随机插补下两线性模型中响应变量分位数差异的经验似然置信区间

设两个样本数据不完全的线性模型,其中协变量的观测值不缺失,响应变量的观测值随机缺失。采用随机回归插补法对响应变量的缺失值进行补足,得到两个线性回归模型的"完全"样本数据,在一定条件下得到两响应变量分位数差异的对数经验似然比统计量的极限分布为加权x_1~2,并利用此结果构造分位数差异的经验似然置信区间。模拟结果表明在随机插补下得到的置信区间具有较高的覆盖精度。     

本刊英文版Vol.37(2021),No.12论文摘要（英文）

<正>Empirical Likelihood of Quantile Difference with Missing Response When High-dimensional Covariates Are Present Cui Juan KONG Han Ying LIANG Abstract We,in this paper,investigate two-sample quantile difference by empirical likelihood method when the responses with high-dimensional covariates of the two populations are missing at random.In particular,based on sufficient dimension reduction technique,we construct three empirical log-likelihood ratios for the quantile difference between two samples by using inverse probability weighting imputation,regression imputation as well as augmented inverse probability weighting imputation,respectively,and prove their asymptotic distributions.     

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本文对两个样本数据不完全的线性模型展开讨论, 其中线性模型协变量的观测值不缺失, 响应变量的观测值随机缺失(MAR). 我们采用逆概率加权填补方法对响应变量的缺失值进行补足, 得到两个线性回归模型``完全'样本数据, 在``完全'样本数据的基础上构造了响应变量分位数差异的对数经验似然比统计量. 与以往研究结果不同的是本文在一定条件下证明了该统计量的极限分布为标准, 降低了由于权系数估计带来的误差, 进一步构造出了精度更高的分位数差异的经验似然置信区间.     

分数填补下两总体分位数差异的经验似然置信区间

设有两个非参数总体,其样本数据不完全,用分数填补法补足缺失数据,得到两总体的"完全"样本数据,在此基础上构造两总体分位数差异的经验似然置信区间.模拟结果显示,分数填补法可以得到更加精确的置信区间.     

分数填补下两总体分位数差异的半经验似然推断

在完全随机缺失机制情形,利用分数填补法填补缺失值,然后用经验似然方法构造两总体分位数差异的半经验似然比统计量,证明其渐近服从加权X~2分布并构造了相应的半经验似然置信区间.     

On nonparametric classification with missing covariates

General procedures are proposed for nonparametric classification in the presence of missing covariates. Both kernel-based imputation as well as Horvitz-Thompson-type inverse weighting approaches are employed to handle the presence of missing covariates. In the case of imputation, it is a certain regression function which is being imputed (and not the missing values). Using the theory of empirical processes, the performance of the resulting classifiers is assessed by obtaining exponential bounds on the deviations of their conditional errors from that of the Bayes classifier. These bounds, in conjunction with the Borel-Cantelli lemma, immediately provide various strong consistency results.     

Estimation and test procedures for composite quantile regression with covariates missing at random

In this paper, we study the weighted composite quantile regression (WCQR) for general linear model with missing covariates. We propose the WCQR estimation and bootstrap test procedures for unknown parameters. Simulation studies and a real data analysis are conducted to examine the finite performance of our proposed methods.     

基于协变量缺失的Horvitz-Thompson加权分位回归估计方法

含有协变量缺失的数据缺失问题是现代统计分析中的热点之一.当缺失数据中同时存在厚尾,偏斜和异方差问题时则更加难以处理.为此,本文提出一种逆概率加权分位回归估计来研究响应和协变量之间的关系.与经典估计方法相比具有明显优势,一方面,该估计量使用了所有可用的数据,并且允许缺失的协变量与响应高度相关;另一方面,该估计量在所有分位数水平上满足一致性和渐近正态性.通过模拟验证了该方法的在有限样本下的有效性,进一步将该方法推广到线性多元回归模型和非参数回归模型.     

Semi-empirical likelihood confidence intervals for the differences of quantiles with missing data

Detecting population （group） differences is useful in many applications, such as medical research. In this paper, we explore the probabilistic theory for identifying the quantile differences .between two populations. Suppose that there are two populations x and y with missing data on both of them, where x is nonparametric and y is parametric. We are interested in constructing confidence intervals on the quantile differences of x and y. Random hot deck imputation is used to fill in missing data. Semi-empirical likelihood confidence intervals on the differences are constructed.     

Joint modeling for mixed-effects quantile regression of longitudinal data with detection limits and covariates measured with error,with application to AIDS studies

It is very common in AIDS studies that response variable (e.g., HIV viral load) may be subject to censoring due to detection limits while covariates (e.g., CD4 cell count) may be measured with error. Failure to take censoring in response variable and measurement errors in covariates into account may introduce substantial bias in estimation and thus lead to unreliable inference. Moreover, with non-normal and/or heteroskedastic data, traditional mean regression models are not robust to tail reactions. In this case, one may find it attractive to estimate extreme causal relationship of covariates to a dependent variable, which can be suitably studied in quantile regression framework. In this paper, we consider joint inference of mixed-effects quantile regression model with right-censored responses and errors in covariates. The inverse censoring probability weighted method and the orthogonal regression method are combined to reduce the biases of estimation caused by censored data and measurement errors. Under some regularity conditions, the consistence and asymptotic normality of estimators are derived. Finally, some simulation studies are implemented and a HIV/AIDS clinical data set is analyzed to to illustrate the proposed procedure.     

因变量缺失下线性变量含误差模型的估计

主要考虑线性模型在自变量测量含误差以及因变量缺失情况下的估计问题.对于模型中的回归系数,我们基于最小二乘方法提出了两类估计,其中一类估计只由完整观测数据构成,而另外一类估计利用的则是利用简单插补方法构造的完整数据.证明了这两类估计是渐近正态性的.     

Weighted local polynomial estimations of a non-parametric function with censoring indicators missing at random and their applications

In this paper, we consider the weighted local polynomial calibration estimation and imputation estimation of a non-parametric function when the data are right censored and the censoring indicators are missing at random, and establish the asymptotic normality of these estimators. As their applications, we derive the weighted local linear calibration estimators and imputation estimations of the conditional distribution function, the conditional density function and the conditional quantile function, and investigate the asymptotic normality of these estimators. Finally, the simulation studies are conducted to illustrate the finite sample performance of the estimators.     

The rate of convergence for sparse and low-rank quantile trace regression

Trace regression models are widely used in applications involving panel data, images, genomic microarrays, etc., where high-dimensional covariates are often involved. However, the existing research involving high-dimensional covariates focuses mainly on the condition mean model. In this paper, we extend the trace regression model to the quantile trace regression model when the parameter is a matrix of simultaneously low rank and row (column) sparsity. The convergence rate of the penalized estimator is derived under mild conditions. Simulations, as well as a real data application, are also carried out for illustration.     

质量调整的价格指数编制中hedonic插补法的应用

在数据缺失的情况下,插补法是一种常用的推断缺失数据的方法。在价格指数的编制中,在基期存在的产品可能在报告期从市面上消失,或者报告期出现了新产品。这都可以看作是数据缺失的情形。同时由于前后时期产品质量发生变化,所编制的价格指数中可能包含"质量变化偏差"。Hedonic插补法将hedonic方法与缺失数据的插补方法结合起来,既处理了缺失数据,又克服了价格指数中的质量变化偏差。本文讨论了hedonic插补法的多种可能形式,并比较了各种方法的特点。本文还利用中国笔记本电脑的数据编制了hedonic插补价格指数,进行了相关的实证分析。     

Empirical likelihood confidence intervals for the differences of quantiles with missing data

Suppose that there are two nonparametric populations x and y with missing data on both of them. We are interested in constructing confidence intervals on the quantile differences of x and y. Random imputation is used. Empirical likelihood confidence intervals on the differences are constructed. Supported by the National Natural Science Foundation of China (No. 10661003) and Natural Science Foundation of Guangxi (No. 0728092).     

Multivariate logistic regression with incomplete covariate and auxiliary information

In this article, we propose and explore a multivariate logistic regression model for analyzing multiple binary outcomes with incomplete covariate data where auxiliary information is available. The auxiliary data are extraneous to the regression model of interest but predictive of the covariate with missing data. Horton and Laird [N.J. Horton, N.M. Laird, Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information, Biometrics 57 (2001) 34–42] describe how the auxiliary information can be incorporated into a regression model for a single binary outcome with missing covariates, and hence the efficiency of the regression estimators can be improved. We consider extending the method of [9] to the case of a multivariate logistic regression model for multiple correlated outcomes, and with missing covariates and completely observed auxiliary information. We demonstrate that in the case of moderate to strong associations among the multiple outcomes, one can achieve considerable gains in efficiency from estimators in a multivariate model as compared to the marginal estimators of the same parameters.     

缺失数据下线性模型回归系数估计的大样本性质

在响应变量满足MAR缺失机制下，我们分别研究了基于观察到的完全样本数据对、基于固定补足后的“完全洋本”和基于分数线性回归填补后的“完全洋本”得到的回归系数的最小二乘估计的弱相合性、强相合性及渐近正态性，我们还通过数值模拟，比较了基于上述估计得到的β的置信区间的优劣。     

Birnbaum‐Saunders quantile regression and its diagnostics with application to economic data

The Birnbaum‐Saunders (BS) distribution is a model that frequently appears in the statistical literature and has proved to be very versatile and efficient across a wide range of applications. However, despite the growing interest in the study of the BS distribution, quantile regression modeling has not been considered for this distribution. To fill this gap, we introduce a class of quantile regression models based on the BS distribution, which allows us to describe positive and asymmetric data when a quantile must be predicted using covariates. We use an approach based on a quantile parameterization to generate the model, permitting us to consider a similar framework to generalized linear models, providing wide flexibility. The methodology proposed includes a thorough study of theoretical properties and practical issues, such as maximum likelihood parameter estimation and diagnostic analytics based on local influence and residuals. The performance of the residuals is evaluated by simulations, whereas an illustrative example of income data is conducted using the methodology to show its potential for applications. The numerical results report an adequate performance of the approach to quantile regression, indicating that the BS distribution is a good modeling choice when dealing with data that have both positive support and asymmetry. The economic implications of our investigation are discussed in the final section. Hence, it can be a valuable addition to the tool kit of applied statisticians and econometricians.     

An Iterative Coordinate Descent Algorithm for High-Dimensional Nonconvex Penalized Quantile Regression

We propose and study a new iterative coordinate descent algorithm (QICD) for solving nonconvex penalized quantile regression in high dimension. By permitting different subsets of covariates to be relevant for modeling the response variable at different quantiles, nonconvex penalized quantile regression provides a flexible approach for modeling high-dimensional data with heterogeneity. Although its theory has been investigated recently, its computation remains highly challenging when p is large due to the nonsmoothness of the quantile loss function and the nonconvexity of the penalty function. Existing coordinate descent algorithms for penalized least-squares regression cannot be directly applied. We establish the convergence property of the proposed algorithm under some regularity conditions for a general class of nonconvex penalty functions including popular choices such as SCAD (smoothly clipped absolute deviation) and MCP (minimax concave penalty). Our Monte Carlo study confirms that QICD substantially improves the computational speed in the p ? n setting. We illustrate the application by analyzing a microarray dataset.     

φ-混合样本下缺失数据情形线性模型回归系数估计的渐近性质

φ-混合样本下,当响应变量满足随机缺失机制时,利用回归填补方法填补缺失的数据,在此基础上给出了线性模型回归系数的估计,并在一定的条件下证明了估计的渐近正态性.