Cross Validation Based Model Averaging for Varying-Coefficient Models with Response Missing at Random

In this paper, a model averaging method is proposed for varying-coefficient models with response missing at random by establishing a weight selection criterion based on cross-validation. Under certain regularity conditions, it is proved that the proposed method is asymptotically optimal in the sense of achieving the minimum squared error.

Testing conditional independence with data missing at random

It is known that conditional independence is a quite basic assumption in many fields of statistics. How to test its validity is of great importance and has been extensively studied by the literature. Nevertheless, all of the existing methods focus on the case that data are fully observed, but none of them seems having taken into account of the scenario when missing data are present. Motivated by this, this paper develops two testing statistics to handle such a situation relying on the idea of inverse probability weighted and augmented inverse probability weighted techniques. The asymptotic distributions of the proposed statistics are also derived under the null hypothesis. The simulation studies indicate that both testing statistics perform well in terms of size and power.

An Efficient Multiple Imputation Approach for Estimating Equations with Response Missing at Random and High-Dimensional Covariates

Empirical-likelihood-based inference for parameters defined by the general estimating equations of Qin and Lawless(1994) remains an active research topic. When the response is missing at random(MAR) and the dimension of covariate is not low, the authors propose a two-stage estimation procedure by using the dimension-reduced kernel estimators in conjunction with an unbiased estimating function based on augmented inverse probability weighting and multiple imputation(AIPW-MI) methods. The authors show that the resulting estimator achieves consistency and asymptotic normality. In addition, the corresponding empirical likelihood ratio statistics asymptotically follow central chi-square distributions when evaluated at the true parameter. The finite-sample performance of the proposed estimator is studied through simulation, and an application to HIV-CD4 data set is also presented.

A class of weighted estimating equations for additive hazards models with covariates missing at random

Missing covariate data arise frequently in biomedical studies.In this article,we propose a class of weighted estimating equations for the additive hazards regression model when some of the covariates are missing at random.Time-specific and subject-specific weights are incorporated into the formulation of weighted estimating equations.Unified results are established for estimating selection probabilities that cover both parametric and non-parametric modelling schemes.The resulting estimators have closed forms and are shown to be consistent and asymptotically normal.Simulation studies indicate that the proposed estimators perform well for practical settings.An application to a mouse leukemia study is illustrated.

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 abstract of doctoral dissertation‘nonlinear wavelet density estimation and hazard rate estimation with data missing at random’

In this thesis,we establish non-linear wavelet density estimators and studying the asymptotic properties of the estimators with data missing at random when covariates are present.The outstanding advantage of non-linear wavelet method is estimating the unsoothed functions,however,the classical kernel estimation cannot do this work.At the same time,we study the larger sample properties of the ISE for hazard rate estimator.

CBPS-Based Inference in Nonlinear Regression Models with Missing Data

In this article, to improve the doubly robust estimator, the nonlinear regression models with missing responses are studied. Based on the covariate balancing propensity score (CBPS), estimators for the regression coefficients and the population mean are obtained. It is proved that the proposed estimators are asymptotically normal. In simulation studies, the proposed estimators show improved performance relative to usual augmented inverse probability weighted estimators.

Using Statistical Learning to Treat Missing Data: A Case of HIV/TB Co-Infection in Kenya

In this study, we investigate the effects of missing data when estimating HIV/TB co-infection. We revisit the concept of missing data and examine three available approaches for dealing with missingness. The main objective is to identify the best method for correcting missing data in TB/HIV Co-infection setting. We employ both empirical data analysis and extensive simulation study to examine the effects of missing data, the accuracy, sensitivity, specificity and train and test error for different approaches. The novelty of this work hinges on the use of modern statistical learning algorithm when treating missingness. In the empirical analysis, both HIV data and TB-HIV co-infection data imputations were performed, and the missing values were imputed using different approaches. In the simulation study, sets of 0% (Complete case), 10%, 30%, 50% and 80% of the data were drawn randomly and replaced with missing values. Results show complete cases only had a co-infection rate (95% Confidence Interval band) of 29% (25%, 33%), weighted method 27% (23%, 31%), likelihood-based approach 26% (24%, 28%) and multiple imputation approach 21% (20%, 22%). In conclusion, MI remains the best approach for dealing with missing data and failure to apply it, results to overestimation of HIV/TB co-infection rate by 8%.

Smoothed Empirical Likelihood Inference for Nonlinear Quantile Regression Models with Missing Response

In this paper, three smoothed empirical log-likelihood ratio functions for the parameters of nonlinear models with missing response are suggested. Under some regular conditions, the corresponding Wilks phenomena are obtained and the confidence regions for the parameter can be constructed easily.

Empirical Likelihood of Quantile Difference with Missing Response When High-dimensional Covariates Are Present

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.At the same time,we give a test to check whether two populations have the same distribution.A simulation study is carried out to investigate finite sample behavior of the proposed methods too.

Empirical Likelihood for Response Differences in Two Linear Regression Models with Missing Data

Oonsider two linear models Xi = U＇β ＋ ei, Yj = V1/2y ＋ ηj with response variables missing at random. In this paper, we assume that X, Y are missing at random （MAR） and use the inverse probability weighted imputation to produce ＇complete＇ data sets for X and Y. Based on these data sets, we construct an empirical likelihood （EL） statistic for the difference of X and Y （denoted as A）, and show that the EL statistic has the limiting distribution of X~, which is used to construct a confidence interval for A. Results of a simulation study on the finite sample performance of EL-based confidence intervals on A are reported.

Integrated Square Error of Hazard Rate Estimation for Survival Data with Missing Censoring Indicators

The problem of hazard rate estimation under right-censored assumption has been investigated extensively.Integrated square error(ISE)of estimation is one of the most widely accepted measurements of the global performance for nonparametric kernel estimation.But there are no results available for ISE of hazard rate estimation under right-censored model with censoring indicators missing at random(MAR)so far.This paper constructs an imputation estimator of the hazard rate function and establish asymptotic normality of the ISE for the kernel hazard rate estimator with censoring indicators MAR.At the same time,an asymptotic representation of the mean integrated square error(MISE)is also presented.The finite sample behavior of the estimator is investigated via one simple simulation.

Estimation for Partially Linear Models with Missing Responses:the Fixed Design Case

Suppose that we have a partially linear model Yi = xiβ ＋ g（ti） ＋εi with independent zero mean errors εi, where （xi,ti, i = 1, ... ,n} are non-random and observed completely and （Yi, i = 1,...,n} are missing at random（MAR）. Two types of estimators of β and g（t） for fixed t are investigated： estimators based on semiparametric regression and inverse probability weighted imputations. Asymptotic normality of the estimators is established, which is used to construct normal approximation based confidence intervals on β and g（t）. Results are reported of a simulation study on the finite sample performance of the estimators and confidence intervals proposed in this paper.

Variable screening with missing covariates: a discussion of ‘statistical inferencefor nonignorable missing data problems: a selective review’ by NianshengTang and Yuanyuan Ju

Feature screening with missing data is a critical problem but has not been well addressed in theliterature. In this discussion we propose a new screening index based on “information value” andapply it to feature screening with missing covariates.

Empirical Likelihood-based Inferences in Varying Coefficient Models with Missing Data

In this paper, we consider the empirical likelihood-based inferences for varying coefficient models Y = X^τα（U） ＋ ε when X are subject to missing at random. Based on the inverse probability-weighted idea, a class of empirical log-likelihood ratios, as well as two maximum empirical likelihood estimators, are developed for α（u）. The resulting statistics are shown to have standard chi-squared or normal distributions asymptotically.Simulation studies are also constructed to illustrate the finite sample properties of the proposed statistics.

EMPIRICAL LIKELIHOOD APPROACH FOR LONGITUDINAL DATA WITH MISSING VALUES AND TIME-DEPENDENT COVARIATES

Missing data and time-dependent covariates often arise simultaneously in longitudinal studies,and directly applying classical approaches may result in a loss of efficiency and biased estimates.To deal with this problem,we propose weighted corrected estimating equations under the missing at random mechanism,followed by developing a shrinkage empirical likelihood estimation approach for the parameters of interest when time-dependent covariates are present.Such procedure improves efficiency over generalized estimation equations approach with working independent assumption,via combining the independent estimating equations and the extracted additional information from the estimating equations that are excluded by the independence assumption.The contribution from the remaining estimating equations is weighted according to the likelihood of each equation being a consistent estimating equation and the information it carries.We show that the estimators are asymptotically normally distributed and the empirical likelihood ratio statistic and its profile counterpart follow central chi-square distributions asymptotically when evaluated at the true parameter.The practical performance of our approach is demonstrated through numerical simulations and data analysis.

Empilrical Likelihood for Non-parametric Regression Models with Missing Responses:Multiple Design Case

Empirical likelihood （EL） ratio statistic on θ=g（x） is constructed based on the inverse probability weighted imputation approach in a nonparametric regression model Y = g（x） ＋ε （x ∈ [0, 1]p） with fixed designs and missing responses, which asymptotically has X1^2 distribution. This result is used to obtain a EL based confidence interval on θ.