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Smoothed quantile regression for censored residual life

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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.

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Data availability statement

The code used in the simulation study is included in the supplementary information. For confidentiality reasons, the dental restoration dataset is available upon request.

References

  • Bai F, Huang J, Zhou Y (2016) Semiparametric inference for the proportional mean residual life model with right-censored length-biased data. Stat Sin 46:1129–1158

    MathSciNet  MATH  Google Scholar 

  • Bai F, Chen X, Chen Y, Huang T (2019) A general quantile residual life model for length-biased right-censored data. Scand J Stat 46(4):1191–1205

    MathSciNet  MATH  Google Scholar 

  • Bang H, Tsiatis AA (2002) Median regression with censored cost data. Biometrics 58(3):643–649

    MathSciNet  MATH  Google Scholar 

  • Brown B, Wang YG (2005) Standard errors and covariance matrices for smoothed rank estimators. Biometrika 92(1):149–158

    MathSciNet  MATH  Google Scholar 

  • Brown B, Wang YG (2007) Induced smoothing for rank regression with censored survival times. Stat Med 26(4):828–836

    MathSciNet  Google Scholar 

  • Cai J, He H, Song X, Sun L (2017) An additive-multiplicative mean residual life model for right-censored data. Biom J 59(3):579–592

    MathSciNet  MATH  Google Scholar 

  • Caplan D, Li Y, Wang W, Kang S, Marchini L, Cowen H, Yan J (2019) Dental restoration longevity among geriatric and special needs patients. JDR Clin Transl Res 4(1):41–48

    Google Scholar 

  • Chen YQ (2007) Additive expectancy regression. J Am Stat Assoc 102(477):153–166

    MathSciNet  MATH  Google Scholar 

  • Chen YQ, Cheng S (2006) Linear life expectancy regression with censored data. Biometrika 93(2):303–313

    MathSciNet  MATH  Google Scholar 

  • Chen Y, Jewell N, Lei X, Cheng S (2005) Semiparametric estimation of proportional mean residual life model in presence of censoring. Biometrics 61(1):170–178

    MathSciNet  MATH  Google Scholar 

  • Cheng YJ, Huang CY (2014) Combined estimating equation approaches for semiparametric transformation models with length-biased survival data. Biometrics 70(3):608–618

    MathSciNet  MATH  Google Scholar 

  • Chiou SH, Kang S, Yan J (2014a) Fast accelerated failure time modeling for case-cohort data. Stat Comput 24(4):559–568

    MathSciNet  MATH  Google Scholar 

  • Chiou SH, Kang S, Yan J et al (2014b) Fitting accelerated failure time models in routine survival analysis with r package aftgee. J Stat Softw 61(11):1–23

    Google Scholar 

  • Chiou S, Kang S, Yan J (2015a) Rank-based estimating equations with general weight for accelerated failure time models: an induced smoothing approach. Stat Med 34(9):1495–1510

    MathSciNet  Google Scholar 

  • Chiou SH, Kang S, Yan J (2015b) Semiparametric accelerated failure time modeling for clustered failure times from stratified sampling. J Am Stat Assoc 110(510):621–629

    MathSciNet  MATH  Google Scholar 

  • Choi S, Kang S, Huang X (2018) Smoothed quantile regression analysis of competing risks. Biom J 60(5):934–946

    MathSciNet  MATH  Google Scholar 

  • Fleming TR, Harrington DP (2011) Counting processes and survival analysis, vol 169. Wiley, Hoboken

    MATH  Google Scholar 

  • Fu L, Wang YG, Bai Z (2010) Rank regression for analysis of clustered data: a natural induced smoothing approach. Comput Anal Data Anal 54(4):1036–1050

    MathSciNet  MATH  Google Scholar 

  • Huang Y (2010) Quantile calculus and censored regression. Ann Stat 38(3):1607

    MathSciNet  MATH  Google Scholar 

  • Johnson LM, Strawderman RL (2009) Induced smoothing for the semiparametric accelerated failure time model: asymptotics and extensions to clustered data. Biometrika 96(3):577–590

    MathSciNet  MATH  Google Scholar 

  • Jung SH (1996) Quasi-likelihood for median regression models. J Am Stat Assoc 91(433):251–257

    MathSciNet  MATH  Google Scholar 

  • Jung SH, Jeong JH, Bandos H (2009) Regression on quantile residual life. Biometrics 65(4):1203–1212

    MathSciNet  MATH  Google Scholar 

  • Kang S (2017) Fitting semiparametric accelerated failure time models for nested case-control data. J Stat Comput Simul 87(4):652–663

    MathSciNet  MATH  Google Scholar 

  • Karatzas I, Shreve S (1988) Brownian motion and stochastic calculus. Springer, New York

    MATH  Google Scholar 

  • Kassebaum N, Bernabé E, Dahiya M, Bhandari B, Murray C, Marcenes W (2015) Global burden of untreated caries: a systematic review and metaregression. J Dent Res 94(5):650–658

    Google Scholar 

  • Kim MO, Zhou M, Jeong JH (2012) Censored quantile regression for residual lifetimes. Lifetime Data Anal 18(2):177–194

    MathSciNet  MATH  Google Scholar 

  • Koenker R (2005) Quantile regression. Econometric society monographs. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  • Koenker R, Bassett G (1978) Regression quantiles. Econometrica 46(1):33–50

    MathSciNet  MATH  Google Scholar 

  • Koenker R, Portnoy S, Ng PT, Zeileis A, Grosjean P, Ripley BD (2012) Package ‘quantreg’

  • Li R, Huang X, Cortes JE (2016) Quantile residual life regression with longitudinal biomarker measurements for dynamic prediction. J R Stat Soc Ser C Appl Stat 65(5):755–773

    MathSciNet  Google Scholar 

  • Lin D (2000) On fitting cox’s proportional hazards models to survey data. Biometrika 87(1):37–47

    MathSciNet  MATH  Google Scholar 

  • Liu S, Ghosh SK (2008) Regression analysis of mean residual life function. Tech. rep., North Carolina State University. Dept. of Statistics

  • Maguluri G, Zhang CH (1994) Estimation in the mean residual life regression model. J R Stat Soc Ser B (Methodological) 56(3):477–489

    MathSciNet  MATH  Google Scholar 

  • Mansourvar Z, Martinussen T, Scheike TH (2016) An additive-multiplicative restricted mean residual life model. Scand J Stat 43(2):487–504

    MathSciNet  MATH  Google Scholar 

  • Oakes D, Dasu T (2003) Inference for the proportional mean residual life model. Lecture Notes-Monogr Ser 43:105–116

    MathSciNet  MATH  Google Scholar 

  • Pang L, Lu W, Wang HJ (2012) Variance estimation in censored quantile regression via induced smoothing. Comput Stat Data Anal 56(4):785–796

    MathSciNet  MATH  Google Scholar 

  • Peng L (2021) Quantile regression for survival data. Annu Rev Stat Appl 8:413–437

    MathSciNet  Google Scholar 

  • Peng L, Huang Y (2008) Survival analysis with quantile regression models. J Am Stat Assoc 103(482):637–649

    MathSciNet  MATH  Google Scholar 

  • Portnoy S (2003) Censored regression quantiles. J Am Stat Assoc 98(464):1001–1012

    MathSciNet  MATH  Google Scholar 

  • Portnoy S, Lin G (2010) Asymptotics for censored regression quantiles. J Nonparametric Stat 22(1):115–130

    MathSciNet  MATH  Google Scholar 

  • Portnoy S, Koenker R et al (1997) The gaussian hare and the Laplacian tortoise: computability of squared-error versus absolute-error estimators. Stat Sci 12(4):279–300

    MathSciNet  MATH  Google Scholar 

  • R Core Team (2021) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org/. Accessed 1 Dec 2021

  • Shorack GR, Wellner JA (2009) Empirical processes with applications to statistics. SIAM, Philadelphia

    MATH  Google Scholar 

  • Sun L, Zhang Z (2009) A class of transformed mean residual life models with censored survival data. J Am Stat Assoc 104(486):803–815

    MathSciNet  MATH  Google Scholar 

  • Sun L, Song X, Zhang Z (2012) Mean residual life models with time-dependent coefficients under right censoring. Biometrika 99(1):185–197

    MathSciNet  MATH  Google Scholar 

  • Wang HJ, Wang L (2009) Locally weighted censored quantile regression. J Am Stat Assoc 104(487):1117–1128

    MathSciNet  MATH  Google Scholar 

  • Wei Y, Pere A, Koenker R, He X (2006) Quantile regression methods for reference growth charts. Stat Med 25(8):1369–1382

    MathSciNet  Google Scholar 

  • Whang YJ (2006) Smoothed empirical likelihood methods for quantile regression models. Econom Theory 22:173–205

    MathSciNet  MATH  Google Scholar 

  • Ying Z, Sit T (2017) Survival analysis: a quantile perspective. In: Koenker R, Chernozhukov V, He X, Peng L (eds) Handbook of quantile regression. Chapman and Hall/CRC, London, pp 69–87. https://www.taylorfrancis.com/chapters/edit/10.1201/9781315120256-6/survival-analysis-quantile-perspective-zhiliang-ying-tony-sit

    Google Scholar 

  • Ying Z, Jung SH, Wei LJ (1995) Survival analysis with median regression models. J Am Stat Assoc 90(429):178–184

    MathSciNet  MATH  Google Scholar 

  • Zhang Z, Zhao X, Sun L (2010) Goodness-of-fit tests for additive mean residual life model under right censoring. Lifetime Data Anal 16(3):385–408

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (2020R1A2C1A0101313911)

Author information

Authors and Affiliations

  1. Department of Applied Statistics, Yonsei University, Seoul, Republic of Korea

    Kyu Hyun Kim & Sangwook Kang

  2. Department of Preventive and Community Dentistry, College of Dentistry, University of Iowa, Iowa City, IA, USA

    Daniel J. Caplan

  3. Department of Statistics and Data Science, Yonsei University, Seoul, Republic of Korea

    Sangwook Kang

Authors
  1. Kyu Hyun Kim
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  2. Daniel J. Caplan
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  3. Sangwook Kang
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Correspondence to Sangwook Kang.

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Supplementary Information

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Supplementary file 1 (pdf 186 KB)

Appendix 1. Proof of Theorem 1

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).

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Kim, K.H., Caplan, D.J. & Kang, S. Smoothed quantile regression for censored residual life. Comput Stat 38, 1001–1022 (2023). https://doi.org/10.1007/s00180-022-01262-z

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