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˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

随机丢失下的分位数回归及其经验似然

沈煜，梁汉营

同济大学数学系，上海 200092

摘要：本文研究反映变量随机丢失下的线性分位数回归模型. 基于逆概率权方法，我们建立了

分位数回归的估计方程并定义未知参数的分位数回归估计量，同时我构造出未知参数的经验似

然比函数，由此定义参数的极大经验似然估计量. 在适当的条件下，我们研究了构造估计量的

渐近正态性，证明经验似然比统计量具有标准的 χ

极限分布.

关键词：渐近正态性，经验似然，极大经验似然，随机丢失，分位数回归

中图分类号： O212.7

Quantile regression and its empirical likelihood

with missing response at random

SHEN Yu, LIANG Han-Ying

Department of Mathematics, Tongji University, Shanghai 200092

Abstract: In this paper, we study the linear quantile regression model when response data

are missing at random. Based on the inverse probability weight, we establish an estimation

equation on quantile regression and deﬁne the quantile regression estimator of unknown

parameter. At the same time, we construct the empirical likelihood (EL) ratio function for

the unknown parameter, and deﬁne the maximum EL estimator of the unknown parameter.

Under suitable assumptions, we investigate the asymptotic normality of the proposed

estimators and prove the EL ratio statistics has a standard chi-squared limiting distribution.

Key words: Asymptotic normality, Empirical likelihood, Maximum empirical likelihood

estimation, Missing at random, Quantile regression

0 Introduction

Quantile regression (QR) introduced by Koenker and Bassett

[1]

has become an increasingly

important tool in statistical analysis. Unlike the mean regression method that relies only on

the central tendency of the data, the QR approach allows the analyst to estimate the functional

dependence between variables for all portions of the conditional distribution of the response

variable. In other words, QR extends the framework of estimating only the behavior of the

基金项目： National Natural Science Foundation of China (11271286)，Specialized Research Fund for the Doctor Program

of Higher Education of China (20120072110007)

作者简介： SHEN Yu（1990-），male，Ph.D. of Tongji University，major research direction：nonparametric statistical

analysis. Correspondence author：LIANG Han-Ying（1964-），male，professor，major research direction：nonparametric

statistical analysis.

- 1 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

central part of a cloud of data points onto all parts of the conditional distribution. In that

sense QR provides a more complete view of relationships between variables of interest. A

comprehensive review can be found in Koenker

[2]

Missing data are common in opinion survey, medical research, experiment science and other

applied ﬁelds. In many practical situations, missing response may occur due to happenstance

or design. For example, in a opinion doll some individuals may refuse to supply personal

information, the responses are so expensive to get that only part of them are available in

an experiment. A simple approach to deal with missing response data is complete-case (CC)

analysis which deletes all incomplete data. However, it has an important drawback that it will

cause biased estimation when the data is not missing completely at random. For more details,

see Little and Rubin

[3]

. Most of the existing literature with missing data have been interested

in the conditional mean of the response given the covariates in regression models. One method

to handle missing data is imputation. Healy and Westmacott

[4]

discussed a linear regression,

Wang and Sun

[5]

developed a semiparametric imputation in partially linear models, Chen and

Van Keilegom

[6]

studied a nonparametric imputation. See also Robins et al.

[7]

, Zhou et al.

[8]

and M¨uller and Van Keilegom

[9]

. The other popular one is the inverse probability weighted

method, such as, Wang et al.

[10]

with missing covariates, and Wang et al.

[11]

, Wang and Sun

[5]

with missing response. Although the missing data analysis has a long history in statistics, most

literature on QR does not contain missing data problems. This is mainly caused by the reason

that the estimator for QR parameters is not linear in responses. In this paper, we focus on the

QR model with missing response at random.

Consider the linear QR model

= X

+ ϵ

, i = 1, 2, . . . , n, (1)

where Y

are response variables, X

are p ×1 random vectors of covariates, β

∈ R

is unknown

parameter and ϵ

are random errors whose τ th conditional quantile given X

equals zero for

τ ∈ (0, 1). Suppose (Y

, X

, δ

) is an iid sample of incomplete observations from (X, Y, δ), where

all the X

’s are observed, and δ

= 0 if Y

is missing, δ

= 1 otherwise. Throughout this paper,

we make the missing at random (MAR) assumption. The MAR assumption means that δ

and

is conditionally independent given X

, that is, P (δ

= 1|Y

, X

) = P (δ

= 1|X

) := ∆

Such missing mechanism is usually used for statistical analysis with missing response data and

is reasonable in many practical problems, see Little and Rubin

[3]

. There are many papers that

analyzed real data examples under MAR situations, such as Zhang and Zhu

[12]

, Liang et al.

[13]

Ciuperca

[14]

, Karimi and Mohammadzadeh

[15]

and so on. Imputation is a common method

for managing missing data under MAR. Yi and He

[16]

studied median regression model for

longitudinal data with dropouts where the covariates are known at all time but the response

may be missing from a certain time point. Wei et al.

[17]

investigated a multiple imputation

for managing missing covariates. Sherwood et al.

[18]

adopted the inverse probability weighted

- 2 -

˖ڍመ᝶஠ڙጲ

http://www.paper.edu.cn

method to develop weighted quantile regression with missing covariates.

For constructing conﬁdence regions, the empirical likelihood (EL) introduced by Owen

[19]

is an attractive method. The EL method shares the merits of parametric likelihood and is more

robust due to its nonparametric feature, see Owen

[20]

for a comprehensive overview, see also

Zhao et al.

[21]

, Fan et al.

[22]

, Zheng et al.

[23]

and Wei et al.

[24]

. Some authors have discussed EL

quantile regression. For example, Whang

[25]

developed a smoothed EL for QR model. Tang and

Leng

[26]

applied EL for QR model with auxiliary information, see also Qin and Tsao

[27]

, Otsu

[28]

and Wang and Zhu

[29]

. For missing data, to our best knowledge, only Lv and Li

[30]

investigated

the EL quantile regression, but they did not adopt imputation or inverse probability weighted

method.

In this paper, we consider the linear QR model (1) with MAR responses. Based on the

inverse probability weight, we establish an estimation equation on quantile regression and give

standard quantile regression estimator of β

. At the same time, we construct the EL ratio

function for β, and deﬁne the maximum EL estimator of β

. Under suitable assumptions, we

establish the asymptotic normality of the proposed estimators and prove the EL ratio statistics

has a standard chi-squared limiting distribution. A simulation study is done to investigate the

ﬁnite sample performance of the proposed method.

The rest of this paper is organized as follows. In Section 1, we deﬁne the estimators and

the empirical log-likelihood function. Main results are described in Section 2. Proofs of main

results are put in Section 3. Proofs of some lemmas, which are used in the proofs of the main

results, are given in Section 4.

1 Estimators

First, we brieﬂy review the quantile regression with complete data. Let

(τ|X

) = inf{y : F

(y|X

) ≥ τ }

be the τth conditional quantile of Y

given X

, where F

(y|X

) is the conditional distribution

function of Y

given X

. Model (1) implies that Q

(τ|X

) = X

. Therefore,

can be

estimated by minimizing the quantile objective function

(β) = n

−1



i=1

− X

β), (2)

where ρ

(t) = t[τ − I(t < 0)] and I(·) is the indicator function. In view of the deﬁnition of

subgradient (see Koenker

[2]

, the minimizer of (2) also satisﬁes that

−1



i=1



τ − I(Y

− X

β < 0)



≈ 0. (3)

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