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On the degree of smoothness and on singularities in distributions of statistical functions

Published online by Cambridge University Press:  24 October 2008

H. P. Mulholland
H. P. Mulholland
Affiliation:
University of Exeter
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Extract

Introduction. When the distribution of a statistical function could not be determined exactly, a suitable form for an approximate formula has sometimes been found by the use of information about the order of smoothness of the distribution and about any singularities it might have (see, for example, Geary ((4))). The object of the present paper is to develop a general theory that will provide information of this kind. For the singularity in the distribution of a function at an end of its range Hotelling ((8)) obtained formulae for cases where the original distribution is uniform over a curved space. However, some special sampling distributions are known to have singularities in the interior of the range, for example, the distribution of √ b1 (see (4)) and those of various serial correlation coefficients (see (6)). In view of this I shall deal below with orders of smoothness and with singularities over the whole range. For simplicity I shall assume that the original distribution has a fairly high order of smoothness: attention will therefore be concentrated on the singularities (if any) that arise from the form of the statistical function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 3 , July 1965 , pp. 721 - 739
Copyright
Copyright © Cambridge Philosophical Society 1965

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