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ERA's: A New Approach to Small Sample Theory

P. C. B. Phillips

Abstract

This article proposes a new approach to small sample theory that achieves a meaningful integration of earlier directions of research in this field. The approach centers on the constructive technique of approximating distributions developed recently by the author in [10]. This technique utilizes extended rational approximants (ERA's) which build on the strengths of alternative, less flexible approximation methods (such as those based on asymptotic expansions) and which simultaneously blend information from diverse analytic, numerical and experimental sources. The first part of the article explores the general theory of approximation of continuous probability distributions by means of ERA's. Existence, characterization, error bound, and uniqueness theorems for these approximants are given and a new proof is provided for the convergence result obtained earlier in [10]. Some further aspects of finding ERA's by modifications to multiple-point Pade approximants are presented and the new approach is applied to the noncircular serial correlation coefficient. The results of this application demonstrate how ERA's provide systematic improvements over Edgeworth and saddlepoint techniques. These results, taken with those of the earlier article [10], suggest that the approach offers considerable potential for empirical application in terms of its reliability, convenience, and generality.