Semiparametric median residual life model and inference

Abstract

For randomly censored data, the authors propose a general class of semiparametric median residual life models. They incorporate covariates in a generalized linear form while leaving the baseline median residual life function completely unspecified. Despite the non‐identifiability of the survival function for a given median residual life function, a simple and natural procedure is proposed to estimate the regression parameters and the baseline median residual life function. The authors derive the asymptotic properties for the estimators, and demonstrate the numerical performance of the proposed method through simulation studies. The median residual life model can be easily generalized to model other quantiles, and the estimation method can also be applied to the mean residual life model. The Canadian Journal of Statistics 38: 665–679; 2010 © 2010 Statistical Society of Canada

Shrinkage estimation of varying covariate effects based on quantile regression

“…For unpenalized estimation of model (1) in the presence of random censoring by C , we consider an adaptation of Ma and Yin (2010)’s estimator, which can be obtained as $${\stackrel{\sim }{\mathrm{bold-italic\beta }}}_{\mathrm{normalC},n}false(\tau false)=\arg \underset{\mathit{\beta }}{\min }{V}_{n}false(\mathrm{bold-italic\beta };\tau false),$$where $$V_{n}false(\mathrm{bold-italic\beta };\tau false)=\sum _{i=1}^n\frac{{\delta }_i}{\stackrel{true^}{G}(X_i)}\rho (X_i-{\mathrm{bold-italicZ}}_i^{\mathrm{normalT}}\mathit{\beta };\tau ).$$Here Ĝ (·) is the Kaplan-Meier estimator of G ( x ) = pr( C ≥ x ). One can use the available software, for example, the R package quantreg , to compute β̃ C, n ( τ ) as weighted regression quantiles.…”

“…Under regularity conditions C1-C2 and some mild assumption on censoring as those in Ma and Yin (2010), we have (selection consistency) ${\lim }_{n\to \infty }\mathrm{italicpr}false(false\{2\le j\le p:{\sup }_{\tau \in \Delta }false|{\stackrel{true^}{\beta }}_{\mathrm{normalC},n,{\mathrm{\lambda }}_{\mathrm{normalC},n}}^{false(jfalse)}false(\tau false)false|=0false\}=false\{s+1,\dots ,pfalse\}false)=1$; (weak convergence) $n^{1/2}false\{{\widehat{\mathrm{bold-italic\beta }}}_{\mathrm{normalC},n,{\mathrm{\lambda }}_{\mathrm{normalC},n}}^{false(1:sfalse)}false(\tau false)-{\mathit{\beta }}_0^{false(1:sfalse)}false(\tau false)false\}$ converges weakly to a mean zero Gaussian process with a covariance matrix Σ C,oracle ( τ, τ′ ). Here, Σ C,oracle ( τ, τ′ ) is the same as the limit covariance matrix of $n^{1/2}\{{\stackrel{true^}{\mathrm{bold-italic\beta }}}_{\mathrm{normalC},\mathrm{oracle}}false(\tau false)-{\mathit{\beta }}_0^{false(1:sfalse)}false(\tau false)\}$ where β̂ C,oracle ( τ ) is the unpenalized estimator of ${\mathit{\beta }}_0^{false(1:sfalse)}false(\tau false)$ obtained as the minimizer of V n ( β ; τ ) with ${\beta }_0^{false(jfalse)}false(\tau false)false(s+1\le j\le pfalse)$ known to be zero functions .…”

Quantile residual lifetime for left-truncated and right-censored data

“…For example, the mean residual life function does not always exist when the distribution is heavy tailed [11].…”

Estimating a Treatment Effect in Residual Time Quantiles Under the Additive Hazards Model

“…In a follow-up paper, Jung, Jeong and Bandos [10] proposed a regression model that allows for modeling of covariate effects on general quantile residual time. A different regression model is proposed by Ma and Yin [17], allowing for estimation of quantiles of residual times in addition to covariate effects on them. In a recent paper, Crouch, May and Chen [4] developed covariate-specific estimators for residual time quantiles based on the Cox model.…”