On structural models and a theorem of d. basu

Abstract

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.

Keywords:

• necessary analysis
• application to distribution theory
• minimum risk equivariant estimators
• statistical independence