Empirical best prediction under a nested error model with log transformation

In regression models involving economic variables such as income, log transformation is typically taken to achieve approximate normality and stabilize the variance. However, often the interest is predicting individual values or means of the variable in the original scale. Under a nested error model for the log transformation of the target variable, we show that the usual approach of back transforming the predicted values may introduce a substantial bias. We obtain the optimal (or “best”) predictors of individual values of the original variable and of small area means under that model. Empirical best predictors are defined by estimating the unknown model parameters in the best predictors. When estimation is desired for subpopulations with small sample sizes (small areas), nested error models are widely used to “borrow strength” from the other areas and obtain estimators with greater efficiency than direct estimators based on the scarce area-specific data. We show that naive predictors of small area means obtained by back-transformation under the mentioned model may even underperform direct estimators. Moreover, assessing the uncertainty of the considered predictor is not straightforward. Exact mean squared errors of the best predictors and second-order approximations to the mean squared errors of the empirical best predictors are derived. Estimators of the mean squared errors that are second-order correct are also obtained. Simulation studies and an example with Mexican data on living conditions illustrate the procedures.Supported by the Spanish Grants SEJ-2007-64500 and MTM2012-37077-C02-01. Supported by the Spanish Grants MTM-2012-33740 and ECO-2011-25706

Nonparametric estimation of mean-squared prediction error in nested-error regression models

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.Comment: Published at http://dx.doi.org/10.1214/009053606000000579 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric estimation of mean-squared prediction error in nested-error regression models

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.Supported in part by NSF Grant SES-03-18184

Small area estimation and prediction problems: spatial models, Bayesian multiple comparisons and robust MSE estimation

We study and partially solve three distinct problems in small area estimation. The problems are loosely connected by a common theme of prediction and (empirical) Bayesian models. In the first part of the thesis we consider prediction in a survey small area context with spatially correlated errors. We introduce a novel asymptotic framework in which the spatially correlated small areas form clusters, the number of such clusters and the number of small areas in each cluster growing with sample size. Under such an asymptotic framework we show consistency and asymptotic normality of the parameter estimators. For empirical predictors based on model estimates, we show through simulation and a real data example, improved prediction over estimates ignoring spatial error-correlations. The second part of the thesis involves using a hierarchical Bayes approach to solve the problem of multiple comparison in small area estimation. In the context of multiple comparison, a new class of moment matching priors is introduced. This class includes the well-known superharmonic prior due to Stein. Through data analysis and simulation we illustrate the use of our class of priors. In the third part of the thesis, for a special case of the nested error regression model, we derive a non-parametric second order unbiased estimator of the mean squared error of the empirical best linear unbiased predictor. For the balanced case, the Prasad-Rao estimator is shown to be second order unbiased when the small area effects are non-normal. Through simulation we show that the Prasad-Rao estimator is robust for departures from normality

Mean squared error of empirical predictor

The term ``empirical predictor'' refers to a two-stage predictor of a linear combination of fixed and random effects. In the first stage, a predictor is obtained but it involves unknown parameters; thus, in the second stage, the unknown parameters are replaced by their estimators. In this paper, we consider mean squared errors (MSE) of empirical predictors under a general setup, where ML or REML estimators are used for the second stage. We obtain second-order approximation to the MSE as well as an estimator of the MSE correct to the same order. The general results are applied to mixed linear models to obtain a second-order approximation to the MSE of the empirical best linear unbiased predictor (EBLUP) of a linear mixed effect and an estimator of the MSE of EBLUP whose bias is correct to second order. The general mixed linear model includes the mixed ANOVA model and the longitudinal model as special cases

Small Area Shrinkage Estimation

The need for small area estimates is increasingly felt in both the public and private sectors in order to formulate their strategic plans. It is now widely recognized that direct small area survey estimates are highly unreliable owing to large standard errors and coefficients of variation. The reason behind this is that a survey is usually designed to achieve a specified level of accuracy at a higher level of geography than that of small areas. Lack of additional resources makes it almost imperative to use the same data to produce small area estimates. For example, if a survey is designed to estimate per capita income for a state, the same survey data need to be used to produce similar estimates for counties, subcounties and census divisions within that state. Thus, by necessity, small area estimation needs explicit, or at least implicit, use of models to link these areas. Improved small area estimates are found by "borrowing strength" from similar neighboring areas.Comment: Published in at http://dx.doi.org/10.1214/11-STS374 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

New important developments in small area estimation

The purpose of this paper is to review and discuss some of the new important developments in small area estimation (SAE) methods. Rao (2003) wrote a very comprehensive book, which covers all the main developments in this topic until that time and so the focus of this review is on new developments in the last 7 years. However, to make the review more self contained, I also repeat shortly some of the older developments. The review covers both design based and model-dependent methods with emphasis on the prediction of the area target quantities and the assessment of the prediction error. The style of the paper is similar to the style of my previous review on SAE published in 2002, explaining the new problems investigated and describing the proposed solutions, but without dwelling on theoretical details, which can be found in the original articles. I am hoping that this paper will be useful both to researchers who like to learn more on the research carried out in SAE and to practitioners who might be interested in the application of the new methods

Uncertainty under a multivariate nested-error regression model with logarithmic transformation

Assuming a multivariate linear regression model with one random factor, we consider the parameters defined as exponentials of mixed effects, i.e., linear combinations of fixed and random effects. Such parameters are of particular interest in prediction problems where the dependent variable is the logarithm of the variable that is the object of inference. We derive bias-corrected empirical predictors of such parameters. A second order approximation for the mean crossed product error of the predictors of two of these parameters is obtained, and an estimator is derived from it. The mean squared error is obtained as a particular case