Robust empirical Bayes small area estimation with density power divergence

Summary A two-stage normal hierarchical model called the Fay–Herriot model and the empirical Bayes estimator are widely used to obtain indirect and model-based estimates of means in small areas. However, the performance of the empirical Bayes estimator can be poor when the assumed normal distribution is misspecified. This article presents a simple modification that makes use of density power divergence and proposes a new robust empirical Bayes small area estimator. The mean squared error and estimated mean squared error of the proposed estimator are derived based on the asymptotic properties of the robust estimator of the model parameters. We investigate the numerical performance of the proposed method through simulations and an application to survey data.

Preliminary Test Estimators and Confidence Intervals for the Parametric Functions of the Moore and Bilikam Family of Lifetime Distributions Based on Records

We consider two measures of reliability functions namely R(t)=P(X>t) and P=P(X>Y) for the Moore and Bilikam (1978) family of lifetime distributions which covers fourteen distributions as specific cases. For record data from this family of distributions, preliminary test estimators (PTEs) and preliminary test confidence interval (PTCI) based on uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), empirical Bayes estimator (EBE) are obtained for the parameter. The bias and mean square error (MSE) (exact and asymptotic) of the proposed estimators are derived to study their relative efficiency and through simulation studies we establish that PTEs perform better than ordinary UMVUE, MLE and EBE. We also obtain the coverage probability (CP) and the expected length of the PTCI of the parameter and establish that the confidence intervals based on MLE are more precise. An application of the ordinary preliminary test estimator is also considered. To the best of the knowledge of the authors, no PTEs have been derived for R(t) and P based on records and thus we define improved PTEs based on MLE and UMVUE of R(t) and P. A comparative study of different methods of estimation done through simulations establishes that PTEs perform better than ordinary UMVUE and MLE.

Robustness to Parametric Assumptions in Missing Data Models

We consider estimation of population averages when data are missing at random. If some cells contain few observations, there can be substantial gains from imposing parametric restrictions on the cell means, but there is also a danger of misspecification. We develop a simple empirical Bayes estimator, which combines parametric and unadjusted estimates of cell means in a data-driven way. We also consider ways to use knowledge of the form of the propensity score to increase robustness. We develop an empirical Bayes extension of a double robust estimator. In a small simulation study, the empirical Bayes estimators perform well. They are similar to fully nonparametric methods and robust to misspecification when cells are moderate to large in size, and when cells are small they maintain the benefits of parametric methods and can have lower sampling variance.