Mean squared error estimators of small area means using survey weights

Using survey weights, You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] proposed a pseudo‐empirical best linear unbiased prediction (pseudo‐EBLUP) estimator of a small area mean under a nested error linear regression model. This estimator borrows strength across areas through a linking model, and makes use of survey weights to ensure design consistency and preserve benchmarking property in the sense that the estimators add up to a reliable direct estimator of the mean of a large area covering the small areas. In this article, a second‐order approximation to the mean squared error (MSE) of the pseudo‐EBLUP estimator of a small area mean is derived. Using this approximation, an estimator of MSE that is nearly unbiased is derived; the MSE estimator of You & Rao [You and Rao, The Canadian Journal of Statistics 2002; 30, 431–439] ignored cross‐product terms in the MSE and hence it is biased. Empirical results on the performance of the proposed MSE estimator are also presented. The Canadian Journal of Statistics 38: 598–608; 2010 © 2010 Statistical Society of Canada     

Bayesian testing for independence of two categorical variables under two-stage cluster sampling with covariates

We consider Bayesian testing for independence of two categorical variables with covariates for a two-stage cluster sample. This is a difficult problem because we have a complex sample (i.e. cluster sample), not a simple random sample. Our approach is to convert the cluster sample with covariates into an equivalent simple random sample without covariates, which provides a surrogate of the original sample. Then, this surrogate sample is used to compute the Bayes factor to make an inference about independence. We apply our methodology to the data from the Trend in International Mathematics and Science Study [30] for fourth grade US students to assess the association between the mathematics and science scores represented as categorical variables. We show that if there is strong association between two categorical variables, there is no significant difference between the tests with and without the covariates. We also performed a simulation study to further understand the effect of covariates in various situations. We found that for borderline cases (moderate association between the two categorical variables), there are noticeable differences in the test with and without covariates.     

Mean-Squared errors of small area estimators under a multivariate linear model for repeated measures data

In this paper, we discuss the derivation of the first and second moments for the proposed small area estimators under a multivariate linear model for repeated measures data. The aim is to use these moments to estimate the mean-squared errors (MSE) for the predicted small area means as a measure of precision. At the first stage, we derive the MSE when the covariance matrices are known. At the second stage, a method based on parametric bootstrap is proposed for bias correction and for prediction error that reflects the uncertainty when the unknown covariance is replaced by its suitable estimator.     

A hierarchical Bayesian model for binary data incorporating selection bias

We consider a Bayesian nonignorable model to accommodate a nonignorable selection mechanism for predicting small area proportions. Our main objective is to extend a model on selection bias in a previously published paper, coauthored by four authors, to accommodate small areas. These authors assume that the survey weights (or their reciprocals that we also call selection probabilities) are available, but there is no simple relation between the binary responses and the selection probabilities. To capture the nonignorable selection bias within each area, they assume that the binary responses and the selection probabilities are correlated. To accommodate the small areas, we extend their model to a hierarchical Bayesian nonignorable model and we use Markov chain Monte Carlo methods to fit it. We illustrate our methodology using a numerical example obtained from data on activity limitation in the U.S. National Health Interview Survey. We also perform a simulation study to assess the effect of the correlation between the binary responses and the selection probabilities.     

Bayesian inference and model comparison for asymmetric smooth transition heteroskedastic models

Inference, quantile forecasting and model comparison for an asymmetric double smooth transition heteroskedastic model is investigated. A Bayesian framework in employed and an adaptive Markov chain Monte Carlo scheme is designed. A mixture prior is proposed that alleviates the usual identifiability problem as the speed of transition parameter tends to zero, and an informative prior for this parameter is suggested, that allows for reliable inference and a proper posterior, despite the non-integrability of the likelihood function. A formal Bayesian posterior model comparison procedure is employed to compare the proposed model with its two limiting cases: the double threshold GARCH and symmetric ARX GARCH models. The proposed methods are illustrated using both simulated and international stock market return series. Some illustrations of the advantages of an adaptive sampling scheme for these models are also provided. Finally, Bayesian forecasting methods are employed in a Value-at-Risk study of the international return series. The results generally favour the proposed smooth transition model and highlight explosive and smooth nonlinear behaviour in financial markets.     

A modified nested-error regression model for small area estimation

A nested-error regression model having both fixed and random effects is introduced to estimate linear parameters of small areas. The model is applicable to data having a proportion of domains where the variable of interest cannot be described by a standard linear mixed model. Algorithms and formulas to fit the model, to calculate EBLUP and to estimate mean-squared errors are given. A Monte Carlo simulation experiment is presented to illustrate the gain of precision obtained by using the proposed model and to obtain some practical conclusions. A motivating application to Spanish Labour Force Survey data is also given.     

Small area estimation of expenditure means and ratios under a unit-level bivariate linear mixed model

Under a unit-level bivariate linear mixed model, this paper introduces small area predictors of expenditure means and ratios, and derives approximations and estimators of the corresponding mean squared errors. For the considered model, the REML estimation method is implemented. Several simulation experiments, designed to analyze the behavior of the introduced fitting algorithm, predictors and mean squared error estimators, are carried out. An application to real data from the Spanish household budget survey illustrates the behavior of the proposed statistical methodology. The target is the estimation of means of food and non-food household annual expenditures and of ratios of food household expenditures by Spanish provinces.     

Properties of the beta regression model for small area estimation of proportions and application to estimation of poverty rates

Abstract

Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. Nonlinear sampling models are becoming increasingly popular for small area estimation problems when the normal model is inadequate. This paper studies the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0, 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0, 1). Properties of a hierarchical Bayesian small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model. Propriety of the posterior distribution using certain noninformative priors is shown, and behavior of the posterior mean as a function of the sampling variance and the model variance is described. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given, and a numerical example studying small sample properties of the model is presented.     

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.     

Akaike Information Criterion for Selecting Variables in the Nested Error Regression Model

The Akaike Information Criterion (AIC) is developed for selecting the variables of the nested error regression model where an unobservable random effect is present. Using the idea of decomposing the likelihood into two parts of “within” and “between” analysis of variance, we derive the AIC when the number of groups is large and the ratio of the variances of the random effects and the random errors is an unknown parameter. The proposed AIC is compared, using simulation, with Mallows' C _p , Akaike's AIC, and Sugiura's exact AIC. Based on the rates of selecting the true model, it is shown that the proposed AIC performs better.