On structural models and a theorem of d. basu

Structural inference as a method of statistical analysis seems to have escaped the attention of many statisticians. This paper focuses on Fraser’s necessary analysis of structural models as a tool to derive classical distribution results.

A structural model analyzed by Zacks (1971) by means of conventional statistical methods and fiducial theory is re-examined by the structural method. It is shown that results obtained by the former methods come as easy consequences of the latter analysis of the structural model. In the process we also simplify Zacks1 methods of obtaining a minimum risk equivariant estimator of a parameter of the model.

A theorem of Basu (1955), often used to prove independence of a complete sufficient statistic and an ancillary statistic, is also reexamined in the light of structural method. It is found that for structural models more can be achieved by necessary analysis without the use of Basu’s theorem. Bain’s (1972) application of Basu’s theorem of constructing confidence intervals for Weibull reliability is given as an example.     

A note on the location parameters of two exponential distributions under order restrictions

The problem of simultaneous estimation of location parameters of two independent exponential distributions is considered when location and/or scale parameters are ordered. We show that the standard estimators in the unrestricted case which use information only from the populations individually can be improved upon when various order restrictions are known to hold. The improved estimators are obtained under the quadratic loss function     

Pitman closeness for hierarchical bayes predictors in mixed linear models

The present article considers the Pitman Closeness (PC) criterion of certain hierarchical Bayes (HB) predictors derived under a normal mixed linear models for known ratios of variance components using a uniform prior for the vector of fixed effects and some proper or improper prior on the error variance. For a generalized Euclidean error, simultaneous HB predictors of several linear combinations of vector of effects are shown to be the Pitman-closest in the frequentist sense in the class of equivariant predictors for location group of transformations. The normality assumption can be relaxed to show that these HB predictors are the Pitman-closest for location-scale group of transformations for a wider family of elliptically symmetric distributions. Also for this family, the HB predictors turn out to be Pitman-closest in the class of all linear unbiased predictors (LUPs). All these results are extended for the HB predictor of finite population mean vector in the context of finite population sampling.