96-F-12. Kubokawa, Tatsuya, A.K.Md. Ehsane Saleh, Yoshihiko Konno and Masaru Ushijima, "Estimation of Variance Components under Kullback-Leibler Loss with Applications", June 1996.

Estimation of variance components in a mixed linear model with two variance components has been discussed for many years. The problems arise in the incompleteness of the minimal sufficient statistics and the drawback of every unbiased estimator taking a negative value with a positive probability for the `between' component of variance. This paper addresses to resolve the latter undesirable property. For evaluating estimators, the expected Kullback-Leibler loss function is utilized instead of the usual mean squared error (MSE). The class of estimators improving on the ANOVA (unbiased) estimators by the Hender-son method (III) are constructed and out of the class, the positive and useful estimators and the empirical Bayes, the generalized Bayes estimators are derived. Applications of the proposed estimators are given (1) to the generalized least squares (GLS) estimation of the regression coefficients, (2) to the GLS F test for a linear hypothesis and (3) to the two-stage prediction which is related to the small-area estimation. Finally, the confidence intervals of the variance components are discussed, and several improvements on the minimum ratio (shortest unbiased) confidence interval related to the Kullback-Leibler loss are proposed.