The perceived framework of a classical statistic: Is the non-invariance of a Wald statistic much ado about null thing?

Abstract

The distinction between a nominal framework for the three classical statistics and a perceived framework for each classical statistic provides more ways to interpret these statistics, and intuitively explains as well as more easily shows some well-known results. In particular, each classical statistic can be viewed in terms of a length in each of four spaces and, since the classical procedures per se are equivalent in a perceived framework, two statistics are identical if their perceived frameworks are identical. This helps to integrate the normally separately treated issues of a reformulation of a null hypothesis and of locally equivalent alternatives. For example, a Wald statistic is not invariant if a reformulation changes its perceived framework, and an appropriate score statistic is invariant as its perceived framework is unaffected by considering a locally equivalent alternative. [During the thirty-four months this paper was under consideration at The Econometrics Journal, the Editor-in-charge (Professor Stephane Gregoir) did not reply to three (of the author's four) requests about the status of the submission, and provided neither a referee's report nor a first decision. Also, when asked to intervene by the author, the new Managing Editor (Professor Richard J Smith) offered the author the possibility of submitting the paper (as a new submission) to the new editorial regime, at which point, the author withdrew the paper.]classical statistic; likelihood ratio statistic; nominal framework; perceived framework; score statistic; Wald statistic