Bootstrapping quantiles in a fixed design regression model with censored data

We consider the problem of estimating the quantiles of a distribution function in a fixed design regression model in which the observations are subject to random right censoring. The quantile estimator is defined via a conditional Kaplan-Meier type estimator for the distribution at a given design point. We establish an a.s. asymptotic representation for this quantile estimator, from which we obtain its asymptotic normality. Because a complicated estimation procedure is necessary for estimating the asymptotic bias and variance, we use a resampling procedure, which provides us, via an asymptotic representation for the bootstrapped estimator, with an alternative for the normal approximation.     

A finite population quantile estimation by unequal probability sampling

Abstract

Using a model-assisted approach, this paper studies asymptotically design-unbiased (ADU) estimation of a population “distribution function” and extends to deriving an asymptotic and approximate unbiased estimator for a population quantile from a sample chosen with varying probabilities. The respective asymptotic standard errors and confidence intervals are then worked out. Numerical findings based on an actual data support the theory with efficient results.     

Quantile regression in functional linear semiparametric model

This paper proposes nonparametric estimation methods for functional linear semiparametric quantile regression, where the conditional quantile of the scalar responses is modelled by both scalar and functional covariates and an additional unknown nonparametric function term. The slope function is estimated using the functional principal component basis and the nonparametric function is approximated by a piecewise polynomial function. The asymptotic distribution of the estimators of slope parameters is derived and the global convergence rate of the quantile estimator of unknown slope function is established under suitable norm. The asymptotic distribution of the estimator of the unknown nonparametric function is also established. Simulation studies are conducted to investigate the finite-sample performance of the proposed estimators. The proposed methodology is demonstrated by analysing a real data from ADHD-200 sample.     

Accelerated Recurrence Time Models

Abstract.  For the analysis with recurrent events, we propose a generalization of the accelerated failure time model to allow for evolving covariate effects. These so-called accelerated recurrence time models postulate that the time to expected recurrence frequency, upon transformation, is a linear function of covariates with frequency-dependent coefficients. This modelling strategy shares the same spirit as quantile regression. An estimation and inference procedure is developed by generalizing the celebrated Powell's ( J. Econometrics 25, 1984, 303; J. Econometrics 32, 1986, 143) estimator for censored quantile regression. Consistency and asymptotic normality of the proposed estimator are established. An algorithm is devised to attain good computational efficiency. Simulations demonstrate that this proposal performs well under practical settings. This methodology is illustrated in an application to the well-known bladder cancer study.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Parameter estimation based upon nonparametric function estimators

By considering the solution to a linear approximation of a nonlinear regression problem, a procedure for developing a para¬meter estimator, based upon a nonpammetric estimator of a para¬metric function, is given. The resulting estimators, which are determinable in closed form, are asymptotically normally distri¬buted and are optimal among the class of estimators based upon the function estimator. Further, in many cases, the estimator will have the same asymptotic distribution theory as the correspond¬ing maximum likelihood estimator. Estimators based upon the Kaplan-Meier quantile function are developed for randomly censored samples.     

Methods of Statistical Inference for Median Regression Models with Doubly Censored Data

Recently, least absolute deviations (LAD) estimator for median regression models with doubly censored data was proposed and the asymptotic normality of the estimator was established. However, it is invalid to make inference on the regression parameter vectors, because the asymptotic covariance matrices are difficult to estimate reliably since they involve conditional densities of error terms. In this article, three methods, which are based on bootstrap, random weighting, and empirical likelihood, respectively, and do not require density estimation, are proposed for making inference for the doubly censored median regression models. Simulations are also done to assess the performance of the proposed methods.     

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.     

Local quantile regression

Quantile regression is a technique to estimate conditional quantile curves. It provides a comprehensive picture of a response contingent on explanatory variables. In a flexible modeling framework, a specific form of the conditional quantile curve is not a priori fixed. This motivates a local parametric rather than a global fixed model fitting approach. A nonparametric smoothing estimator of the conditional quantile curve requires to balance between local curvature and stochastic variability. In this paper, we suggest a local model selection technique that provides an adaptive estimator of the conditional quantile regression curve at each design point. Theoretical results claim that the proposed adaptive procedure performs as good as an oracle which would minimize the local estimation risk for the problem at hand. We illustrate the performance of the procedure by an extensive simulation study and consider a couple of applications: to tail dependence analysis for the Hong Kong stock market and to analysis of the distributions of the risk factors of temperature dynamics.     

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.     

Extreme Quantile Estimation for Autoregressive Models

ABSTRACT

A quantile autoregresive model is a useful extension of classical autoregresive models as it can capture the influences of conditioning variables on the location, scale, and shape of the response distribution. However, at the extreme tails, standard quantile autoregression estimator is often unstable due to data sparsity. In this article, assuming quantile autoregresive models, we develop a new estimator for extreme conditional quantiles of time series data based on extreme value theory. We build the connection between the second-order conditions for the autoregression coefficients and for the conditional quantile functions, and establish the asymptotic properties of the proposed estimator. The finite sample performance of the proposed method is illustrated through a simulation study and the analysis of U.S. retail gasoline price.     

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.     

Estimation with Interval Censored Data and Covariates

In biostatistical applications interest often focuses on the estimation of the distribution of time T between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed monitoring time C, then the data conforms to the well understood singly-censored current status model, also known as interval censored data, case I. Additional covariates can be used to allow for dependent censoring and to improve estimation of the marginal distribution of T. Assuming a wrong model for the conditional distribution of T, given the covariates, will lead to an inconsistent estimator of the marginal distribution. On the other hand, the nonparametric maximum likelihood estimator of FT requires splitting up the sample in several subsamples corresponding with a particular value of the covariates, computing the NPMLE for every subsample and then taking an average. With a few continuous covariates the performance of the resulting estimator is typically miserable. In van der Laan, Robins (1996) a locally efficient one-step estimator is proposed for smooth functionals of the distribution of T, assuming nothing about the conditional distribution of T, given the covariates, but assuming a model for censoring, given the covariates. The estimators are asymptotically linear if the censoring mechanism is estimated correctly. The estimator also uses an estimator of the conditional distribution of T, given the covariates. If this estimate is consistent, then the estimator is efficient and if it is inconsistent, then the estimator is still consistent and asymptotically normal. In this paper we show that the estimators can also be used to estimate the distribution function in a locally optimal way. Moreover, we show that the proposed estimator can be used to estimate the distribution based on interval censored data (T is now known to lie between two observed points) in the presence of covariates. The resulting estimator also has a known influence curve so that asymptotic confidence intervals are directly available. In particular, one can apply our proposal to the interval censored data without covariates. In Geskus (1992) the information bound for interval censored data with two uniformly distributed monitoring times at the uniform distribution (for T has been computed. We show that the relative efficiency of our proposal w.r.t. this optimal bound equals 0.994, which is also reflected in finite sample simulations. Finally, the good practical performance of the estimator is shown in a simulation study. This revised version was published online in July 2006 with corrections to the Cover Date.     

Efficient estimation for time-varying coefficient longitudinal models

For estimation of time-varying coefficient longitudinal models, the widely used local least-squares (LS) or covariance-weighted local LS smoothing uses information from the local sample average. Motivated by the fact that a combination of multiple quantiles provides a more complete picture of the distribution, we investigate quantile regression-based methods to improve efficiency by optimally combining information across quantiles. Under the working independence scenario, the asymptotic variance of the proposed estimator approaches the Cramér–Rao lower bound. In the presence of dependence among within-subject measurements, we adopt a prewhitening technique to transform regression errors into independent innovations and show that the prewhitened optimally weighted quantile average estimator asymptotically achieves the Cramér–Rao bound for the independent innovations. Fully data-driven bandwidth selection and optimal weights estimation are implemented through a two-step procedure. Monte Carlo studies show that the proposed method delivers more robust and superior overall performance than that of the existing methods.     

Median regression analysis from data with left and right censored observations

Median regression models provide a robust alternative to regression based on the mean. We propose a methodology for fitting a median regression model from data with both left and right censored observations, in which the left censoring variable is always observed. First we set up an adjusted least absolute deviation estimating function using the inverse censoring weighted approach, whose solution specifies the estimator. We derive the consistency and asymptotic normality of the proposed estimator and describe the inference procedure for the regression parameter. Finally, we check the finite sample performance of the proposed procedure through simulation.     

Quantile regression estimation for distortion measurement error data

We study the quantile estimation methods for the distortion measurement error data when variables are unobserved and distorted with additive errors by some unknown functions of an observable confounding variable. After calibrating the error-prone variables, we propose the quantile regression estimation procedure and composite quantile estimation procedure. Asymptotic properties of the proposed estimators are established, and we also investigate the asymptotic relative efficiency compared with the least-squares estimator. Simulation studies are conducted to evaluate the performance of the proposed methods, and a real dataset is analyzed as an illustration.     

Central quantile subspace

Quantile regression (QR) is becoming increasingly popular due to its relevance in many scientific investigations. There is a great amount of work about linear and nonlinear QR models. Specifically, nonparametric estimation of the conditional quantiles received particular attention, due to its model flexibility. However, nonparametric QR techniques are limited in the number of covariates. Dimension reduction offers a solution to this problem by considering low-dimensional smoothing without specifying any parametric or nonparametric regression relation. The existing dimension reduction techniques focus on the entire conditional distribution. We, on the other hand, turn our attention to dimension reduction techniques for conditional quantiles and introduce a new method for reducing the dimension of the predictor $$\mathbf {X}$$. The novelty of this paper is threefold. We start by considering a single index quantile regression model, which assumes that the conditional quantile depends on $$\mathbf {X}$$ through a single linear combination of the predictors, then extend to a multi-index quantile regression model, and finally, generalize the proposed methodology to any statistical functional of the conditional distribution. The performance of the methodology is demonstrated through simulation examples and real data applications. Our results suggest that this method has a good finite sample performance and often outperforms the existing methods.     

On multivariate quantile regression analysis

This paper investigates the estimation of parameters in a multivariate quantile regression model when the investigator wants to evaluate the associated distribution function. It proposes a new directional quantile estimator with the following properties: (1) it applies to an arbitrary number of random variables; (2) it is equivalent to estimating the distribution function allowing for non-convex distribution contours; (3) it satisfies nice equivariance properties; (4) it has desirable statistical properties (i.e., consistency and asymptotic normality); and (5) its implementation involves a modest computational burden: our proposed estimator can be obtained by solving parametric linear programming problems. As such, this paper expands the range of applications of quantile estimation for multivariate regression models.     

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     

Bias reduction for high quantiles

High quantile estimation is of importance in risk management. For a heavy-tailed distribution, estimating a high quantile is done via estimating the tail index. Reducing the bias in a tail index estimator can be achieved by using either the same order or a larger order of number of the upper order statistics in comparison with the theoretical optimal one in the classical tail index estimator. For the second approach, one can either estimate all parameters simultaneously or estimate the first and second order parameters separately. Recently, the first method and the second method via external estimators for the second order parameter have been applied to reduce the bias in high quantile estimation. Theoretically, the second method obviously gives rise to a smaller order of asymptotic mean squared error than the first one. In this paper we study the second method with simultaneous estimation of all parameters for reducing bias in high quantile estimation.