A Discussion of Some Basic Concepts in Statistical Inference

Abstract

Despite its’ distinctive past history, the problem of statistical inference in a broad sense, besides being controversial among statisticians, is continuously stimulating vigorous and rigorous research. There are some interesting recent advances but many problems either remain open and unresolved or need improvements. In order to give a reasonably comprehensive discussion of the basic concepts, ideas and principles, in this paper we concentrate on the various underlying assumptions, statistical decision error structure, various measures of information, sufficiency and ancillarity, and some important and most commonly used principles of statistical inference. It is suggested that sphericity and exchangeability, besides being weaker than normality, are more plausible and tangible, at least from a practical viewpoint. Thus they are perhaps more natural and appealing as a basis for statistical inference. For a large class of standard decisions it is proposed that instead of the classical testing approach, one should minimize a linear combination, with nonnegative weights, of the first and second type errors. After discussing various measures of information, it is pointed out that the family of likelihood functions of various types essentially contain relevant information for statistical problems. Next, along with sufficiency and ancillarity concepts, various inferential principles are discussed. Without assuming discreteness or finiteness of underlying universes, with due regard to the measure-theoretic difficulties, implications, interplay and consequences of various principles such as conditionality, invariance, sufficiency, likelihoods (all types) and their weak versions are clarified.