Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T05:41:45.818Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

(1)Daniels, H. E.The approximate distribution of serial correlation coefficients. Biometrika, 43 (1956), 170–185.CrossRefGoogle Scholar
(2)Erdélyi, A.Asymptotic expansions (New York, 1956).Google Scholar
(3)Erdélyi, A. and Sneddon, I. N.Fractional integration and dual integral equations. Canadian J. Math. 14 (1962), 685–693.CrossRefGoogle Scholar
(4)Geary, R. C.The frequency distribution of √ b 1 for samples of all sizes drawn at random from a normal population. Biometrika, 34 (1947), 68–97.Google ScholarPubMed
(5)Graves, L. M.The theory of functions of real variables (New York and London, 1946).Google Scholar
(6)Grenander, U. and Rosenblatt, M.Statistical analysis of stationary time series (New York and Stockholm, 1957).CrossRefGoogle Scholar
(7)Gurland, J.Distribution of quadratic forms and ratios of quadratic forms. Ann. Math. Statist. 24 (1953), 416–427.CrossRefGoogle Scholar
(8)Hotelling, H.Tubes and spheres in n-space, and a class of statistical problems. American J. Math. 61 (1939), 440–460.CrossRefGoogle Scholar
(9)Littman, W.Fourier transforms of surface-carried measures and differentiability of sur face averages. Bull. American Math. Soc. 69 (1963), 766–770.CrossRefGoogle Scholar
(10)Marston., MorseRelations between the critical points of a real function of n independent variables. Trans. American Math. Soc. 27 (1925), 345–396.Google Scholar
(11)Ostrowski, A.Vorlesungen über Differential und Integralrechnung, III (Basle, 1954).Google Scholar
(12)Robbins, H.The distribution of a definite quadratic form. Ann. Math. Statist. 19 (1948), 266–270.CrossRefGoogle Scholar
(13)Siddiqui, M. M.Distribution of a serial correlation coefficient near the ends of the range. Ann. Math. Statist. 29 (1958), 852–861.CrossRefGoogle Scholar