Smoothed quantile regression for censored residual life

Abstract

We consider a regression modeling of the quantiles of residual life, remaining lifetime at a specific time. We propose a smoothed induced version of the existing non-smooth estimating equations approaches for estimating regression parameters. The proposed estimating equations are smooth in regression parameters, so solutions can be readily obtained via standard numerical algorithms. Moreover, the smoothness in the proposed estimating equations enables one to obtain a robust sandwich-type covariance estimator of regression estimators aided by an efficient resampling method. To handle data subject to right censoring, the inverse probability of censoring distribution is used as a weight. The consistency and asymptotic normality of the proposed estimator are established. Extensive simulation studies are conducted to validate the proposed estimator’s performance in various finite samples settings. We apply the proposed method to dental study data evaluating the longevity of dental restorations.

Appendix 1. Proof of Theorem 1

In this Appendix, we provide a proof of Theorem 1: consistency and asymptotic normality of the proposed induced smoothed estimator.

First, we establish the consistency of the proposed estimator $\hat{\beta }_{IS}$. The consistency of the non-smooth counterpart, $\hat{\beta }_{NS}$, is shown in (Li et al. 2016). Based on this consistency result, it suffices to we prove that, as $n \rightarrow \infty$, the difference between $\tilde{U}_{t_0}(\beta , \tau , H)$ and $U_{t_0}(\beta , \tau )$ scaled by $n^{1/2}$ converges uniformly to zero in probability for $\beta$ in the compact neighborhood of $\beta _0$.

Let $\sigma _i=(X_i^\top H X_i)^{1/2}$, $\epsilon _i(\beta ) = X_i\beta - \log (Z_i-t_0)$ and $d_i(\beta ) = \mathop {\mathrm{sign}}\nolimits (\epsilon _i^\beta )\Phi (-|\epsilon _i^\beta /\sigma _i|)$. Then,

$$\begin{aligned}&n^{1/2}\left\{ \tilde{U}_{t_0}(\beta , \tau , H) - U_{t_0}(\beta , \tau ))\right\} \\&\quad = {n^{-1/2}} \sum _{i=1}^{n}I\left[ Z_i \ge t_0\right] X_i \delta _i \frac{\hat{G}(t_0)}{\hat{G}(Z_i)} \left\{ \Phi \left( \frac{-\epsilon _i(\beta )}{\sigma _i}\right) -I\left[ \epsilon _i(\beta ) < 0\right] \right\} \\&\quad = n^{-1/2} \sum _{i=1}^{n}I[Z_i \ge t_0] X_i \delta _i \frac{G(t_0)}{G(Z_i)}d_i(\beta ) + n^{-1/2} \sum _{i=1}^{n}I[Z_i \ge t_0] X_i \delta _i \left\{ \frac{\hat{G}(t_0)}{\hat{G}(Z_i)} - \frac{G(t_0)}{G(Z_i)}\right\} d_i(\beta )\\&q = \ D^{(1)}_{n}(\beta ) + D^{(2)}_{n}(\beta ) \end{aligned}$$

To show $\Vert D^{(1)}_n(\beta ) \Vert \xrightarrow {p} 0$ as $n \rightarrow \infty$, we first note that

$$\begin{aligned} \mathop {\mathrm{E}}\nolimits \left\{ D^{(1)}_n(\beta )\right\}&= \mathop {\mathrm{E}}\nolimits \left\{ n^{-1/2} \sum _{i=1}^{n}I[Z_i \ge t_0] X_i \delta _i \frac{G(t_0)}{G(Z_i)}d_i(\beta )\right\} \\&= n^{-1/2} \sum _{i=1}^{n} X_i \mathop {\mathrm{E}}\nolimits \left\{ d_i(\beta ) | T_i \ge t_0\right\} . \end{aligned}$$

Let $\omega _{1i}^*$ be the line segment lying between $X_i^{\top } (\beta -\beta _0)$ and $X_i^{\top } (\beta -\beta _0)+\sigma _i t$. Then,

$$\begin{aligned} \mathop {\mathrm{E}}\nolimits \left\{ d_i(\beta ) | T_i \ge t_0 \right\}&= \int _{-\infty }^{\infty }d_i(\beta )g_{T_i - t_0} \left\{ \epsilon _i(\beta )+ X_i^{\top }(\beta -\beta _0) | T_i \ge t_0 \right\} d\epsilon _i(\beta )\\&= \sigma _i \int _{-\infty }^{\infty }\Phi (-|t |)\left\{ 2I[t>0]-1\right\} \left[ g_{T_i - t_0}\left\{ \sigma _i t + X_i^{\top }(\beta -\beta _0) \right\} + g_{T_i - t_0}^\prime \left\{ \omega _i^*(t) \right\} \sigma _i t \right] dt \end{aligned}$$

It follows from Conditions C1 and C3 that $\sup _i g_{T_i - t_0} \{\sigma _i t + X_i^{\top } (\beta -\beta _0)\}<\infty$. Since $\int _{-\infty }^{\infty } \Phi (-|t |)\{2I[t>0]-1\}dt=0$, we have

$$\begin{aligned} \int _{-\infty }^{\infty } \Phi (-|t |)\left\{ 2I[t>0]-1\right\} g_{T_i - t_0} \left\{ X_i^{\star \top } (\beta -\beta _0)\right\} dt=0. \end{aligned}$$

Again, by Condition C1,

$$\begin{aligned} \exists M > 0 \quad \text {such that} \quad \sup _i \left|g_i^\prime \left\{ \omega _i^*(t)\right\} \right|< M. \end{aligned}$$

Thus,

$$\begin{aligned} \left|E\left\{ d_i(\beta )\right\} \right|\le \int _{-\infty }^{\infty } |t |\Phi (|t |) |g_i^\prime \{\omega _i^*(t)\}|dt \le M \sigma _i^2 /2. \end{aligned}$$

Note that $\sum _{i=1}^{n} \sigma _i^2 = \text{ tr }(X H X^{\top }) = \text{ tr }(H X^{\top }X)$ is bounded by $H=O(n^{-1})$ and Condition C2. Then, $\sum _{i=1}^{n} |E\{d_i(\beta )\} |\le M \sum _{i=1}^{n}\sigma _i^2 /2$ is also bounded. Therefore,

$$\begin{aligned} \left\Vert \mathop {\mathrm{E}}\nolimits \{D_n^{(1)}(\beta )\} \right\Vert \le n^{-1/2} \sqrt{p} \sup \limits _{i,j} \left|X_{ij} \right|\sum _{i=1}^{n} \left|\mathop {\mathrm{E}}\nolimits \left\{ d_i(\beta ) | T_i \ge t_0 \right\} \right|\rightarrow 0 \quad \text {as}\ n\rightarrow 0. \end{aligned}$$

(10)

By applying Condition C3, we have

$$\begin{aligned} \mathop {\mathrm{Var}}\nolimits \left\{ D_n^{(1)}(\beta )\right\}&= \mathop {\mathrm{Var}}\nolimits \left\{ n^{-1/2} \sum _{i=1}^{n} X_i X_i^{\top } I[Z_i \ge t_0]\delta _i \frac{G(t_0)}{G(Z_i)} d_i(\beta ) \right\} \\&\le n^{-1}\sum _{i=1}^{n} \frac{X_i X_i^{\top }}{c_0}\mathop {\mathrm{E}}\nolimits \left\{ d_i^2(\beta ) | T_i \ge t_0\right\} . \end{aligned}$$

It follows from the arguments similar to evaluating $\mathop {\mathrm{E}}\nolimits \{d_i(\beta ) | T_i \ge t_0\}$ combining with Conditions C1 and C2, we have, as $n \rightarrow \infty$, $\Vert \mathop {\mathrm{E}}\nolimits \{d_i^2(\beta ) | T_i \ge t_0 \} \Vert \rightarrow 0$. This implies $\Vert \mathop {\mathrm{Var}}\nolimits \{D_n^{(1)}(\beta )\} \Vert \rightarrow 0$. Then, by the Weak Law of Large Numbers,

$$\begin{aligned} \left\Vert D_n^{(1)}(\beta ) \right\Vert \xrightarrow {p} 0, \quad \text {as}\ n\rightarrow \infty . \end{aligned}$$

(11)

for $\beta$ in a compact neighborhood of $\beta _0$.

To show $\Vert D^{(2)}_n(\beta ) \Vert \xrightarrow {p} 0$ as $n \rightarrow \infty$, we use the martingale representation of the Kaplan–Meier estimator (Fleming and Harrington 2011). Specifically, $\hat{G}(t)$ can be represented as

$$\begin{aligned} \frac{\hat{G}(t) - G(t)}{G(t)} = -\sum _{i=1}^n \int _0^t \left\{ \frac{\hat{G}(u^-)}{G(u)}\right\} \frac{dM_{i}^C(u)}{Y(u)} \end{aligned}$$

where $M_i^C(u) = N_i^C(u)-\int _{0}^{t}Y_i(u)d\Lambda ^C(s)$, $N_i^C(u)=(1-\delta _i)I[Z_i\le u], \Lambda ^C(u) = -\log \{G(u)\}$, $Y(u) = \sum _{i=1}^n Y_i(u)$, and $Y_i(u) = I[Z_i \ge u]$. By combining this with an application of the functional delta method and the uniform convergence result of $\hat{G}(\cdot )$ to $G(\cdot )$, we have

$$\begin{aligned} D^{(2)}_{n}(\beta )&= n^{-1/2} \sum _{i=1}^{n}I[Z_i \ge t_0] X_i \delta _i n^{-1}\sum _{j=1}^n\left\{ \frac{h_j(t_0)}{G(Z_i)} - \frac{h_j(Z_i)G(t_0)}{G^2(Z_i)}\right\} d_i(\beta ) + o_p(1)\\&= n^{-1/2} \sum _{j=1}^{n} \int _{t_{0}}^{\nu } \left\{ n^{-1}\sum _{i=1}^n I[Z_i \ge t_0] X_i \delta _i Y_i(u) \frac{G(t_0)}{G(Z_i)}d_i(\beta )\right\} \frac{dM_{j}^C(u)}{y(u)} + o_p(1) \end{aligned}$$

where

$$\begin{aligned} h_{j}(t) = G(t)\int _0^t \frac{dM_{j}^C(u)}{Y(u)} \text{ and } y(t) = \lim \limits _{n \rightarrow \infty } n^{-1} Y(t). \end{aligned}$$

Using the arguments similar to those used to establish $\Vert D_n^{(1)}(\beta ) \Vert \xrightarrow {p} 0$, as $n\rightarrow \infty$, it can be shown that $\mathop {\mathrm{E}}\nolimits \left\{ \left| Y_i(u)d_i(\beta ) \right| |\ T_i \ge t_0 \right\} = O(n^{-1/2})$. Thus,

$$\begin{aligned}&\left\Vert \mathop {\mathrm{E}}\nolimits \left\{ n^{-1}\sum _{i=1}^n I[Z_i \ge t_0] X_i \delta _i Y_i(u) \frac{G(t_0)}{G(Z_i)}d_i(\beta )\right\} \right\Vert \\&\quad = \left\Vert n^{-1}\sum _{i=1}^n X_i \mathop {\mathrm{E}}\nolimits \left\{ Y_i(u)d_i(\beta ) | T_i \ge t_0 \right\} \right\Vert \\&\quad \le \sqrt{p} \sup \limits _{i,j} |X_{ij} |n^{-1} \sum _{i=1}^n \mathop {\mathrm{E}}\nolimits \left\{ |Y_i(u)d_i(\beta ) |\ | \ T_i \ge t_0 \right\} \rightarrow 0. \end{aligned}$$

It then follows that, as $n \rightarrow \infty$

$$\begin{aligned}&\left\Vert n^{-1}\sum _{i=1}^n I[Z_i \ge t_0] X_i \delta _i Y_i(u) \frac{G(t_0)}{G(Z_i)}d_i(\beta ) \right. \\&\left. \quad - \mathop {\mathrm{E}}\nolimits \left\{ n^{-1}\sum _{i=1}^n I[Z_i \ge t_0] X_i \delta _i Y_i(u) \frac{G(t_0)}{G(Z_i)}d_i(\beta )\right\} \right\Vert \xrightarrow {p} 0 \end{aligned}$$

uniformly in $\beta$ for $\beta$ in the compact neighborhood of $\beta _0$. By applying the martingale central limit theorem and the Kolmogorov–Centsov Theorem (Karatzas and Shreve 1988, p53),

$$\begin{aligned}&n^{-1/2} \sum _{j=1}^{n} \frac{dM_{j}^C(u)}{y(u)} \text {converges weakly to a zero-mean Gaussian process}\\&\quad \text {with continuous sample paths.} \end{aligned}$$

By combining these results, it follows from Lemma 1 in Lin (2000) that

$$\begin{aligned} \left\Vert n^{-1/2} \sum _{j=1}^{n} \int _{t_0}^{\nu } \left\{ n^{-1}\sum _{i=1}^n I[Z_i \ge t_0] X_i \delta _i Y_i(u) \frac{G(t_0)}{G(Z_i)}d_i(\beta )\right\} \frac{dM_j^c(u)}{y(u)} \right\Vert \xrightarrow {p} 0. \end{aligned}$$

(12)

By combining (11) and (12), we have

$$\begin{aligned} \left\Vert {n^{1/2}} \left\{ \tilde{U}_n(\beta , \tau , H) - U_{t_0}(\beta , \tau ) \right\} \right\Vert \xrightarrow {p} 0. \end{aligned}$$

(13)

Note that both $\tilde{U}_{t_0}(\beta , \tau , H)$ and $U_{t_0}(\beta , \tau )$ are monotone functions, thus the point-wise convergence could be strengthened to uniform convergence Shorack and Wellner (2009).

To establish the asymptotic equivalence of $n^{1/2}(\hat{\beta }_{IS} - \beta _0)$ and $n^{1/2}(\hat{\beta }_{NS} - \beta _0)$, it suffices to show that the following two convergence results hold: As $n \rightarrow \infty$,

$$\begin{aligned}&\text{(i) } \left\Vert \hat{A}(\beta _0, H) - A\right\Vert \rightarrow 0 \text{ and } \\&\quad \text{(ii) } \left\Vert n^{1/2}\left\{ \tilde{U}_{t_0}(\beta _0, \tau , H) - U_{t_0}(\beta _0, \tau )\right\} \right\Vert \rightarrow 0. \end{aligned}$$

Note that Eq. (13) implies (ii). Thus, we prove (i). For any vectors $a, b \in R^p$,

$$\begin{aligned} \mathop {\mathrm{E}}\nolimits \left[ a^{\top } \hat{A}(\beta _0, H) b \right]&= a^{\top } \mathop {\mathrm{E}}\nolimits \left[ n^{-1}\sum _{i=1}^{n} I[Z_i> t_0] X_i X_i^{\top } \frac{\hat{G}(t_0) \delta _i}{\hat{G}(Z_i)} \phi \left( -\frac{\epsilon _i(\beta _0)}{\sigma _i}\right) \left( \frac{1}{\sigma _i}\right) \right] b \\&= a^{\top } \left[ n^{-1}\sum _{i=1}^{n} X_iX_i^{\top } \mathop {\mathrm{E}}\nolimits \left\{ \phi \left( -\frac{\epsilon _i(\beta _0)}{\sigma _i}\right) \left( \frac{1}{\sigma _i}\right) \bigg |\ T_i > t_0, X_i \right\} \right] b \end{aligned}$$

It follows from the variable transformation $t = \epsilon (\beta _0)/\sigma _i$ and the Taylor expansion at 0 that

$$\begin{aligned}&\mathop {\mathrm{E}}\nolimits \left\{ \phi \left( -\frac{\epsilon _i(\beta _0)}{\sigma _i}\right) \left( \frac{1}{\sigma _i}\right) \bigg |T_i > t_0, X_i \right\} \\&\quad = \int _{-\infty }^{\infty } \phi \bigg (-\frac{\epsilon _i(\beta _0)}{\sigma _i}\bigg )\bigg (\frac{1}{\sigma _i}\bigg )g_{T_i - t_0}(\epsilon _i)d\epsilon _i \\&= \int _{-\infty }^{\infty } \phi \left( -t \right) g_{T_i - t_0}(0)dt + \sigma _i\int _{-\infty }^{\infty } t\phi \left( -t \right) g^{\prime }_{T_i - t_0}\left( \omega _{2i}^{*}\right) dt \end{aligned}$$

where $\omega _{2i}^*$ is some value lying between 0 and $\sigma _i t$.

By Condition C1, we have

$$\begin{aligned} \sigma _i\int _{-\infty }^{\infty } t\phi \left( -t \right) g^{\prime }_{T_i - t_0}\left( \omega _{2i}^{*}\right) dt \le M\sigma _i\int _{-\infty }^{\infty } |t|\phi \left( -|t| \right) dt \rightarrow 0. \end{aligned}$$

Since $\int _{-\infty }^{\infty } \phi \left( -t \right) g_{T_i - t_0}(0)dt = 0$, we have $\mathop {\mathrm{E}}\nolimits \left\{ \phi \Big (-\frac{\epsilon _i(\beta _0)}{\sigma _i}\Big )\Big (\frac{1}{\sigma _i}\Big ) \bigg |T_i > t_0, X_i \right\} \rightarrow g_{T_i - t_0}(0)$ and, therefore,

$$\begin{aligned} \lim _{n \rightarrow \infty } \mathop {\mathrm{E}}\nolimits \left[ a^{\top } \hat{A}(\beta _0, H) b \right]&= a^{\top } \left\{ \lim _{n \rightarrow \infty } n^{-1}\sum _{i=1}^n X_i X_i^{\top } g_{T_i - t_0}(0) \right\} b \nonumber \\&= a^{\top } A b. \end{aligned}$$

(14)

By Condition C1 and applying the arguments in Pang et al. (2012, p 795, Appendix), it can be shown that

$$\begin{aligned} \mathop {\mathrm{E}}\nolimits \left[ \left\{ \phi \left( -\frac{\epsilon _i(\beta _0)}{\sigma _i}\right) \left( \frac{1}{\sigma _i}\right) \right\} ^2 \bigg |\ T_i > t_0, X_i \right] = O\left( n^{1/2}\right) . \end{aligned}$$

Then,

$$\begin{aligned}&\mathop {\mathrm{Var}}\nolimits \left[ a^{\top }\left\{ n^{-1}\sum _{i=1}^{n} I[Z_i> t_0] X_iX_i^{\top } \frac{\hat{G}(t_0) \delta _i}{\hat{G}(Z_i)} \phi \bigg (-\frac{\epsilon _i(\beta _0)}{\sigma _i}\bigg )\bigg (\frac{1}{\sigma _i}\bigg )\right\} b\right] \nonumber \\&\quad \le \ \frac{1}{n^2 c_0}\sum _{i=1}^n \left( a^{\top }X_i X_i^{\top }\right) ^2 \mathop {\mathrm{E}}\nolimits \left[ \left\{ \phi \bigg (-\frac{\epsilon _i(\beta _0)}{\sigma _i}\bigg ) \bigg (\frac{1}{\sigma _i}\bigg )\right\} ^2 \bigg |\ T_i > t_0, X_i \right] \rightarrow 0. \end{aligned}$$

(15)

By combining the results in Eqs. (14) and (15), we have $\left\Vert \hat{A}(\beta _0, H) - A\right\Vert \rightarrow 0$ as $n \rightarrow \infty$. This completes the proof of (i).