Generalized Estimation of the BLUP in Mixed-Effects Models: A Comparison with ML and REML

The Best Linear Unbiased Predictor (BLUP) in mixed models is a function of the variance components and they are estimated using maximum likelihood (ML) or restricted ML methods. Nonconvergence of BLUP would occur due to a drawback of the standard likelihood-based approaches. In such situations, ML and REML either do not provide any BLUPs or all become equal. To overcome this drawback, we provide a generalized estimate (GE) of BLUP that does not suffer from the problem of negative or zero variance components, and compare its performance against the ML and REML estimates of BLUP. Simulated and published data are used to compare BLUP.     

Testing the difference of two exponential means using generalized p-values

There are no exact fixed-level tests for testing the null hypothesis that the difference of two exponential means is less than or equal to a prespecified value θ₀. For this testing problem, there are several approximate testing procedures available in the literature. Using an extended definition of p-values, Tsui and Weerahandi (1989) gave an exact significance test for this testing problem. In this paper, the performance of that procedure is investigated and is compared with approximate procedures. A size and power comparison is carried out using a simulation study. Its findings show that the test based on the generalized p-value guarantees the intended size and that it is either as good as or outperforms approximate procedures available in the literature, both in power and in size.     

Size performance of some tests in one-way anova

Tsui and Weerahandi (1989) introduced the notion of generalized p-values and since then this idea is used to solve many statistical testing problems. Heteroskedasticity is one of the major practical problems encountered in ANOVA problems. To compare the means of several groups under heteroskedasticity approximate tests are used in the literature. Weerahandi (1995a) introduced a test using the notion of generalized p-values for comparing the means of several populations when the variances are not equal. This test is referred to as a generalized F-test.

In this paper we compare the size performance of the Generalized F-test and four other widely used procedures: the Classical F-test for ANOVA, the F-test obtained by the weighted least-squares to adjust for heteroskedasticity, the Brown-Forsythe-test, and the Welch-test. The comparison is based on a simulation study of size performance of tests applied to the balanced one-way model. The intended level of the tests is set at 0.05. While the Generalized F-test was found to have size not exceeding the intended level, as heteroskedasticity becomes severe the other tests were found to have poor size performance. With mild heteroskedasticity the Welch-test and the classical ANOVA F-test have the intended levels, and the Welch-test was found to perform better than the latter. Widely used (due to computational convenience) weighted F-test was found to have very serious size problems. The size advantage of the generalized F-test was also found to be robust even under severe deviations from the assumption of normality.     

A general method of inference for two-parameter continuous distributions

Abstract

This article presents a general method of inference of the parameters of a continuous distribution with two unknown parameters. Except in a few distributions such as the normal distribution, the classical approach fails in this context to provide accurate inferences with small samples.Therefore, by taking the generalized approach to inference (cf. Weerahandi, 1995 Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis. New York: Springer Verlag. [Google Scholar]), in this article we present a general method of inference to tackle practically useful two-parameter distributions such as the gamma distribution as well as distributions of theoretical interest such as the two-parameter uniform distribution. The proposed methods are exact in the sense that they are based on exact probability statements and exact expected values. The advantage of taking the generalized approach over the classical approximate inferences is shown via simulation studies.

This article has the potential to motivate much needed further research in non normal regressions, multiparameter problems, and multivariate problems for which basically there are only large sample inferences available. The approach that we take should pave the way for researchers to solve a variety of non normal problems, including ANOVA and MANOVA problems, where even the Bayesian approach fails. In the context of testing of hypotheses, the proposed method provides a superior alternative to the classical generalized likelihood ratio method.     

A characterization of the general mean

A characterization is given of the general mean which is (?X^p)^1/p or exp[log X] for X > 0 according as ρ ≠0 or ρ = 0, where ρ is an unspecified parameter. The approach is axiomatic. The general mean is defined as any quantity satisfying certain axioms which embrace the minimal requirements that any reasonable candidate for the role of mean or most typical value should satisfy. The quantity, indexed by p and given above, is uniquely implied by these axioms.     

A note reconciling ANOVA tests under unequal error variances

The purpose of this note is to derive simple testing procedures for ANOVA under heteroscedasticity by a single approach that are equivalent to the prior art in the literature obtained by the Parametric Bootstrap and the Generalized Fiducial approach. By similar approach, researchers are encouraged to derive generalized tests in other applications, as alternative to parametric bootstrap tests and fiducial tests, including ANCOVA and MANOVA under heteroscedasticity, especially in Mixed Model applications, where the bootstrap approach fails.     

Bayesian nonparametric smoothers for regular processes

This report is about the analysis of stochastic processes of the form R = S + N, where S is a “smooth” functional and N is noise. The proposed methods derive from the assumption that the observed R-values and unobserved values of R, the assumed inferential objectives of the analysis, are linearly related through Taylor series expansions of observed about unobserved values. The expansion errors and all other priori unspecified quantities have a joint multivariate normal distribution which expresses the prior uncertainty about their values. The results include interpolators, predictors, and derivative estimates, with credibility-interval estimates automatically generated in each case. An analysis of an acid-rain wet-deposition time series is included to indicate the efficacy of the proposed method. It was this problem which led to the methodological developments reported in this paper.     

Generalized Point Estimation with Application to Small Response Estimation

Motivated by a number of drawbacks of classical methods of point estimation, we generalize the definitions of point estimation, and address such notions as unbiasedness and estimation under constraints. The utility of the extension is shown by deriving more reliable estimates for small coefficients of regression models, and for variance components and random effects of mixed models. The extension is in the spirit of generalized confidence intervals introduced by Weerahandi (1993 Weerahandi , S. ( 1993 ). Generalized confidence intervals . J. Amer. Statist. Assoc. 88 : 899 – 905 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) and should encourage much needed further research in point estimation in unbalanced models, multi-variate models, non normal models, and nonlinear models.