimators were established and numerical studies demonstrated the usefulness of the proposed methodology in finite samples.

In the preceding sections, we only considered time-independent covariates and it is apparent that one may encounter time-dependent covariates in some applications. For the latter case, one needs to redefine the model (1) as $\lambda (t \mid {{\varvec{Z}}}^{*}(t))=\lambda (t)+\varvec{\beta }^{\top } {{\varvec{Z}}}^{*}(t)$, where ${{\varvec{Z}}}^{*}(t)$ is a p-dimensional vector of time-dependent covariates, and the estimation procedures proposed above can be readily adapted to handle this situation. Note that, in the proposed methods, we assumed that the failure times of interest are conditionally independent of the observation times given the covariates, which is usually referred to as non-informative censoring in the literature. However, this assumption may not hold in some applications, and it is helpful to extend the proposed methods to the case of informative censoring (Zhang et al. 2005; Ma et al. 2015; Li et al. 2017). Model checking is often of great interest and Ghosh (2003) proposed a formal goodness-of-fit test procedure for the additive hazards model under current status data or case I interval-censored data. Future work will be devoted to generalize the method proposed by Ghosh (2003) to the situation of left-truncated and interval-censored data. In addition, left-truncated and multivariate interval-censored failure time data can also be encountered in many scientific fields and it is useful to extend the proposed methods to the analysis of such data.

Appendices

Appendix A: Proofs of Theorems 1 and 2

Let $S^{(k)}(\varvec{\beta },t)=\sum _{j=1}^n R_j(t) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_jt\}( {{\varvec{Z}}}_jt)^{\otimes {k}}$ for $k=0$, 1, 2. For a matrix ${{\varvec{A}}}$ or a vector ${{\varvec{a}}}$, $||{{\varvec{A}}}||=\sup _{i,j}|a_{ij}|$ and $||{{\varvec{a}}}||=\sup _{i}|a_{i}|$, where $a_{ij}$ is the (i, j)th element of ${{\varvec{A}}}$ and $a_{i}$ is the ith component of ${{\varvec{a}}}$. For a vector ${{\varvec{a}}}$, $|{{\varvec{a}}}| = ({{\varvec{a}}}^{\top }{{\varvec{a}}})^{1/2}$. In what follows, “$\overset{P}{\rightarrow }$” and “$\overset{D}{\rightarrow }$” denote the convergence in probability and the convergence in distribution, respectively.

To establish the asymptotic properties of proposed estimators of $\varvec{\beta }$ under left-truncated and case I interval-censored data, we need the following regularity conditions:

(C1):

$P(T^*>A^* \mid {{\varvec{Z}}}^*)>0$, $P(A+C \ge \tau \mid {{\varvec{Z}}})>0$ and $P(R(t)=1 \mid {{\varvec{Z}}})>0$ for all $t \in [A, \tau ]$, where $R(t) =I(A \le t \le A+C)$.

(C2):

${{\varvec{Z}}}_i$ is bounded, and $H_0(\tau ) <\infty$, where $H_{0}(t)$ is the true value of H(t).

(C3):

The true regression vector $\varvec{\beta }_0$ lies in the interior of a compact set $\mathcal {B}$ and there exist functions $s^{(k)}(\varvec{\beta },t)$ with $k=0,1,2$ defined on $\mathcal {B}\times (0,\tau ]$ satisfying

(a):

$\sup _{\varvec{\beta }\in \mathcal {B},t\in (0,\tau ]}||n^{-1}S^{(k)}(\varvec{\beta },t)-s^{(k)}(\varvec{\beta },t)||\overset{P}{\rightarrow }0$ as $n\rightarrow \infty$.

(b):

$s^{(0)}(\varvec{\beta },t)$ is bounded away from 0 for $t\in (0,\tau ]$.

(c):

For $k=0,1,2$, $s^{(k)}(\varvec{\beta },t)$ is a continuous function of $\varvec{\beta }$ uniformly in $t\in (0,\tau ]$, where $s^{(1)}(\varvec{\beta },t)=\partial s^{(0)}(\varvec{\beta },t)/\partial \varvec{\beta }$ and $s^{(2)}(\varvec{\beta },t)=\partial ^2\,s^{(0)}(\varvec{\beta },t)/\partial \varvec{\beta }\partial \varvec{\beta }^{\top }$.

(d):

For $\varvec{\beta }\in \mathcal {B}$, let $v(\varvec{\beta },t)=s^{(2)}(\varvec{\beta },t)/s^{(0)}(\varvec{\beta },t)-e(\varvec{\beta },t)e(\varvec{\beta },t)^{\top }$ and $e(\varvec{\beta },t)=s^{(1)}(\varvec{\beta },t)/s^{(0)}(\varvec{\beta },t)$. $\Sigma (\varvec{\beta }_0)=\int _0^{\tau }v(\varvec{\beta }_0,u)s^{(0)}(\varvec{\beta }_0,u) d H_{0}(u)$ is positive definite.

Proof of Theorem 1:

To prove the asymptotic properties of $\hat{\varvec{\beta }}_U$, we mainly follow the arguments given in Andersen and Gill (1982) and Kalbfleisch and Prentice (2002). Note that the objective function corresponding to the equation (3) is

$$\begin{aligned} l(\varvec{\beta }) = -\sum _{i=1}^n \int _0^{\tau } \left\{ \varvec{\beta }^{\top } {{\varvec{Z}}}_i u + \log [S^{(0)}(\varvec{\beta }, u)] \right\} d N_i(u). \end{aligned}$$

For $t \le \tau$, define

$$\begin{aligned} l(\varvec{\beta }, t) = -\sum _{i=1}^n \int _0^{t} \left\{ \varvec{\beta }^{\top } {{\varvec{Z}}}_i u + \log [S^{(0)}(\varvec{\beta }, u)] \right\} d N_i(u). \end{aligned}$$

Consider the process

$$\begin{aligned} X(\varvec{\beta },t)= & {} n^{-1}(l(\varvec{\beta }, t) -l(\varvec{\beta }_0, t) )\\ {}= & {} -n^{-1}\sum _{i=1}^n \left\{ \int _0^t (\varvec{\beta }-\varvec{\beta }_0)^{\top }{{\varvec{Z}}}_i u + \log \left[ \frac{S^{(0)}(\varvec{\beta }, u)}{S^{(0)}(\varvec{\beta }_0, u)}\right] \right\} dN_i(u). \end{aligned}$$

Then the compensator of $X(\varvec{\beta },t)$ is

$$\begin{aligned} \widetilde{X}(\varvec{\beta },t)=-n^{-1} \int _0^t \left\{ (\varvec{\beta }-\varvec{\beta }_0)^{\top } S^{(1)}(\varvec{\beta }_0,u) + \log \left[ \frac{S^{(0)}(\varvec{\beta },u)}{S^{(0)}(\varvec{\beta }_0,u)}\right] S^{(0)}(\varvec{\beta }_0,u) \right\} dH_0(u). \end{aligned}$$

Under Conditions (C1)–(C3), we can conclude that $\widetilde{X}(\varvec{\beta }, \tau )$ converges to a function of $\varvec{\beta }$ denoted as $f(\varvec{\beta })$, where

$$\begin{aligned} f(\varvec{\beta }) = - \int _0^{\tau } \left\{ (\varvec{\beta }-\varvec{\beta }_0)^{\top } s^{(1)}(\varvec{\beta }_0,u) + \log \left[ \frac{s^{(0)}(\varvec{\beta },u)}{s^{(0)}(\varvec{\beta }_0,u)}\right] s^{(0)}(\varvec{\beta }_0,u) \right\} dH_0(u). \end{aligned}$$

The first and second derivatives of $f(\varvec{\beta })$ with respect to $\varvec{\beta }$ are given by

$$\begin{aligned} \partial f(\varvec{\beta }) / \partial \varvec{\beta }= - \int _0^{\tau } \left\{ s^{(1)}(\varvec{\beta }_0,u) - s^{(1)}(\varvec{\beta },u) \frac{s^{(0)}(\varvec{\beta }_0,u)}{s^{(0)}(\varvec{\beta },u)} \right\} dH_0(u), \end{aligned}$$

and

$$\begin{aligned} \partial ^2 f(\varvec{\beta })/\partial \varvec{\beta }\partial \varvec{\beta }^{\top } =&\int _0^{\tau } \left\{ - \frac{s^{(2)}(\varvec{\beta },u)}{s^{(0)}(\varvec{\beta },u)} + \frac{s^{(1)}(\varvec{\beta },u)^{\otimes {2}} }{s^{(0)}(\varvec{\beta },u)^2} \right\} s^{(0)}(\varvec{\beta }_0,u)dH_0(u)\\&=- \int _0^{\tau } v(\varvec{\beta },u) s^{(0)}(\varvec{\beta }_0,u)dH_0(u) = -\Sigma (\varvec{\beta }), \end{aligned}$$

respectively.

It is easy to find that $\partial f(\varvec{\beta })/\partial \varvec{\beta }|_{\varvec{\beta }=\varvec{\beta }_0}=0$, and $\partial ^2 f(\varvec{\beta })/\partial \varvec{\beta }\partial \varvec{\beta }^{\top }$ is minus a positive definite matrix. Therefore, the unique maximum of $f(\varvec{\beta })$ is at $\varvec{\beta }=\varvec{\beta }_0$. By following Andersen and Gill (1982), it can be verified that the predictable variation process of $X(\varvec{\beta },t)-\widetilde{X}(\varvec{\beta },t)$ converges to 0 on interval $(0,\tau ]$. Then by using the Lenglart’s inequality given in Andersen and Gill (1982), we can conclude that $X({\varvec{\beta }},\tau ) \overset{P}{\rightarrow }\ f(\varvec{\beta })$ with probability one. Note that $\hat{\varvec{\beta }}_U$ maximizes the function $X(\varvec{\beta },\tau )$. Applying the convex analysis to $X(\varvec{\beta },\tau )$ and $f(\varvec{\beta })$ as Andersen and Gill (1982) leads to the conclusion that $\hat{\varvec{\beta }}_U {\rightarrow }\varvec{\beta }_0$ with probability one as $n \rightarrow \infty$.

By applying Taylor expansion about $\varvec{\beta }_0$ to $\Phi (\hat{\varvec{\beta }}_U)$, we have

$$\begin{aligned} 0=n^{-1/2}\Phi (\hat{\varvec{\beta }}_U) = n^{-1/2}\Phi (\varvec{\beta }_0) - n^{1/2}(\hat{\varvec{\beta }}_U-\varvec{\beta }_0)[n^{-1}I(\varvec{\beta }^*)] \end{aligned}$$

where $\varvec{\beta }^*$ is between $\hat{\varvec{\beta }}_U$ and $\varvec{\beta }_0$ and $I(\varvec{\beta }) =-\partial ^2\,l(\varvec{\beta })/\partial \varvec{\beta }\partial \varvec{\beta }^{\top }$. Then the asymptotic distribution of $n^{1/2}(\hat{\varvec{\beta }}_U-\varvec{\beta }_0)$ can be established if we can show the probability limit of $n^{-1}I(\varvec{\beta }^{*})$ and the asymptotic distribution of $\Phi (\varvec{\beta }_0)$.

By the consistency of $\hat{\varvec{\beta }}_U$ and the arguments given in (Andersen and Gill 1982), it can be shown show that $n^{-1}I(\varvec{\beta }^{*})\overset{P}{\rightarrow }\ \Sigma (\varvec{\beta }_0)$ for any $\varvec{\beta }^*$ that converges in probability to $\varvec{\beta }_0$. Next, we prove that $n^{-1/2}\Phi (\varvec{\beta }_0)\overset{D}{\rightarrow } N[0,\Sigma (\varvec{\beta }_0)]$. Following Lin et al. (1998), for each i, we can conclude that $M_i(t)$ is martingale with respect to the $\sigma$-filtration: $\sigma \{N_i(s), R_i(s), {{\varvec{Z}}}_i; s\le t,i=1, \ldots , n\}$. Under Conditions (C2) and (C3), we know that $n^{-1/2}G_i^{(n)}(u)=n^{-1/2}\left\{ {{\varvec{Z}}}_i u - S^{(1)}(\varvec{\beta }_0,u)/S^{(0)}(\varvec{\beta }_0,u)\right\}$ is a vector of predictable process for each i. Simple algebraic manipulation shows that $n^{-1/2}\Phi (\varvec{\beta }_0, t)=\sum _{i=1}^n \int _0^{t} G_i^{(n)}(u) dM_i(u)$ is a martingale with the predictable covariation process

$$\begin{aligned} \begin{aligned} \langle n^{-1/2}\Phi (\varvec{\beta }_0)\rangle (t)&= \sum _{i=1}^n \int _0^t G_i^{(n)}(u)^{\top } G_i^{(n)}(u) R_i(u) \exp [-\varvec{\beta }_0^{\top }{{\varvec{Z}}}_i u]dH_0(u)\\&= \int _0^t V(\varvec{\beta }_0,u) S^{(0)}(\varvec{\beta }_0,u) dH_0(u). \end{aligned} \end{aligned}$$

Under Condition (C3), we have

$$\begin{aligned} \langle n^{-1/2}\Phi (\varvec{\beta }_0)\rangle (t) \overset{P}{\rightarrow } \int _0^t v(\varvec{\beta }_0,u) s^{(0)}(\varvec{\beta }_0,u) dH_0(u). \end{aligned}$$

Furthermore, note that Conditions (C2) and (C3) are sufficient to imply the Lindeberg condition of Rebolledo’s Central Limit Theorem (Andersen and Gill 1982). By applying the Rebolledo’s Central Limit Thoerem, we can conclude that $n^{-1/2}\Phi (\varvec{\beta }_0)\overset{D}{\rightarrow } N[0,\Sigma (\varvec{\beta }_0)]$. Thus, $n^{1/2}(\hat{\varvec{\beta }}_U-\varvec{\beta }_0)\overset{D}{\rightarrow }\ N(0,\Sigma (\varvec{\beta }_0)^{-1})$.

Proof of Theorem 2:

To prove the asymptotic properties of $\hat{\varvec{\beta }}_{\epsilon }$ obtained by solving (7), we mainly follow the arguments given in Huang and Qin (2013). Note that the pairwise pseudo-likelihood of the observed $(A_i, A_j)$ conditional on $A_i \le \tau$, $A_j \le \tau$ and $({{\varvec{Z}}}_i, {{\varvec{Z}}}_j)$ is

$$\begin{aligned} \sum _{1 \le i<j \le n} - \log \left\{ 1+\exp (\varvec{\beta }^{\top } \rho _{ij})\right\} , \end{aligned}$$

which achieves the maximum at the true parameter value as $n\rightarrow \infty$. By applying the conditional Kullback–Leibler information inequality (Andersen 1970), the maximum pairwise pseudo-likelihood estimator $\hat{\varvec{\beta }}_{\psi }$ converges to $\varvec{\beta }_0$ with probability one. Note that $\psi (\varvec{\beta })$ is a U-statistic of order two since $\psi _{ij}(\varvec{\beta })$ is permutation-symmetric in the augments $(A_i, {{\varvec{Z}}}_i)$ and $(A_j, {{\varvec{Z}}}_j)$. By Conditions (C2) and Conditions (C3), we know that $E\{\psi _{ij}(\varvec{\beta }_0)\}=0$ and $E\{\psi _{ij}(\varvec{\beta }_0)^{\otimes 2}\}<\infty$. By the projection method proposed by Hoeffding (1948), we know that $n^{1/2}\psi (\varvec{\beta }_0)$ converges to the normal distribution with mean zero and covariance matrix $V_1=4E\{\psi _{12}(\varvec{\beta }_0)^{\top }\psi _{13}(\varvec{\beta }_0)\}$. Applying the delta method, $n^{1/2}(\hat{\varvec{\beta }_{\psi }}-\varvec{\beta }_0)$ converges to the normal distribution with mean zero and covariance matrix $V_2(\varvec{\beta }_0)^{-1} V_1(\varvec{\beta }_0) V_2(\varvec{\beta }_0)^{-1}$, where $V_1(\varvec{\beta }_0)=4E\{\psi _{12}(\varvec{\beta }_0)\psi _{13}(\varvec{\beta }_0)\}$ and $V_2(\varvec{\beta }_0)=-E\{\partial \psi _{12}(\varvec{\beta }_0)/\partial \varvec{\beta }\}=E[\rho _{12}^{\otimes 2}\exp (-\varvec{\beta }^{\top }_0\rho _{12})/\{1+\exp (-\varvec{\beta }_0^{\top }\rho _{12})^2\}]$.

Because $U(\varvec{\beta })$ is linear of $\varvec{\beta }$, the consistency of $\hat{\varvec{\beta }}_{\epsilon }$ follows directly from the consistency of $\hat{\varvec{\beta }}_{\psi }$. Since we have shown that $\hat{\varvec{\beta }}_{U}$ and $\hat{\varvec{\beta }}_{\psi }$ are asymptotic normal, the asymptotic normality of $\hat{\varvec{\beta }}_{\epsilon }$ follows from the asymptotic independence of $U(\varvec{\beta })$ and $\psi (\varvec{\beta })$ (van der Vaart and Wellner 1996). Because $E[dM_i(t) \mid A_i,{{\varvec{Z}}}_i]=0$ and $\psi _{ij}(\varvec{\beta }_0)$ only involves $(A_i, {{\varvec{Z}}}_i)$ and $(A_j, {{\varvec{Z}}}_j)$, by double expectation we have that $E[U(\varvec{\beta }_0)^{\top }\psi (\varvec{\beta }_0)]=0$ and $\textrm{var}[\epsilon (\varvec{\beta }_0)]=\textrm{var}[U(\varvec{\beta }_0)]+\textrm{var}[\psi (\varvec{\beta }_0)]$. By the central limit theorem for U-statistics, $n^{1/2}\epsilon (\varvec{\beta }_0)$ converges in distribution to a normal distribution with mean zero and covariance matrix $\textrm{var}[n^{1/2}U(\varvec{\beta }_0)]+\textrm{var}[n^{1/2}\psi (\varvec{\beta }_0)]=\Sigma (\varvec{\beta }_0)+V_1(\varvec{\beta }_0)$. Applying the Taylor expansion to $\epsilon (\varvec{\beta })$, we can conclude that, as $n\rightarrow \infty$, $\sqrt{n} (\hat{\varvec{\beta }}_{\epsilon }-\varvec{\beta }_0)$ converges to a normal distribution with mean zero and covariance matrix $[\Sigma (\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}[\Sigma (\varvec{\beta }_0)+V_1(\varvec{\beta }_0)][\Sigma (\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}$.

Appendix B: Proofs of Theorems 3 and 4

Define

$$\begin{aligned} S_1^{(k)}(\varvec{\beta },t)=\sum _{j=1}^n R^{(1)}_{j}(t) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_jt\}( {{\varvec{Z}}}_jt)^{\otimes {k}}, \end{aligned}$$

and

$$\begin{aligned} S_2^{(k)}(\varvec{\beta },t)=\sum _{j=1}^n R^{(2)}_{j}(t) \exp \{-\varvec{\beta }^{\top } {{\varvec{Z}}}_jt\}( {{\varvec{Z}}}_jt)^{\otimes {k}}, \end{aligned}$$

for $k=0$, 1, 2. To establish the asymptotic properties of the proposed estimators of $\varvec{\beta }$ under left-truncated and general interval-censored data, we need the following regularity conditions:

(C4):

$P(T^*>A^* \mid {{\varvec{Z}}}^*)>0$, $P(A+U \ge \tau \mid {{\varvec{Z}}})>0$, $P(A+V \ge \tau \mid {{\varvec{Z}}})>0$, $P(R^{(1)}(t)=1 \mid {{\varvec{Z}}})>0$ and $P(R^{(2)}(t)=1 \mid {{\varvec{Z}}})>0$ for all $t \in [A, \tau ]$, where $R^{(1)}(t) =I(A \le t \le A+U)$ and $R^{(2)}(t) =I(A \le t \le A+V)$.

(C5):

${{\varvec{Z}}}_i$ is bounded. $H_{10}(\tau ) <\infty$ and $H_{20}(\tau ) <\infty$, where $H_{10}(t)$ and $H_{20}(t)$ are the true values of $H_1(t)$ and $H_2(t)$, respectively.

(C6):

The true regression vector $\varvec{\beta }_0$ lies in the interior of a compact set $\mathcal {B}$ and there exist functions $s_j^{(k)}(\varvec{\beta },t)$ with $k=0,1,2$ and $j=1,2$ defined on $\mathcal {B}\times (0, \tau ]$ satisfying

(a):

$\sup _{\varvec{\beta }\in \mathcal {B},t\in (0, \tau ]}||n^{-1}S_j^{(k)}(\varvec{\beta },t)-s_j^{(k)}(\varvec{\beta },t)||\overset{P}{\rightarrow }0$ as $n\rightarrow \infty$ with $j=1,2$.

(b):

For $j=1,2$, $s_j^{(0)}(\varvec{\beta },t)$ is bounded away from 0 for $t\in (0, \tau ]$.

(c):

For $k=0,1,2$, $s_j^{(k)}(\varvec{\beta },t)$ is a continuous function of $\varvec{\beta }$ uniformly in $t\in (0, \tau ]$, where $s_j^{(1)}(\varvec{\beta },t)=\partial s_j^{(0)}(\varvec{\beta },t)/\partial \varvec{\beta }$ and $s_j^{(2)}(\varvec{\beta },t)=\partial ^2 s_j^{(0)}(\varvec{\beta },t)/\partial \varvec{\beta }\partial \varvec{\beta }^{\top }$ with $j = 1, 2$.

(d):

For $\varvec{\beta }\in \mathcal {B}$, let $v_j(\varvec{\beta },t)=s_j^{(2)}(\varvec{\beta },t)/s_j^{(0)}(\varvec{\beta },t)-e_j(\varvec{\beta },t)e_j(\varvec{\beta },t)^{\top }$ and $e_j(\varvec{\beta },t)=s_j^{(1)}(\varvec{\beta },t)/s_j^{(0)}(\varvec{\beta },t)$ with $j = 1, 2$. $\Sigma (\varvec{\beta }_0)=\sum _{j=1}^2\int _0^{\tau }v_j(\varvec{\beta }_0,u)s_j^{(0)}(\varvec{\beta }_0,u) d H_{j0}(u)$ is positive definite.

Proof of Theorem 3:

The objective function corresponding to the equation (8) is

$$\begin{aligned} l(\varvec{\beta })= & {} -\sum _{i=1}^n \int _0^{\tau } \left\{ \varvec{\beta }^{\top } {{\varvec{Z}}}_i t + \log [S_1^{(0)}(\varvec{\beta }, t)] \right\} d N_i^{(1)}(t)\\{} & {} -\sum _{i=1}^n \int _0^{\tau } \left\{ \varvec{\beta }^{\top } {{\varvec{Z}}}_i t + \log [S_2^{(0)}(\varvec{\beta }, t)]\right\} d N_i^{(2)}(t). \end{aligned}$$

Note that $\hat{\varvec{\beta }}_{U2}$ is obtained by maximizing $l(\varvec{\beta })$ or solving the estimating Eq. (10), the consistency of $\hat{\varvec{\beta }}_{U2}$ can be established by applying the arguments in the proof of Theorem 1. Following Lin et al. (1998), for each i, we know that

$$\begin{aligned} M^{(1)}_{i}(t) = N^{(1)}_{i}(t) - \int _0^t R^{(1)}_{i}(u) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i u\}dH_{10}(u), \end{aligned}$$

and

$$\begin{aligned} M^{(2)}_{i}(t) = N^{(2)}_{i}(t) - \int _0^t R^{(2)}_{i}(u) \exp \{-\varvec{\beta }^{\top }{{\varvec{Z}}}_i u\}dH_{20}(u) \end{aligned}$$

are both martingales.

Also note that $n^{-1/2}\Phi (\varvec{\beta })$ can be equivalently expressed as

$$\begin{aligned} n^{-1/2}\Phi (\varvec{\beta }) = n^{-1/2} \sum _{i=1}^n \{a_{i}(\varvec{\beta })+b_{i}(\varvec{\beta })\}, \end{aligned}$$

where $\Phi (\varvec{\beta })$ is given by (8),

$$\begin{aligned} a_i(\varvec{\beta }) = \int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_1^{(1)}(\varvec{\beta },t)}{S_1^{(0)}(\varvec{\beta },t)}\right\} dM^{(1)}_{i}(t), \end{aligned}$$

and

$$\begin{aligned} b_i(\varvec{\beta })=\int _0^{\tau }\left\{ {{\varvec{Z}}}_it-\frac{S_2^{(1)}(\varvec{\beta },t)}{S_2^{(0)}(\varvec{\beta },t)}\right\} dM^{(2)}_{i}(t), \end{aligned}$$

for $i = 1, \ldots , n$.

We apply the Taylor expansion about $\varvec{\beta }_0$ to $n^{-1/2}\Phi (\hat{\varvec{\beta }}_{U2})$ and have

$$\begin{aligned} 0=n^{-1/2}\Phi (\hat{\varvec{\beta }}_{U2}) = n^{-1/2}\Phi (\varvec{\beta }_0) -n^{1/2}(\hat{\varvec{\beta }}_{U2}-\varvec{\beta }_0)[n^{-1}I(\varvec{\beta }^*)], \end{aligned}$$

where $\varvec{\beta }^*$ is between $\hat{\varvec{\beta }}_{U2}$ and $\varvec{\beta }_0$ and $I(\varvec{\beta }) =-\partial ^2\,l(\varvec{\beta })/\partial \varvec{\beta }\partial \varvec{\beta }^{\top }$.

By applying the similar arguments in the proof of Theorem 1 and using Conditions (C4)–(C6), we have

$$\begin{aligned} n^{1/2}(\hat{\varvec{\beta }}_{U2}-\varvec{\beta }_0)\overset{D}{\rightarrow } N(0,\Sigma _B(\varvec{\beta }_0)^{-1} \Sigma _A(\varvec{\beta }_0) \Sigma _B(\varvec{\beta }_0)^{-1}), \end{aligned}$$

where $\Sigma _A(\varvec{\beta }_0)$ is the probability limit of $\frac{1}{n}\sum _{i=1}^n [a_i(\varvec{\beta }_0) + b_i(\varvec{\beta }_0)] [ a_i(\varvec{\beta }_0) + b_i(\varvec{\beta }_0)]^{\top }$ and $\Sigma _B(\varvec{\beta }_0)$ is the probability limit of $n^{-1}I(\varvec{\beta }_0)$.

Proof of Theorem 4:

The proof follows the similar augments in the proofs of Theorems 2 and 3, and we can conclude that the maximum pairwise pseudo-likelihood estimator $\hat{\varvec{\beta }}_{\psi }$ converges to $\varvec{\beta }_0$ with probability one, and $n^{1/2}(\hat{\varvec{\beta }_{\psi }}-\varvec{\beta }_0)$ converges to the normal distribution with mean zero and covariance matrix $V_2(\varvec{\beta }_0)^{-1}V_1(\varvec{\beta }_0) V_2(\varvec{\beta }_0)^{-1}$, where $V_1(\varvec{\beta }_0)=4E\{\psi _{12}(\varvec{\beta }_0)\psi _{13}(\varvec{\beta }_0)\}$ and $V_2(\varvec{\beta }_0)=-E\{\partial \psi _{12}(\varvec{\beta }_0)/\partial \varvec{\beta }\}=E[\rho _{12}^{\otimes 2}\exp (-\varvec{\beta }^{\top }_0\rho _{12})/\{1+\exp (-\varvec{\beta }_0^{\top }\rho _{12})^2\}]$.

Then, by following the similar arguments as in the proof of Theorem 2, we know that $n^{1/2}\epsilon (\varvec{\beta }_0)$ converges in distribution to a normal distribution with mean zero and covariance matrix $\textrm{var}[n^{1/2}U(\varvec{\beta }_0)]+\textrm{var}[n^{1/2}\psi (\varvec{\beta }_0)]=\Sigma _A(\varvec{\beta }_0)+V_1(\varvec{\beta }_0)$. Under Conditions (C4)–(C6), we apply the Taylor expansion about $\varvec{\beta }_0$ to $\epsilon (\hat{\varvec{\beta }}_{\epsilon 2})$, and can conclude that, as $n\rightarrow \infty$, $\sqrt{n} (\hat{\varvec{\beta }}_{\epsilon 2}-\varvec{\beta }_0)$ converges to a normal distribution with mean zero and covariance matrix $[\Sigma _B(\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}[\Sigma _A(\varvec{\beta }_0)+V_1(\varvec{\beta }_0)][\Sigma _B(\varvec{\beta }_0)+V_2(\varvec{\beta }_0)]^{-1}$.