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The nested-error regression model is one of the best-known models in small area estimation. A small area mean is often expressed as a linear combination of fixed effects and realized values of random effects. In such analyses, prediction is made by borrowing strength from other related areas or sources and mean-squared prediction error (MSPE) is often used as a measure of uncertainty. In this article, we propose a bias-corrected analytical estimation of MSPE as well as a moment-match jackknife method to estimate the MSPE without specific assumptions about the distributions of the data. Theoretical and empirical studies are carried out to investigate performance of the proposed methods with comparison to existing procedures. Le modèle de régression à erreur imbriquée est l’un des mieux connus pour l’estimation sur des petits domaines. La moyenne d’un petit domaine est souvent exprimée comme une combinaison linéaire d’effets fixes et de valeurs réalisées d’effets aléatoires. Pour de telles analyses, les prévisions sont effectuées en empruntant de l’information d’autres domaines associés ou d’autres sources, et l’erreur quadratique moyenne de prévision (EQMP) sert souvent à mesurer l’incertitude. Les auteurs proposent une estimation analytique de l’EQMP corrigée pour le biais ainsi qu’une méthode jackknife d’appariement des moments afin d’estimer l’EQMP sans formuler d’hypothèses spécifiques sur la distribution des données. Ils présentent des études théoriques et empiriques comparant la performance des méthodes proposées aux procédures existantes.

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.

A low-degree polynomial model for a response curve is used commonly in practice. It generally incorporates a linear or quadratic function of the covariate. In this paper we suggest methods for testing the goodness of fit of a general polynomial model when there are errors in the covariates. There, the true covariates are not directly observed, and conventional bootstrap methods for testing are not applicable. We develop a new approach, in which deconvolution methods are used to estimate the distribution of the covariates under the null hypothesis, and a \"wild\" or moment-matching bootstrap argument is employed estimate the distribution of the experimental errors (distinct from the distribution of the errors in covariates). Most of our attention is directed at the case where the distribution of the errors in covariates is known, although we also discuss methods for estimation and testing when the covariate error distribution is estimated. No assumptions are made about the distribution of experimental error, and, in particular, we depart substantially from conventional parametric models for errors-in-variables problems.