Test Equating, Scaling, and Linking: Methods and Practices

This text, which covers the theory and applications of preliminary test estimators and Stein-type estimation, is intended for graduate students, researchers, and teachers. The basic prerequisite is knowledge of statistical theory at the level of Casella and Berger (2002). The author’s stated aim is “to provide the readers with the mathematical and statistical foundations of the preliminary test and shrinkage estimation.” His pathway to this end, within each of several general standard models mentioned subsequently, is to present and study the analytic properties of five different estimators: the unrestricted estimator (UE), the restricted estimator (RE), the preliminary test estimator (PTE), the Stein-type shrinkage estimator (SE), and the positive-rule shrinkage estimator. The class of models for which estimators are developed and studied includes linear models, parallelism models, analysis of variance (ANOVA) models, multiple regression models, multivariate models, and discrete models. In general terms, one may think of the unrestricted estimator as the “usual” classical [maximum likelihood estimator (MLE) or least squares estimator (LSE)] estimator for the (unrestricted) model at hand. The restricted estimator may be, for example, the MLE under some “restriction” (which restriction may be thought of as being given by a null hypothesis that represents perhaps, imperfect prior knowledge). The preliminary test estimator (PTE) is equal to the UE if the null hypothesis is rejected and is equal to RE if the null hypothesis is accepted (at some level of significance to be determined). The Stein-type estimator is viewed in this book, at least in part, as related to the PTE by replacing the indicator function implicit in the definition of the PTE by a decreasing (smooth) function of the test statistic used in the PTE. For example, if the UE is the unrestricted MLE ̃θ , the likelihood ratio statistic for the null hypothesis (the restriction) is Ln, and the RE is the MLE ̂θ , under the restriction that the null hypothesis is true, then PTE = ̃θ − ( ̃θ −̂θ)I [Ln < Ln,α], where Ln,α is the α-level critical value of Ln. A Stein-type estimator is given by PTE = ̃θ − ( ̃θ − ̂θ)ψ(Ln), where ψ(·) is nonnegative and decreasing (e.g., c/Ln). Also, the related positive rule is given by PTE = ̃θ − ( ̃θ − θ)ψ(Ln)I [ψ(Ln) ≤ 1]. The author also presents alternative developments of some Stein-type estimators, in particular, Stein’s risk difference (or, perhaps more appropriately, Stein’s unbiased estimate of risk difference) and the empirical Bayes approach of Efron and Morris. In general, however, it is the representation of the Stein-type estimator as a smoothed PTE that serves as a unifying theme throughout most of the text and that motivates many of the Stein-type estimators in the various models. For (almost) each model/restriction (hypothesis) considered, the author evaluates the bias, quadratic bias, the mean squared error matrix, and quadratic risk of each of the five estimators. The development relies on extensive use of properties of the chi-squared and noncentral chi-squared distributions, and on their multivariate counterparts. The following list of chapters indicates the coverage and flow of material: