A general quantile residual life model for length‐biased right‐censored data

The quantile residual lifetime function provides comprehensive quantitative measures for residual life, especially when the distribution of the latter is skewed or heavy‐tailed and/or when the data contain outliers. In this paper, we propose a general class of semiparametric quantile residual life models for length‐biased right‐censored data. We use the inverse probability weighted method to correct the bias due to length‐biased sampling and informative censoring. Two estimating equations corresponding to the quantile regressions are constructed in two separate steps to obtain an efficient estimator. Consistency and asymptotic normality of the estimator are established. The main difficulty in implementing our proposed method is that the estimating equations associated with the quantiles are nondifferentiable, and we apply the majorize–minimize algorithm and estimate the asymptotic covariance using an efficient resampling method. We use simulation studies to evaluate the proposed method and illustrate its application by a real‐data example.     

Estimation for the censored partially linear quantile regression models

In this article, we develop estimation procedures for partially linear quantile regression models, where some of the responses are censored by another random variable. The nonparametric function is estimated by basis function approximations. The estimation procedure is easy to implement through existing weighted quantile regression, and it requires no specification of the error distributions. We show the large-sample properties of the resulting estimates, the proposed estimator of the regression parameter is root-n consistent and asymptotically normal and the estimator of the functional component achieves the optimal convergence rate of the nonparametric function. The proposed method is studied via simulations and illustrated with the analysis of a primary biliary cirrhosis (BPC) data.     

Linear censored quantile regression: A novel minimum-distance approach

In this article, we investigate a new procedure for the estimation of a linear quantile regression with possibly right-censored responses. Contrary to the main literature on the subject, we propose in this context to circumvent the formulation of conditional quantiles through the so-called “check” loss function that stems from the influential work of Koenker and Bassett (1978). Instead, our suggestion is here to estimate the quantile coefficients by minimizing an alternative measure of distance. In fact, our approach could be qualified as a generalization in a parametric regression framework of the technique consisting in inverting the conditional distribution of the response given the covariates. This is motivated by the knowledge that the main literature for censored data already relies on some nonparametric conditional distribution estimation as well. The ideas of effective dimension reduction are then exploited in order to accommodate for higher dimensional settings as well in this context. Extensive numerical results then suggest that such an approach provides a strongly competitive procedure to the classical approaches based on the check function, in fact both for complete and censored observations. From a theoretical prospect, both consistency and asymptotic normality of the proposed estimator for linear regression are obtained under classical regularity conditions. As a by-product, several asymptotic results on some “double-kernel” version of the conditional Kaplan–Meier distribution estimator based on effective dimension reduction, and its corresponding density estimator, are also obtained and may be of interest on their own. A brief application of our procedure to quasar data then serves to further highlight the relevance of the latter for quantile regression estimation with censored data.     

Copula‐based semiparametric analysis for time series data with detection limits

The analysis of time series data with detection limits is challenging due to the high‐dimensional integral involved in the likelihood. Existing methods are either computationally demanding or rely on restrictive parametric distributional assumptions. We propose a semiparametric approach, where the temporal dependence is captured by parametric copula, while the marginal distribution is estimated non‐parametrically. Utilizing the properties of copulas, we develop a new copula‐based sequential sampling algorithm, which provides a convenient way to calculate the censored likelihood. Even without full parametric distributional assumptions, the proposed method still allows us to efficiently compute the conditional quantiles of the censored response at a future time point, and thus construct both point and interval predictions. We establish the asymptotic properties of the proposed pseudo maximum likelihood estimator, and demonstrate through simulation and the analysis of a water quality data that the proposed method is more flexible and leads to more accurate predictions than Gaussian‐based methods for non‐normal data. The Canadian Journal of Statistics 47: 438–454; 2019 © 2019 Statistical Society of Canada     

Quantile regression based on a weighted approach under semi-competing risks data

In this article, we investigate the quantile regression analysis for semi-competing risks data in which a non-terminal event may be dependently censored by a terminal event. Due to the dependent censoring, the estimation of quantile regression coefficients on the non-terminal event becomes difficult. In order to handle this problem, we assume Archimedean Copula to specify the dependence of the non-terminal event and the terminal event. Portnoy [Censored regression quantiles. J Amer Statist Assoc. 2003;98:1001–1012] considered the quantile regression model under right-censoring data. We extend his approach to construct a weight function, and then impose the weight function to estimate the quantile regression parameter for the non-terminal event under semi-competing risks data. We also prove the consistency and asymptotic properties for the proposed estimator. According to the simulation studies, the performance of our proposed method is good. We also apply our suggested approach to analyse a real data.     

Asymptotic normality for a non parametric estimator of conditional quantile with left-truncated data

In this paper, we construct a non parametric estimator of conditional distribution function by the double-kernel local linear approach for left-truncated data, from which we derive the weighted double-kernel local linear estimator of conditional quantile. The asymptotic normality of the proposed estimators is also established. Finite-sample performance of the estimator is investigated via simulation.     

Focused information criterion and model averaging based on weighted composite quantile regression

We study the focused information criterion and frequentist model averaging and their application to post‐model‐selection inference for weighted composite quantile regression (WCQR) in the context of the additive partial linear models. With the non‐parametric functions approximated by polynomial splines, we show that, under certain conditions, the asymptotic distribution of the frequentist model averaging WCQR‐estimator of a focused parameter is a non‐linear mixture of normal distributions. This asymptotic distribution is used to construct confidence intervals that achieve the nominal coverage probability. With properly chosen weights, the focused information criterion based WCQR estimators are not only robust to outliers and non‐normal residuals but also can achieve efficiency close to the maximum likelihood estimator, without assuming the true error distribution. Simulation studies and a real data analysis are used to illustrate the effectiveness of the proposed procedure.     

A semi-parametric estimation of a survival function from incomplete and doubly censored data

The purpose of this paper is to present a semi-parametric estimation of a survival function when analyzing incomplete and doubly censored data. Under the assumption that the chance of censoring is not related to the individual's survivorship, we propose a consistent estimation of survival. The derived estimator treats the uncensored observations nonparametrically and uses parametric models for both right and left censored data. Some asymptotic properties and simulation studies are also presented in order to analyze the behavior of the proposed estimator.     

Nonparametric Quantile Estimation with Correlated Failure Time Data

In biomedical studies, correlated failure time data arise often. Although point and confidence interval estimation for quantiles with independent censored failure time data have been extensively studied, estimation for quantiles with correlated failure time data has not been developed. In this article, we propose a nonparametric estimation method for quantiles with correlated failure time data. We derive the asymptotic properties of the quantile estimator and propose confidence interval estimators based on the bootstrap and kernel smoothing methods. Simulation studies are carried out to investigate the finite sample properties of the proposed estimators. Finally, we illustrate the proposed method with a data set from a study of patients with otitis media.     

Quantile regression for panel data models with fixed effects under random censoring

Abstract

The locally weighted censored quantile regression approach is proposed for panel data models with fixed effects, which allows for random censoring. The resulting estimators are obtained by employing the fixed effects quantile regression method. The weights are selected either parametrically, semi-parametrically or non-parametrically. The large panel data asymptotics are used in an attempt to cope with the incidental parameter problem. The consistency and limiting distribution of the proposed estimator are also derived. The finite sample performance of the proposed estimators are examined via Monte Carlo simulations.     

Single-index composite quantile regression

In this paper, we extend the composite quantile regression (CQR) method to a single-index model. The unknown link function is estimated by local composite quantile regression and the parametric index is estimated through the linear composite quantile. It is shown that the proposed estimators are consistent and asymptotically normal. The simulation studies and real data applications are conducted to illustrate the finite sample performance of the proposed methods.     

Semiparametric Quantile Regression Analysis of Right‐censored and Length‐biased Failure Time Data with Partially Linear Varying Effects

Right‐censored and length‐biased failure time data arise in many fields including cross‐sectional prevalent cohort studies, and their analysis has recently attracted a great deal of attention. It is well‐known that for regression analysis of failure time data, two commonly used approaches are hazard‐based and quantile‐based procedures, and most of the existing methods are the hazard‐based ones. In this paper, we consider quantile regression analysis of right‐censored and length‐biased data and present a semiparametric varying‐coefficient partially linear model. For estimation of regression parameters, a three‐stage procedure that makes use of the inverse probability weighted technique is developed, and the asymptotic properties of the resulting estimators are established. In addition, the approach allows the dependence of the censoring variable on covariates, while most of the existing methods assume the independence between censoring variables and covariates. A simulation study is conducted and suggests that the proposed approach works well in practical situations. Also, an illustrative example is provided.     

Simultaneous estimation for non-crossing multiple quantile regression with right censored data

In this paper, we consider the estimation problem of multiple conditional quantile functions with right censored survival data. To account for censoring in estimating a quantile function, weighted quantile regression (WQR) has been developed by using inverse-censoring-probability weights. However, the estimated quantile functions from the WQR often cross each other and consequently violate the basic properties of quantiles. To avoid quantile crossing, we propose non-crossing weighted multiple quantile regression (NWQR), which estimates multiple conditional quantile functions simultaneously. We further propose the adaptive sup-norm regularized NWQR (ANWQR) to perform simultaneous estimation and variable selection. The large sample properties of the NWQR and ANWQR estimators are established under certain regularity conditions. The proposed methods are evaluated through simulation studies and analysis of a real data set.     

A software reliability model using mean residual quantile function

In this paper, we propose a class of distributions with the inverse linear mean residual quantile function. The distributional properties of the family of distributions are studied. We then discuss the reliability characteristics of the family of distributions. Some characterizations of the class of distributions are also discussed. The parameters of the class of distributions are estimated using the method of L-moments. The proposed class of distributions is applied to a real data set.     

The estimation of treatment effect for the censored bivariate data under the univariate censoring

This paper establishes a nonparametric estimator for the treatment effect on censored bivariate data under unvariate censoring. This proposed estimator is based on the one from Lin and Ying(1993)'s nonparametric bivariate survival function estimator, which is itself a generalized version of Park and Park(1995)' quantile estimator. A Bahadur type representation of quantile functions were obtained from the marginal survival distribution estimator of Lin and Ying' model. The asymptotic property of this estimator is shown below and the simulation studies are also given     

Parametric Estimation of Menarcheal Age Distribution Based on Recall Data

Menarche, the onset of menstruation, is an important maturational event of female childhood. Most of the studies of age at menarche make use of dichotomous (status quo) data. More information can be harnessed from recall data, but such data are often censored in a informative way. We show that the usual maximum likelihood estimator based on interval censored data, which ignores the informative nature of censoring, can be biased and inconsistent. We propose a parametric estimator of the menarcheal age distribution on the basis of a realistic model of the recall phenomenon. We identify the additional information contained in the recall data and demonstrate theoretically as well as through simulations the advantage of the maximum likelihood estimator based on recall data over that based on status quo data.     

Fast Censored Linear Regression

Weighted log‐rank estimating function has become a standard estimation method for the censored linear regression model, or the accelerated failure time model. Well established statistically, the estimator defined as a consistent root has, however, rather poor computational properties because the estimating function is neither continuous nor, in general, monotone. We propose a computationally efficient estimator through an asymptotics‐guided Newton algorithm, in which censored quantile regression methods are tailored to yield an initial consistent estimate and a consistent derivative estimate of the limiting estimating function. We also develop fast interval estimation with a new proposal for sandwich variance estimation. The proposed estimator is asymptotically equivalent to the consistent root estimator and barely distinguishable in samples of practical size. However, computation time is typically reduced by two to three orders of magnitude for point estimation alone. Illustrations with clinical applications are provided.     

The Robustness of Several Estimators of the Survivorship Function with Randomly Censored Data

The problem of estimating the survivorship function, R(t) = P(T > t), arises frequently in both engineering and biomedical sciences. In many applications the data one sees are censored due to the occurrence of some competing cause of failure such as withdrawal from the study, failure from some cause not under study, etc. In the biomedical sciences the distribution free estimator suggested by Kaplan and Meier (JASA 1958) is routinely used, while in the engineering sciences a parametric approach is more commonly used. In this report we study the efficiency of these two techniques when a particular parametric model such as the exponential, Weibull, normal, log normal, exponential power, Pareto, Gompertz, gamma, or bathtub shaped hazard distribution is assumed under a variety of censoring schemes and underlying failure models. We conclude that in most cases the parametric estimators outperform the distribution free estimator. The results are particularly striking if the Weibull forms of these estimators are used routinely.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.