On empirical likelihood for linear models with missing responses

Suppose that we have a linear regression model Y=X^′β+_ν0(X)ε$Y=X^{\prime }\beta +{\nu }_0(X)\epsilon$ with random error ε$\epsilon$, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random (MAR). In this paper, based on the ‘complete’ data set for Y after inverse probability weighted imputation, we construct empirical likelihood statistics on EY   and β$\beta$ which have the χ²${\chi }^2$-type limiting distributions under some new conditions compared with Xue (2009). Our results broaden the applicable scope of the approach combined with Xue (2009).     

Empirical likelihood for generalized linear models with missing responses

The paper uses the empirical likelihood method to study the construction of confidence intervals and regions for regression coefficients and response mean in generalized linear models with missing response. By using the inverse selection probability weighted imputation technique, the proposed empirical likelihood ratios are asymptotically chi-squared. Our approach is to directly calibrate the empirical likelihood ratio, which is called as a bias-correction method. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Empirical likelihood for the response mean of generalized linear models with missing at random responses

In this article, we present a new empirical likelihood ratio for constructing the confidence interval of the response mean of generalized linear models with missing at random responses. Compared with the existing methods, the proposal can avoid the so-called “curse of dimensionality” problem when the dimension of covariates is high, and is still chi-squared distributed asymptotically, nevertheless. Simulation studies are also provided to illustrate the performance of the developed method.     

Empirical likelihood inference for partial functional linear model with missing responses

In this paper, we consider the empirical likelihood inferences of the partial functional linear model with missing responses. Two empirical log-likelihood ratios of the parameters of interest are constructed, and the corresponding maximum empirical likelihood estimators of parameters are derived. Under some regularity conditions, we show that the proposed two empirical log-likelihood ratios are asymptotic standard Chi-squared. Thus, the asymptotic results can be used to construct the confidence intervals/regions for the parameters of interest. We also establish the asymptotic distribution theory of corresponding maximum empirical likelihood estimators. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals. An example of real data is also used to illustrate our proposed methods.     

Empirical likelihood for nonlinear regression models with nonignorable missing responses

This article develops three empirical likelihood (EL) approaches to estimate parameters in nonlinear regression models in the presence of nonignorable missing responses. These are based on the inverse probability weighted (IPW) method, the augmented IPW (AIPW) method and the imputation technique. A logistic regression model is adopted to specify the propensity score. Maximum likelihood estimation is used to estimate parameters in the propensity score by combining the idea of importance sampling and imputing estimating equations. Under some regularity conditions, we obtain the asymptotic properties of the maximum EL estimators of these unknown parameters. Simulation studies are conducted to investigate the finite sample performance of our proposed estimation procedures. Empirical results provide evidence that the AIPW procedure exhibits better performance than the other two procedures. Data from a survey conducted in 2002 are used to illustrate the proposed estimation procedure. The Canadian Journal of Statistics 48: 386–416; 2020 © 2020 Statistical Society of Canada     

Empirical likelihood for nonlinear models with missing responses

In this paper, a nonlinear model with response variables missing at random is studied. In order to improve the coverage accuracy for model parameters, the empirical likelihood (EL) ratio method is considered. On the complete data, the EL statistic for the parameters and its approximation have a χ² asymptotic distribution. When the responses are reconstituted using a semi-parametric method, the empirical log-likelihood on the response variables associated with the imputed data is also asymptotically χ². The Wilks theorem for EL on the parameters, based on reconstituted data, is also satisfied. These results can be used to construct the confidence region for the model parameters and the response variables. It is shown via Monte Carlo simulations that the EL methods outperform the normal approximation-based method in terms of coverage probability for the unknown parameter, including on the reconstituted data. The advantages of the proposed method are exemplified on real data.     

Confidence Intervals for Nonparametric Regression Functions with Missing Data

Suppose that we have a nonparametric regression model Y = m(X) + ε with X ∈ R^p, where X is a random design variable and is observed completely, and Y is the response variable and some Y-values are missing at random. Based on the “complete” data sets for Y after nonaprametric regression imputation and inverse probability weighted imputation, two estimators of the regression function m(x₀) for fixed x₀ ∈ R^p are proposed. Asymptotic normality of two estimators is established, which is used to construct normal approximation-based confidence intervals for m(x₀). We also construct an empirical likelihood (EL) statistic for m(x₀) with limiting distribution of χ²₁, which is used to construct an EL confidence interval for m(x₀).     

A comparison of multiple imputation and doubly robust estimation for analyses with missing data

Summary.  Multiple imputation is now a well-established technique for analysing data sets where some units have incomplete observations. Provided that the imputation model is correct, the resulting estimates are consistent. An alternative, weighting by the inverse probability of observing complete data on a unit, is conceptually simple and involves fewer modelling assumptions, but it is known to be both inefficient (relative to a fully parametric approach) and sensitive to the choice of weighting model. Over the last decade, there has been a considerable body of theoretical work to improve the performance of inverse probability weighting, leading to the development of 'doubly robust' or 'doubly protected' estimators. We present an intuitive review of these developments and contrast these estimators with multiple imputation from both a theoretical and a practical viewpoint.     

Empirical likelihood for single index model with missing covariates at random

In this paper, we investigate the empirical-likelihood-based inference for the construction of confidence intervals and regions of the parameters of interest in single index models with missing covariates at random. An augmented inverse probability weighted-type empirical likelihood ratio for the parameters of interest is defined such that this ratio is asymptotically standard chi-squared. Our approach is to directly calibrate the empirical log-likelihood ratio, and does not need multiplication by an adjustment factor for the original ratio. Our bias-corrected empirical likelihood is self-scale invariant and no plug-in estimator for the limiting variance is needed. Some simulation studies are carried out to assess the performance of our proposed method.     

Model-based confidence bands for survival functions

This paper focuses on a novel method of developing one-sample confidence bands for survival functions from right censored data. The approach is model-based, relying on a parametric model for the conditional expectation of the censoring indicator given the observed minimum, and derives its strength from easy access to a good-fitting model among a plethora of choices available for binary response data. The substantive methodological contribution is in exploiting a semiparametric estimator of the survival function to produce improved simultaneous confidence bands. To obtain critical values for computing the confidence bands, a two-stage bootstrap approach that combines the classical bootstrap with the more recent model-based regeneration of censoring indicators is proposed and a justification of its asymptotic validity is also provided. Several different confidence bands are studied using the proposed approach. Numerical studies, including robustness of the proposed bands to misspecification, are carried out to check efficacy. The method is illustrated using two lung cancer data sets.     

Empirical likelihood and Wilks phenomenon for data with nonignorable missing values

Wilks's theorem is useful for constructing confidence regions. When applying the popular empirical likelihood to data with nonignorable nonresponses, Wilks's phenomenon does not hold. This paper unveils that this is caused by the extra estimation of the nuisance parameter in the nonignorable nonresponse propensity. Motivated by this result, we propose an adjusted empirical likelihood for which Wilks's theorem holds. Asymptotic results are presented and supplemented by simulation results for finite sample performance of the point estimators and confidence regions. An analysis of a data set is included for illustration.     

Empirical likelihood inference for probability density functions under association

In this paper, we consider to apply the empirical likelihood method to a probability density function under an associated sample. It is shown that the empirical likelihood ratio statistic is asymptotically χ²-type distributed under some mild conditions. The result is used to construct empirical likelihood-based confidence intervals on the probability density function.     

Empirical likelihood based confidence regions for first order parameters of heavy-tailed distributions

Let X₁,…,X_n be some i.i.d. observations from a heavy-tailed distribution F, i.e. the common distribution of the excesses over a high threshold u_n can be approximated by a generalized Pareto distribution G_γ,σn with γ>0. This paper deals with the problem of finding confidence regions for the couple (γ,σ_n): combining the empirical likelihood methodology with estimation equations (close but not identical to the likelihood equations) introduced by Zhang (2007), asymptotically valid confidence regions for (γ,σ_n) are obtained and proved to perform better than Wald-type confidence regions (especially those derived from the asymptotic normality of the maximum likelihood estimators). By profiling out the scale parameter, confidence intervals for the tail index are also derived.     

The weighted least square based estimators with censoring indicators missing at random

In this paper, we study linear regression analysis when some of the censoring indicators are missing at random. We define regression calibration estimate, imputation estimate and inverse probability weighted estimate for the regression coefficient vector based on the weighted least squared approach due to Stute (1993), and prove all the estimators are asymptotically normal. A simulation study was conducted to evaluate the finite properties of the proposed estimators, and a real data example is provided to illustrate our methods.     

Empirical likelihood confidence intervals under imputation for missing survey data from stratified simple random sampling

Missing observations due to non‐response are commonly encountered in data collected from sample surveys. The focus of this article is on item non‐response which is often handled by filling in (or imputing) missing values using the observed responses (donors). Random imputation (single or fractional) is used within homogeneous imputation classes that are formed on the basis of categorical auxiliary variables observed on all the sampled units. A uniform response rate within classes is assumed, but that rate is allowed to vary across classes. We construct confidence intervals (CIs) for a population parameter that is defined as the solution to a smooth estimating equation with data collected using stratified simple random sampling. The imputation classes are assumed to be formed across strata. Fractional imputation with a fixed number of random draws is used to obtain an imputed estimating function. An empirical likelihood inference method under the fractional imputation is proposed and its asymptotic properties are derived. Two asymptotically correct bootstrap methods are developed for constructing the desired CIs. In a simulation study, the proposed bootstrap methods are shown to outperform traditional bootstrap methods and some non‐bootstrap competitors under various simulation settings. The Canadian Journal of Statistics 47: 281–301; 2019 © 2019 Statistical Society of Canada     

Empirical Likelihood in a Semi-Parametric Model for Missing Response Data

Let Y be a response and, given covariate X,Y has a conditional density f(y | x, θ), where θ is a unknown p-dimensional vector of parameters and the marginal distribution of X is unknown. When responses are missing at random, with auxiliary information and imputation, we define an adjusted empirical log-likelihood ratio for the mean of Y and obtain its asymptotic distribution. A simulation study is conducted to compare the adjusted empirical log-likelihood and the normal approximation method in terms of coverage accuracies.     

A Weighting Approach for GEE Analysis with Missing Data

We propose a new weighting (WT) method to handle missing categorical outcomes in longitudinal data analysis using generalized estimating equations (GEE). The proposed WT provides a valid GEE estimator when the data are missing at random (MAR), and has more stable weights and shows advantage in efficiency compared to the inverse probability weighing method in the presence of small observation probabilities. The WT estimator is similar to the stabilized weighting (SWT) estimator under mild conditions, but it is more stable and efficient than SWT when the associations of the outcome with the observation probabilities and the covariate are strong.     

Empirical likelihood for linear regression models under imputation for missing responses

The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log‐likelihood ratio is asymptotically a weighted sum of chi‐square variables with unknown weights. They obtain an adjusted empirical log‐likelihood ratio which is asymptotically standard chi‐square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log‐likelihood ratio and use its distribution to approximate that of the empirical log‐likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method.     

Penalized empirical likelihood for quantile regression with missing covariates and auxiliary information

Based on the inverse probability weight method, we, in this article, construct the empirical likelihood (EL) and penalized empirical likelihood (PEL) ratios of the parameter in the linear quantile regression model when the covariates are missing at random, in the presence and absence of auxiliary information, respectively. It is proved that the EL ratio admits a limiting Chi-square distribution. At the same time, the asymptotic normality of the maximum EL and PEL estimators of the parameter is established. Also, the variable selection of the model in the presence and absence of auxiliary information, respectively, is discussed. Simulation study and a real data analysis are done to evaluate the performance of the proposed methods.