Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

Latent mixture modeling for clustered data

Statistics and Computing - This article proposes a mixture modeling approach to estimating cluster-wise conditional distributions in clustered (grouped) data. We adapt the mixture-of-experts model...     

Bayesian Estimators for Small Area Models Shrinking Both Means and Variances

For small area estimation of area‐level data, the Fay–Herriot model is extensively used as a model‐based method. In the Fay–Herriot model, it is conventionally assumed that the sampling variances are known, whereas estimators of sampling variances are used in practice. Thus, the settings of knowing sampling variances are unrealistic, and several methods are proposed to overcome this problem. In this paper, we assume the situation where the direct estimators of the sampling variances are available as well as the sample means. Using this information, we propose a Bayesian yet objective method producing shrinkage estimation of both means and variances in the Fay–Herriot model. We consider the hierarchical structure for the sampling variances, and we set uniform prior on model parameters to keep objectivity of the proposed model. For validity of the posterior inference, we show under mild conditions that the posterior distribution is proper and has finite variances. We investigate the numerical performance through simulation and empirical studies.     

Hierarchical Bayes small‐area estimation with an unknown link function

Area‐level unmatched sampling and linking models have been widely used as a model‐based method for producing reliable estimates of small‐area means. However, one practical difficulty is the specification of a link function. In this paper, we relax the assumption of a known link function by not specifying its form and estimating it from the data. A penalized‐spline method is adopted for estimating the link function, and a hierarchical Bayes method of estimating area means is developed using a Markov chain Monte Carlo method for posterior computations. Results of simulation studies comparing the proposed method with a conventional approach based on a known link function are presented. In addition, the proposed method is applied to data from the Survey of Family Income and Expenditure in Japan and poverty rates in Spanish provinces.     

Adaptively transformed mixed‐model prediction of general finite‐population parameters

For estimating area‐specific parameters (quantities) in a finite population, a mixed‐model prediction approach is attractive. However, this approach strongly depends on the normality assumption of the response values, although we often encounter a non‐normal case in practice. In such a case, transforming observations to make them suitable for normality assumption is a useful tool, but the problem of selecting a suitable transformation still remains open. To overcome the difficulty, we here propose a new empirical best predicting method by using a parametric family of transformations to estimate a suitable transformation based on the data. We suggest a simple estimating method for transformation parameters based on the profile likelihood function, which achieves consistency under some conditions on transformation functions. For measuring the variability of point prediction, we construct an empirical Bayes confidence interval of the population parameter of interest. Through simulation studies, we investigate the numerical performance of the proposed methods. Finally, we apply the proposed method to synthetic income data in Spanish provinces in which the resulting estimates indicate that the commonly used log transformation would not be appropriate.     

Conditional Akaike information under covariate shift with application to small area estimation

In this study, we consider the problem of selecting explanatory variables of fixed effects in linear mixed models under covariate shift, which is when the values of covariates in the model for prediction differ from those in the model for observed data. We construct a variable selection criterion based on the conditional Akaike information introduced by Vaida & Blanchard (2005). We focus especially on covariate shift in small area estimation and demonstrate the usefulness of the proposed criterion. In addition, numerical performance is investigated through simulations, one of which is a design‐based simulation using a real dataset of land prices. The Canadian Journal of Statistics 46: 316–335; 2018 © 2018 Statistical Society of Canada