The particularly wide range of applications of small area prediction, e.g. in policy making decisions, has meant that this topic has received substantial attention in recent years. The problems of estimating mean-squared predictive error, of correcting that estimator for bias and of constructing prediction intervals have been addressed by various workers, although existing methodology is still restricted to a narrow range of models. To overcome this difficulty we develop new, bootstrap-based methods, which are applicable in very general settings, for constructing bias-corrected estimators of mean-squared error and for computing prediction regions. Unlike existing techniques, which are based largely on Taylor expansions, our bias-corrected mean-squared error estimators do not require analytical calculation. They also have the property that they are non-negative. Our prediction intervals have a high degree of coverage accuracy, "O"("n"-super- - 3), where "n" is the number of areas, if double-bootstrap methods are employed. The techniques do not depend on the form of the small area estimator and are applicable to general two-level, small area models, where the variables at either level can be discrete or continuous and, in particular, can be non-normal. Most importantly, the new methods are simple and easy to apply. Copyright 2006 Royal Statistical Society.

BackgroundMicroarray gene expression time-course experiments provide the opportunity to observe the evolution of transcriptional programs that cells use to respond to internal and external stimuli. Most commonly used methods for identifying differentially expressed genes treat each time point as independent and ignore important correlations, including those within samples and between sampling times. Therefore they do not make full use of the information intrinsic to the data, leading to a loss of power.ResultsWe present a flexible random-effects model that takes such correlations into account, improving our ability to detect genes that have sustained differential expression over more than one time point. By modeling the joint distribution of the samples that have been profiled across all time points, we gain sensitivity compared to a marginal analysis that examines each time point in isolation. We assign each gene a probability of differential expression using an empirical Bayes approach that reduces the effective number of parameters to be estimated.ConclusionsBased on results from theory, simulated data, and application to the genomic data presented here, we show that BETR has increased power to detect subtle differential expression in time-series data. The open-source R package betr is available through Bioconductor. BETR has also been incorporated in the freely-available, open-source MeV software tool available from http://www.tm4.org/mev.html.

Summary. The problem of accurately estimating the mean-squared error of small area estimators within a Fay-Herriot normal error model is studied theoretically in the common setting where the model is fitted to a logarithmically transformed response variable. For bias-corrected empirical best linear unbiased predictor small area point estimators, mean-squared error formulae and estimators are provided, with biases of smaller order than the reciprocal of the number of small areas. The performance of these mean-squared error estimators is illustrated by a simulation study and a real data example relating to the county level estimation of child poverty rates in the US Census Bureau's on-going 'Small area income and poverty estimation' project.

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Nested-error regression models are widely used for analyzing clustered data. For example, they are often applied to two-stage sample surveys, and in biology and econometrics. Prediction is usually the main goal of such analyses, and mean-squared prediction error is the main way in which prediction performance is measured. In this paper we suggest a new approach to estimating mean-squared prediction error. We introduce a matched-moment, double-bootstrap algorithm, enabling the notorious underestimation of the naive mean-squared error estimator to be substantially reduced. Our approach does not require specific assumptions about the distributions of errors. Additionally, it is simple and easy to apply. This is achieved through using Monte Carlo simulation to implicitly develop formulae which, in a more conventional approach, would be derived laboriously by mathematical arguments. . This reprint differs from the original in pagination and typographic detail. 1 2 P. HALL AND T. MAITI method has several attractive properties. First, it does not require specific distributional assumptions about error distributions. Second, it produces positive, bias-corrected estimators of mean-squared prediction errors. (See [2] and [5] for discussion of possible negativity.) Third, it is easy to apply. Although our emphasis is on small-area prediction, our methodology is equally useful for other applications, such as estimating subject-or cluster-specific random effects. Standard mixed-effects prediction involves two steps. First, a best linear unbiased predictor, or BLUP, is derived under the assumption that model parameters are known. Then, the model parameters are replaced by estimators, producing an empirical version of BLUP. This approach is popular because it is straightforward and, at this level, does not require distributional assumptions.However, estimation of mean-squared prediction error is significantly more challenging. The variability of parameter estimators can substantially influence mean-squared error, to a much greater extent than a conventional asymptotic analysis suggests. Moreover, the nature and extent of this influence is intimately connected to the values of the design variables and to properties of the two error distributions.In this paper we point out that, in terms of the biases of estimators of mean-squared prediction error, the two error distributions influence results predominantly through their second and fourth moments. This observation leads to a surprisingly simple, moment-matching, double-bootstrap algorithm for estimating, and correcting for, bias. We show that this approach substantially reduces the large degree of underestimation by the naive approach.Kackar and Harville [20] and Harville and Jeske [18] studied various approximations to the mean-squared prediction error of the empirical BLUP, assuming normality in both stages. Prasad and Rao [27] pointed out that if unknown model parameters are replaced by their estimators, then significant underestimation of true mean-squared prediction error c...

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Small area estimation has long been a popular and important research topic due to its growing demand in public and private sectors. We consider here the basic area level model, popularly known as the Fay-Herriot model. Although much of current research is predominantly focused on second order unbiased estimation of mean squared prediction errors, we concentrate on developing confidence intervals (CIs) for the small area means that are second order correct. The corrected CI can be readily implemented, because it only requires quantities that are already estimated as part of the mean squared error estimation. We extend the approach to a CI for the difference of two small area means. The findings are illustrated with a simulation study.

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