Empirical Likelihood Confidence Intervals for Response Mean with Data Missing at Random

Abstract

A kernel regression imputation method for missing response data is developed. A class of bias-corrected empirical log-likelihood ratios for the response mean is defined. It is shown that any member of our class of ratios is asymptotically chi-squared, and the corresponding empirical likelihood confidence interval for the response mean is constructed. Our ratios share some of the desired features of the existing methods: they are self-scale invariant and no plug-in estimators for the adjustment factor and asymptotic variance are needed; when estimating the non-parametric function in the model, undersmoothing to ensure root-"n" consistency of the estimator for the parameter is avoided. Since the range of bandwidths contains the optimal bandwidth for estimating the regression function, the existing data-driven algorithm is valid for selecting an optimal bandwidth. We also study the normal approximation-based method. A simulation study is undertaken to compare the empirical likelihood with the normal approximation method in terms of coverage accuracies and average lengths of confidence intervals. Copyright (c) 2009 Board of the Foundation of the Scandinavian Journal of Statistics.

Empirical likelihood for nonlinear models with missing responses

“…Xue [19] constructed a weight-corrected empirical log-likelihood ratio for θ 0 , which is also asymptotically chi-squared. Now, let us consider the following functions, constructed using the reconstituted response:…”

Empirical likelihood and Wilks phenomenon for data with nonignorable missing values

“…When some Y i 's are missing, even if we modify the function ( X , Y , θ ) in to address the missing values, Wilks's theorem may break down unless a particular modified is used in . For example, when missingness is ignorable, that is, the propensity P ( δ = 1 | X , Y ) = π ( X ) is a function of X only, where δ is the indicator of whether Y is observed, Xue () showed that Wilks's theorem holds if in is replaced by where and are nonparametric regression estimators of π ( X ) and ( X , θ ) = E { ( X , Y , θ ) | X , δ = 1}, respectively. It was also shown that if we omit the second term on the right‐hand side of , then Wilks's theorem does not hold, although using such a in leads to an asymptotically valid estimator of θ 0 .…”

Empirical likelihood for single index model with missing covariates at random

“…In the following simulations, the kernel function K(·) is taken to be K(t) = 3 4 (1 − t 2 ), if |t| ≤ 1, 0 otherwise, L(·) and Q(·) are taken to be K 1 (y)K 2 (v) with K 1 (y) = 15 16 (1 − y 2 ) 2 if |y| ≤ 1, 0 otherwise and K 2 (v) = − 15 8 v 2 + 9 8 if |v| ≤ 1, 0 otherwise. The bandwidths are taken to be b = h 2 = 3 5 n −1/6 and h = h 1 = 2 7 n −1/5 , which satisfy Condition (C5).…”

“…They also considered the case with auxiliary information and defined an empirical likelihood-based estimator. Xue [16] reexamined the problem of empirical likelihood for the mean of Y in the situation with missing data. They proposed a class of bias-corrected empirical log-likelihood ratios and showed that any member of their class of ratios is asymptotically chi-squared and avoided the plug-in estimators for the adjustment factor and asymptotic variance.…”