An empirical likelihood method for quantile regression models with censored data
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An empirical likelihood method for quantile regression models with censored data Qibing Gao1 · Xiuqing Zhou1
· Yanqin Feng2 · Xiuli Du1 · XiaoXiao Liu1
Received: 16 May 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract An estimation for censored quantile regression models, which is based on an inversecensoring-probability weighting method, is studied in this paper, and asymptotic distribution of the parameter vector estimator is obtained. Based on the parameter estimation and asymptotic distribution of the estimator, an empirical likelihood inference method is proposed for censored quantile regression models and asymptotic property of empirical likelihood ratio is proved. Since the limiting distribution of the empirical likelihood ratio statistic is a mixture of chi-squared distributions, adjustment methods are also proposed to make the statistic converge to standard chi-squared distribution. The weighting scheme used in the parameter estimation is simple and the loss function is continuous and convex, and therefore, compared with empirical likelihood methods for quantile regression models with completely observed data, the methods proposed in this paper will not increase the computational complexity. This makes it especially useful for data with medium or high dimensional covariates. Simulation studies are developed to illustrate the performance of proposed methods. Keywords Quantile regression model · Empirical likelihood method · Censored data · Asymptotic properties
1 Introduction As an alternative of the mean regression model, quantile regression can describe the relationships between response variable Ti and the covariate vector X i more specifically, especially in some cases where upper and lower quantiles are of essential concern. The traditional quantile regression model can be written as
B
Xiuqing Zhou [email protected]
1
School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University, Nanjing 210023, China
2
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
123
Q. Gao et al.
Ti = βτT X i + i ,
i = 1, . . . , n,
(1)
where X i is the covariate vector of dimension k, βτ ∈ R k is the unknown true parameter vector, and i is the regression error whose τ th quantile conditional on X i is zero. In medical science, reliability studies and other applications, Ti may be the survival time or a transformation of the survival time which is exposed to possible censoring. In such cases, we can only observe {(Yi , X i , δi ), i = 1, . . . , n} instead of {(Ti , X i ), i = 1, . . . , n}, where Yi = min(Ti , Ci ), δi = I (Yi ≤ Ci ), and Ci is a censoring variable. Censored quantile regression models have been studied by many authors. The median regression model with all the censoring variables equal zero, which is called “Tobit” model in economics, was studied by Powell (1984), Pollard (1990), and Rao and Zhao (1993) among others. Zhou and Wang (2004, 2005), Zhong and Cui (2010) studied the median regression model
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