Under a single-index regression assumption, we introduce a new semiparametric procedure to estimate a conditional density of a censored response. The regression model can be seen as a generalization of Cox regression model and also as a profitable tool to perform dimension reduction under censoring. This technique extends the results of Delecroix et al. (2003). We derive consistency and asymptotic normality of our estimator of the index parameter by proving its asymptotic equivalence with the (uncomputable) maximum likelihood estimator, using martingales results for counting processes and arguments of empirical processes theory. Furthermore, we provide a new adaptive procedure which allows us both to chose the smoothing parameter involved in our approach and to circumvent the weak performances of Kaplan-Meier estimator (1958) in the right-tail of the distribution. Through a simulation study, we study the behavior of our estimator for small samples.
In this article, we propose some new generalizations of M-estimation procedures for single-index regression models in presence of randomly right-censored responses. We derive consistency and asymptotic normality of our estimates. The results are proved in order to be adapted to a wide range of techniques used in a censored regression framework (e.g. synthetic data or weighted least squares). As in the uncensored case, the estimator of the single-index parameter is seen to have the same asymptotic behavior as in a fully parametric scheme. We compare these new estimators with those based on the …
In this paper, we study the local constant and the local linear estimators of the conditional density function with right-censored data which exhibit some type of dependence. It is assumed that the observations form a stationary $α-$mixing sequence. The asymptotic normality of the two estimators is established, which combined with the condition that $\lim\limits_{n\rightarrow\infty}nh_nb_n=\infty$ implies the consistency of the two estimators and can be employed to construct confidence intervals for the conditional density function. The result on the local linear estimator of the conditional d…
We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
This paper considers inference in a linear regression model with random right censoring and outliers. The number of outliers can grow with the sample size while their proportion goes to zero. The model is semiparametric and we make only very mild assumptions on the distribution of the error term, contrary to most other existing approaches in the literature. We propose to penalize the estimator proposed by Stute for censored linear regression by the l1-norm. We derive rates of convergence and establish asymptotic normality of the estimator of the regression coefficients. Our estimator has the s…
The stratified proportional intensity model generalizes Cox's proportional intensity model by allowing different groups of the population under study to have distinct baseline intensity functions. In this article, we consider the problem of estimation in this model when the variable indicating the stratum is unobserved for some individuals in the studied sample. In this setting, we construct nonparametric maximum likelihood estimators for the parameters of the stratified model and we establish their consistency and asymptotic normality. Consistent estimators for the limiting variances are also…
Censored quantile regression (CQR) has become a valuable tool to study the heterogeneous association between a possibly censored outcome and a set of covariates, yet computation and statistical inference for CQR have remained a challenge for large-scale data with many covariates. In this paper, we focus on a smoothed martingale-based sequential estimating equations approach, to which scalable gradient-based algorithms can be applied. Theoretically, we provide a unified analysis of the smoothed sequential estimator and its penalized counterpart in increasing dimensions. When the covariate dimen…
We consider quantile regression processes from censored data under dependent data structures and derive a uniform Bahadur representation for those processes. We also consider cases where the dimension of the parameter in the quantile regression model is large. It is demonstrated that traditional penalized estimators such as the adaptive lasso yield sub-optimal rates if the coefficients of the quantile regression cross zero. New penalization techniques are introduced which are able to deal with specific problems of censored data and yield estimates with an optimal rate. In contrast to most of t…
This paper, investigates the conditional quantile estimation of a scalar random response and a functional random covariate (i.e. valued in some infinite-dimensional space) whenever {\it functional stationary ergodic data with random censorship} are considered. We introduce a kernel type estimator of the conditional quantile function. We establish the strong consistency with rate of this estimator as well as the asymptotic normality which induces a confidence interval that is usable in practice since it does not depend on any unknown quantity. An application to electricity peak demand interval …
In this paper, we are concerned with nonparametric estimation of the multivariate regression function in the presence of right censored data. More precisely, we propose a statistic that is shown to be asymptotically normally distributed under the additive assumption, and that could be used to test for additivity in the censored regression setting.
We develop inference procedures for longitudinal data where some of the measurements are censored by fixed constants. We consider a semi-parametric quantile regression model that makes no distributional assumptions. Our research is motivated by the lack of proper inference procedures for data from biomedical studies where measurements are censored due to a fixed quantification limit. In such studies the focus is often on testing hypotheses about treatment equality. To this end, we propose a rank score test for large sample inference on a subset of the covariates. We demonstrate the importance …
In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tildeθ_n,\tilde{F}_n)$ for the underlying parameter $θ_0$ and distribution $F_0$ is derived, and the strong consistency of $(\tildeθ_n,\tilde{F}_n)$…
Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile cal…
This note develops a simple two-stage least squares (2SLS) procedure to estimate the causal effect of some endogenous regressors on a randomly right censored outcome in the linear model. The proposal replaces the usual ordinary least squares regressions of the standard 2SLS by weighted least squares regressions. The weights correspond to the inverse probability of censoring. We show consistency and asymptotic normality of the estimator. The estimator exhibits good finite sample performances in simulations.
In many semiparametric models that are parameterized by two types of parameters---a Euclidean parameter of interest and an infinite-dimensional nuisance parameter---the two parameters are bundled together, that is, the nuisance parameter is an unknown function that contains the parameter of interest as part of its argument. For example, in a linear regression model for censored survival data, the unspecified error distribution function involves the regression coefficients. Motivated by developing an efficient estimating method for the regression parameters, we propose a general sieve M-theorem…
Interval-censored multi-state data arise in many studies of chronic diseases, where the health status of a subject can be characterized by a finite number of disease states and the transition between any two states is only known to occur over a broad time interval. We formulate the effects of potentially time-dependent covariates on multi-state processes through semiparametric proportional intensity models with random effects. We adopt nonparametric maximum likelihood estimation (NPMLE) under general interval censoring and develop a stable expectation-maximization (EM) algorithm. We show that …
Department of Mathematics University of California San Diego（加利福尼亚大学圣地亚哥分校数学系）
Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on new estimating equations that adapt to censoring and lead to quantile score whenever the data do not exhibit censoring. The proposed procedure named censored quantile regression f…
This paper develops robust confidence intervals in high-dimensional and left-censored regression. Type-I censored regression models are extremely common in practice, where a competing event makes the variable of interest unobservable. However, techniques developed for entirely observed data do not directly apply to the censored observations. In this paper, we develop smoothed estimating equations that augment the de-biasing method, such that the resulting estimator is adaptive to censoring and is more robust to the misspecification of the error distribution. We propose a unified class of robus…
University of California San Diego（加州大学圣地亚哥分校）；Alexa AI（Alexa人工智能部门）
Random forests are powerful non-parametric regression method but are severely limited in their usage in the presence of randomly censored observations, and naively applied can exhibit poor predictive performance due to the incurred biases. Based on a local adaptive representation of random forests, we develop its regression adjustment for randomly censored regression quantile models. Regression adjustment is based on a new estimating equation that adapts to censoring and leads to quantile score whenever the data do not exhibit censoring. The proposed procedure named {\it censored quantile regr…
We consider the problem of nonparametric quantile regression for twice censored data. Two new estimates are presented, which are constructed by applying concepts of monotone rearrangements to estimates of the conditional distribution function. The proposed methods avoid the problem of crossing quantile curves. Weak uniform consistency and weak convergence is established for both estimates and their finite sample properties are investigated by means of a simulation study. As a by-product, we obtain a new result regarding the weak convergence of the Beran estimator for right censored data on the…
Two-step estimators often called upon to fit censored regression models in many areas of science and engineering. Since censoring incurs a bias in the naive least-squares fit, a two-step estimator first estimates the bias and then fits a corrected linear model. We develop a framework for performing valid /post-correction inference/ with two-step estimators. By exploiting recent results on post-selection inference, we obtain valid confidence intervals and significance tests for the fitted coefficients.
We consider linear regression model estimation where the covariate of interest is randomly censored. Under a non-informative censoring mechanism, one may obtain valid estimates by deleting censored observations. However, this comes at a cost of lost information and decreased efficiency, especially under heavy censoring. Other methods for dealing with censored covariates, such as ignoring censoring or replacing censored observations with a fixed number, often lead to severely biased results and are of limited practicality. Parametric methods based on maximum likelihood estimation as well as sem…
The present paper deals with a nonparametric M-estimation for right censored regression model with stationary ergodic data. Defined as an implicit function, a kernel type estimator of a family of robust regression is considered when the covariate take its values in R^d (d >= 1) and the data are sampled from stationary ergodic process. The strong consistency (with rate) and the asymptotic distribution of the estimator are established under mild assumptions. Moreover, a usable confidence interval is provided which does not depend on any unknown quantity. Our results hold without any mixing condi…
Outlying observations, which significantly deviate from other measurements, may distort the conclusions of data analysis. Therefore, identifying outliers is one of the important problems that should be solved to obtain reliable results. While there are many statistical outlier detection algorithms and software programs for uncensored data, few are available for censored data. In this article, we propose three outlier detection algorithms based on censored quantile regression, two of which are modified versions of existing algorithms for uncensored or censored data, while the third is a newly d…
With the availability of high dimensional genetic biomarkers, it is of interest to identify heterogeneous effects of these predictors on patients' survival, along with proper statistical inference. Censored quantile regression has emerged as a powerful tool for detecting heterogeneous effects of covariates on survival outcomes. To our knowledge, there is little work available to draw inference on the effects of high dimensional predictors for censored quantile regression. This paper proposes a novel procedure to draw inference on all predictors within the framework of global censored quantile …
Interpretable Deep Regression Models with Interval-Censored Failure Time Data
School of Mathematics, Jilin University（吉林大学数学学院）；School of Economics and Statistics, Guangzhou University（广州大学经济与统计学院）；Department of Statistics, Chinese University of Hong Kong（香港中文大学统计系）；School of Data Science, Fudan University（复旦大学数据科学学院）
We establish asymptotic normality for estimators of the additive regression components under random censorship. To build our estimators, we couple the marginal integration method (Newey (1994)) with an initial Inverse Probability of Censoring Weighted estimator of the multivariate censored regression function introduced by Carbonez et al. (1995) and Kohler et al. (2002). Asymptotic confidence bands are derived from our result.
When facing multivariate covariates, general semiparametric regression techniques come at hand to propose flexible models that are unexposed to the curse of dimensionality. In this work a semiparametric copula-based estimator for conditional quantiles is investigated for complete or right-censored data. In spirit, the methodology is extending the recent work of Noh et al. (2013) and Noh et al. (2015), as the main idea consists in appropriately defining the quantile regression in terms of a multivariate copula and marginal distributions. Prior estimation of the latter and simple plug-in lead to…
In many applications, heterogeneous treatment effects on a censored response variable are of primary interest, and it is natural to evaluate the effects at different quantiles (e.g., median). The large number of potential effect modifiers, the unknown structure of the treatment effects, and the presence of right censoring pose significant challenges. In this paper, we develop a hybrid forest approach called Hybrid Censored Quantile Regression Forest (HCQRF) to assess the heterogeneous effects varying with high-dimensional variables. The hybrid estimation approach takes advantage of the random …
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 subje…
We discuss semiparametric regression when only the ranks of responses are observed. The model is $Y_i = F (\mathbf{x}_i'{\boldsymbolβ}_0 + \varepsilon_i)$, where $Y_i$ is the unobserved response, $F$ is a monotone increasing function, $\mathbf{x}_i$ is a known $p-$vector of covariates, ${\boldsymbolβ}_0$ is an unknown $p$-vector of interest, and $\varepsilon_i$ is an error term independent of $\mathbf{x}_i$. We observe $\{(\mathbf{x}_i,R_n(Y_i)) : i = 1,\ldots ,n\}$, where $R_n$ is the ordinal rank function. We explore a novel estimator under Gaussian assumptions. We discuss the literature, ap…
Tsinghua University（清华大学）；Microsoft Research（微软研究院）；University of Pittsburgh（匹兹堡大学）
This paper considers doing quantile regression on censored data using neural networks (NNs). This adds to the survival analysis toolkit by allowing direct prediction of the target variable, along with a distribution-free characterisation of uncertainty, using a flexible function approximator. We begin by showing how an algorithm popular in linear models can be applied to NNs. However, the resulting procedure is inefficient, requiring sequential optimisation of an individual NN at each desired quantile. Our major contribution is a novel algorithm that simultaneously optimises a grid of quantile…
We calculate finite sample and asymptotic distributions for the largest censored and uncensored survival times, and some related statistics, from a sample of survival data generated according to an iid censoring model. These statistics are important for assessing whether there is sufficient followup in the sample to be confident of the presence of immune or cured individuals in the population. A key structural result obtained is that, conditional on the value of the largest uncensored survival time, and knowing the number of censored observations exceeding this time, the sample partitions into…
This paper proposes a new test for the comparison of conditional quantile curves when the outcome of interest, typically a duration, is subject to right censoring. The test can be applied both in the case of two independent samples and for paired data, and can be used for the comparison of quantiles at a fixed quantile level, a finite set of levels or a range of quantile levels. The asymptotic distribution of the proposed test statistics is obtained both under the null hypothesis and under local alternatives. We describe a bootstrap procedure in order to approximate the critical values, and pr…
Interval-censored competing risks data arise when each study subject may experience an event or failure from one of several causes and the failure time is not observed exactly but rather known to lie in an interval between two successive examinations. We formulate the effects of possibly time-varying covariates on the cumulative incidence or sub-distribution function (i.e., the marginal probability of failure from a particular cause) of competing risks through a broad class of semiparametric regression models that captures both proportional and non-proportional hazards structures for the sub-d…
In this paper, we study a novel approach for the estimation of quantiles when facing potential right censoring of the responses. Contrary to the existing literature on the subject, the adopted strategy of this paper is to tackle censoring at the very level of the loss function usually employed for the computation of quantiles, the so-called "check" function. For interpretation purposes, a simple comparison with the latter reveals how censoring is accounted for in the newly proposed loss function. Subsequently, when considering the inclusion of covariates for conditional quantile estimation, by…
We consider identification and estimation of nonseparable sample selection models with censored selection rules. We employ a control function approach and discuss different objects of interest based on (1) local effects conditional on the control function, and (2) global effects obtained from integration over ranges of values of the control function. We derive the conditions for the identification of these different objects and suggest strategies for estimation. Moreover, we provide the associated asymptotic theory. These strategies are illustrated in an empirical investigation of the determin…
In this paper, we develop a new censored quantile instrumental variable (CQIV) estimator and describe its properties and computation. The CQIV estimator combines Powell (1986) censored quantile regression (CQR) to deal with censoring, with a control variable approach to incorporate endogenous regressors. The CQIV estimator is obtained in two stages that are non-additive in the unobservables. The first stage estimates a non-additive model with infinite dimensional parameters for the control variable, such as a quantile or distribution regression model. The second stage estimates a non-additive …
We investigate the performance of model based bootstrap methods for constructing point-wise confidence intervals around the survival function with interval censored data. We show that bootstrapping from the nonparametric maximum likelihood estimator of the survival function is inconsistent for both the current status and case 2 interval censoring models. A model based smoothed bootstrap procedure is proposed and shown to be consistent. In addition, simulation studies are conducted to illustrate the (in)-consistency of the bootstrap methods. Our conclusions in the interval censoring model would…
The computational prediction algorithm of neural network, or deep learning, has drawn much attention recently in statistics as well as in image recognition and natural language processing. Particularly in statistical application for censored survival data, the loss function used for optimization has been mainly based on the partial likelihood from Cox's model and its variations to utilize existing neural network library such as Keras, which was built upon the open source library of TensorFlow. This paper presents a novel application of the neural network to the quantile regression for survival…
In this paper, we study the behavior of a kernel estimator of the regression function in the right censored model with $α$-mixing data . The uniform strong consistency over a real compact set of the estimate is established along with a rate of convergence. Some simulations are carried out to illustrate the behavior of the estimate with different examples for finite sample sizes.
Regression models for continuous outcomes often require a transformation of the outcome, which the user either specify {\it a priori} or estimate from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation and are thus a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where the outcomes are censored at the ends. We prove uniform co…
In this paper, we propose a class of high breakdown point estimators for the linear regression model when the response variable contains censored observations. These estimators are robust against high-leverage outliers and they generalize the LMS (least median of squares), S, MM and $τ$-estimators for linear regression. An important contribution of this paper is that we can define consistent estimators using a bounded loss function (or equivalently, a redescending score function). Since the calculation of these estimators can be computationally costly, we propose an efficient algorithm to comp…
It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kernel-type Inverse Probability of Censoring Weighted estimators of the multivariate regression function \citeDebbarhViallon3. In this paper, we get the exact rate of strong uniform consistency for such estimators. Our uniform limit laws especially lead to the construction of asymptotic simultaneous 100% confidence bands for the true regression function.
We propose a unified framework for likelihood-based regression modeling when the response variable has finite support. Our work is motivated by the fact that, in practice, observed data are discrete and bounded. The proposed methods assume a model which includes models previously considered for interval-censored variables with log-concave distributions as special cases. The resulting log-likelihood is concave, which we use to establish asymptotic normality of its maximizer as the number of observations $n$ tends to infinity with the number of parameters $d$ fixed, and rates of convergence of $…
Let $ (T_i)_i$ be a sequence of independent identically distributed (i.i.d.) random variables (r.v.) of interest distributed as $ T$ and $(X_i)_i$ be a corresponding vector of covariates taking values on $ \mathbb{R}^d$. In censorship models the r.v. $T$ is subject to random censoring by another r.v. $C$. In this paper we built a new kernel estimator based on the so-called synthetic data of the mean squared relative error for the regression function. We establish the uniform almost sure convergence with rate over a compact set and its asymptotic normality. The asymptotic variance is explicitly…
Probabilistic survival predictions from models trained with Maximum Likelihood Estimation (MLE) can have high, and sometimes unacceptably high variance. The field of meteorology, where the paradigm of maximizing sharpness subject to calibration is popular, has addressed this problem by using scoring rules beyond MLE, such as the Continuous Ranked Probability Score (CRPS). In this paper we present the \emph{Survival-CRPS}, a generalization of the CRPS to the survival prediction setting, with right-censored and interval-censored variants. We evaluate our ideas on the mortality prediction task us…
We develop a unified approach for classification and regression support vector machines for data subject to right censoring. We provide finite sample bounds on the generalization error of the algorithm, prove risk consistency for a wide class of probability measures, and study the associated learning rates. We apply the general methodology to estimation of the (truncated) mean, median, quantiles, and for classification problems. We present a simulation study that demonstrates the performance of the proposed approach.
This work was motivated by observational studies in pregnancy with spontaneous abortion (SAB) as outcome. Clearly some women experience the SAB event but the rest do not. In addition, the data are left truncated due to the way pregnant women are recruited into these studies. For those women who do experience SAB, their exact event times are sometimes unknown. Finally, a small percentage of the women are lost to follow-up during their pregnancy. All these give rise to data that are left truncated, partly interval and right-censored, and with a clearly defined cured portion. We consider the non-…
