On Mean Squared Prediction Error Estimation in Small Area Estimation Problems

For a general linear mixed normal model, a new linearized weighted jackknife method is proposed to estimate the mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) of a general mixed effect. Different MSPE estimators are compared using a Monte Carlo simulation study.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

Assessing different uncertainty measures of EBLUP: a resampling-based approach

The empirical best linear unbiased prediction approach is a popular method for the estimation of small area parameters. However, the estimation of reliable mean squared prediction error (MSPE) of the estimated best linear unbiased predictors (EBLUP) is a complicated process. In this paper we study the use of resampling methods for MSPE estimation of the EBLUP. A cross-sectional and time-series stationary small area model is used to provide estimates in small areas. Under this model, a parametric bootstrap procedure and a weighted jackknife method are introduced. A Monte Carlo simulation study is conducted in order to compare the performance of different resampling-based measures of uncertainty of the EBLUP with the analytical approximation. Our empirical results show that the proposed resampling-based approaches performed better than the analytical approximation in several situations, although in some cases they tend to underestimate the true MSPE of the EBLUP in a higher number of small areas.     

Optimal prediction in finite populations under matrix loss

Optimal prediction problems in finite population are investigated. Under matrix loss, we provide necessary and sufficient conditions for the linear predictor of a general linearly predictable variable to be the best linear unbiased predictor (BLUP). The essentially unique BLUP of a linearly predictable variable is obtained in the general superpopulation model. Surprisingly, the both BLUPs under matrix and quadratic loss functions are equivalent to each other. Next, we prove that the BLUP is admissible in the class of linear predictors. Conditions for optimality of the simple projection predictor (SPP) are given. Furthermore, the robust SPP and the robust BLUP are characterized on the misspecification of the covariance matrix.     

Linear predictors of future order statistics

In this paper we establish an optimal asymptotic linear predictor which does not involve the finite-sample variance-covariance structure. Extensions to the problem of finding the best linear unbiased and simple linear unbiased predictors for k samples are given. Moreover, we obtain alternative linear predictors by modifying the covariance matrix by either an identity matrix or a diagonal matrix. For normal, logistic and Rayleigh samples of size 10, the alternative linear predictors with these modifications have high efficiency when compared with the best linear unbiased predictor.     

A Further Study of Predictions in Linear Mixed Models

This article is concerned with the prediction problems in linear mixed models (LMM). Both biased predictors and restricted predictors are introduced. It was found that the mean square error matrix (MSEM) of a predictor strongly depends on the MSEM of corresponding estimator of the fixed effects and precise formulas are obtained. As an application, we propose three new predictors to improve the best linear unbiased predictor (BLUP). The performance of the new predictors can be examined easily with the help of vast literature on the linear regression models (LM). We also illustrate our findings with a Monte Carlo simulation and a numerical example.     

General ridge predictors in a mixed linear model

This paper mainly aims to put forward two estimators for the linear combination of fixed effects and random effects, and to investigate their properties in a general mixed linear model. First, we define the notion of a Type-I general ridge predictor (GRP) and obtain two sufficient conditions for a Type-I GRP to be superior over the best linear unbiased predictor (BLUP). Second, we establish the relationship between a Type-I GRP and linear admissibility, which results in the notion of Type-II GRP. We show that a linear predictor is linearly admissible if and only if it is a Type-II GRP. The superiority of a Type-II GRP over the BLUP is also obtained. Third, the problem of confidence ellipsoids based on the BLUP and Type-II GRP is investigated.     

Simple least squares estimation versus best linear unbiased prediction

Necessary and sufficient conditions are developed for the simple least squares estimator to coincide with the best linear unbiased predictor. The conditions obtained are valid for a general linear model and are generalizations of the condition given by Watson (1972). Also, as a preliminary result, a new representation of the best linear unbiased predictor is established.     

On equality of ordinary least squares estimator,best linear unbiased estimator and best linear unbiased predictor in the general linear model

The equality of ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE) and best linear unbiased predictor (BLUP) in the general linear model with new observations is investigated through matrix rank method, some new necessary and sufficient conditions are given.     

The concept of risk unbiasedness in statistical prediction

This paper extends the concept of risk unbiasedness for applying to statistical prediction and nonstandard inference problems, by formalizing the idea that a risk unbiased predictor should be at least as close to the “true” predictant as to any “wrong” predictant, on the average. A novel aspect of our approach is measuring closeness between a predicted value and the predictant by a regret function, derived suitably from the given loss function. The general concept is more relevant than mean unbiasedness, especially for asymmetric loss functions. For squared error loss, we present a method for deriving best (minimum risk) risk unbiased predictors when the regression function is linear in a function of the parameters. We derive a Rao–Blackwell type result for a class of loss functions that includes squared error and LINEX losses as special cases. For location-scale families, we prove that if a unique best risk unbiased predictor exists, then it is equivariant. The concepts and results are illustrated with several examples. One interesting finding is that in some problems a best unbiased predictor does not exist, but a best risk unbiased predictor can be obtained. Thus, risk unbiasedness can be a useful tool for selecting a predictor.     

Tolerance regions for random vectors and best linear predictors

In the case that vectors X and Y have a joint multivariate normal distribution, tolerance regions are found for the best linear predictor of Y using X if samples are used to estimate the regression coeffierante. Tolerance regions are also found for Y. In addition, simultaneous tolerance intervals for all linear functions of Y or of the best linear predictor of Y using X are found.     

Fundamental Equations of BLUE and BLUP in the Multivariate Linear Model with Applications

Ordinary least squares estimator (OLSE), best linear unbiased estimator (BLUE), and best linear unbiased predictor (BLUP) in the general linear model with new observations are generalized to the general multivariate linear model. The fundamental equations of BLUE and BLUP in the multivariate linear model are derived by two methods, including the vectorization method and projection method. By using the matrix rank method, some new results of linear BLUE-sufficiency, linear BLUP-sufficiency, and the equality of OLSE, BLUE, and BLUP are given in the multivariate linear model.     

Point predictors of the latent failure times of censored units in progressively Type-II censored competing risks data from the exponential distributions

In this paper, we discuss the problem of predicting times to the latent failures of units censored in multiple stages in a progressively Type-II censored competing risks model. It is assumed that the lifetime distribution of the latent failure times are independent and exponential-distributed with the different scale parameters. Several classical point predictors such as the maximum likelihood predictor, the best unbiased predictor, the best linear unbiased predictor, the median unbiased predictor and the conditional median predictor are obtained. The Bayesian point predictors are derived under squared error loss criterion. Moreover, the point estimators of the unknown parameters are obtained using the observed data and different point predictors of the latent failure times. Finally, Monte-Carlo simulations are carried out to compare the performances of the different methods of prediction and estimation and one real data is used to illustrate the proposed procedures.     

Equivariant prediction in finite populations

A theory of equivariant prediction is developed for predicting the population total in finite populations. Minimum risk equivariant predictors (MREP) are derived under the location, scale and locationscale superpopulation models. Under the general linear model, it is shown that the best(linear) unbiased predictor (B(L)UP) is an MREP.     

Prediction in a class of mixed models with two variance components

A best unbiased predictor (BUP) of an arbitrary linear combination of fixed and random effects in mixed linear models is available when the true values of the variance ratios are known. When the true values are unknown, a two-stage predictor, obtained from the BUP by replacing the true values by estimated values, can be used. In this article, exact mean squared errors of two-stage predictors are obtained for a class of mixed models with two variance components that includes the balanced one-way random model and other analysis-of-variance models with proportional frequencies and one balanced random factor.     

Pitman comparisons of predictors of censored observations from progressively censored samples for exponential distribution

On the basis of a progressively censored sample, Basak et al. [On some predictors of times to failure of censored items in progressively censored samples. Comput Statist Data Anal. 2006;50:1313 –1337] considered the problem of predicting the unobserved censored units at various stages of progressive censoring. They then discussed several different point predictors of these censored units and compared them with respect to mean square prediction error. In this work, we use the Pitman closeness (PC) criterion to compare the maximum likelihood, best linear unbiased, best linear equivariant, and conditional median predictors (CMPs) of these progressively censored units. Next, we compare all these with respect to the median unbiased predictor in terms of PC. Numerical computations are then performed to compare all these predictors. By comparing our results to those of Basak et al. (2006), we note that our findings in the sense of PC are similar to theirs in which the CMP competes well when compared to all other predictors.     

Likelihood Prediction for Generalized Linear Mixed Models under Covariate Uncertainty

This article presents the techniques of likelihood prediction for the generalized linear mixed models. Methods of likelihood prediction are explained through a series of examples; from a classical one to more complicated ones. The examples show, in simple cases, that the likelihood prediction (LP) coincides with already known best frequentist practice such as the best linear unbiased predictor. This article outlines a way to deal with the covariate uncertainty while producing predictive inference. Using a Poisson errors-in-variable generalized linear model, it has been shown in certain cases that LP produces better results than already known methods.     

Prediction of Times to Failure of Censored Items for a Simple Step-Stress Model with Regular and Progressive Type I Censoring from the Exponential Distribution

The problem of predicting times to failure of units from the Exponential Distribution which are censored under a simple step-stress model is considered in this article. We discuss two types of censoring—regular and progressive Type I—and two kinds of predictors—the maximum likelihood predictors (MLP) and the conditional median predictors (CMP) for each type of censoring. Numerical examples are used to illustrate the prediction methods. Using simulation studies, mean squared prediction error (MSPE) and prediction intervals are generated for these examples. MLP and the CMP are then compared with respect to MSPE and the prediction interval.