Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-12T17:15:27.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Characterizations of the normal distribution by suitable transformations

Published online by Cambridge University Press:  14 July 2016

S. Beer  and
E. Lukacs
S. Beer
Affiliation:
The Institute of Technology of Vienna
E. Lukacs*
Affiliation:
The Catholic University of America
*
∗ Now at Bowling Green State University, Ohio.
Get access Rights & Permissions [Opens in a new window]

Abstract

Yu. V. Linnik showed that certain transformations, given by Formulae (1.1), (1.6) and (1.7) transform a normal sample into itself. The transformations (1.1) and (1.7) apply to samples of size 2 while (1.6) admits an arbitrary sample size. It is also assumed that the population mean is zero.

In the present paper the converse theorems are proven so that characterizations of the normal distribution are obtained. The problem leads to the functional equations (2.3) and (2.13) whose solution yields the desired results.

Keywords

CHARACTERIZATIONSNORMAL DISTRIBUTIONTRANSFORMATIONS OF NORMAL SAMPLESFUNCTIONAL EQUATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 1 , March 1973 , pp. 100 - 108
Copyright
Copyright © Applied Probability Trust 1973 

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.)

Footnotes

This paper was partly written during Professor Lukacs' visit to the Institute of Technology, Vienna, and also while Miss Beer subsequently visited the Catholic University of America. The work was supported by the National Science Foundation under grant NSF-GP-22585.

1

Personal communication of Professor Linnik.

References

[1] Cramér, H. (1946) Mathematical Methods of Statistics. Princeton University Press, Princeton, N. J. Google Scholar
[2] Lancaster, H. O. (1960) The characterization of the normal distribution. J. Aust . Math. Soc. 1, 368–383.Google Scholar
[3] Muir, T. (1906-1923) The Theory of Determinants in the Historical Order of Development. Longmans-Green, London (1963). Dover reprint, New York (1960).Google Scholar
[4] Skitovic, V. P. (1954) Linear combinations of independent random variables and thenormal distribution law. Izv. Akad. Nauk SSSR Ser. Mat. 18, 185–200. English translation in Selected Translations in Mathematical Statistics and Probability. Vol. 2, 211–228. Amer. Math. Soc. (1962), Providence, R. I. Google Scholar
[5] Gröbner, W. and Hofreiter, N. (1950) Integraltafel, zweiter Teil, bestimmte Integrale. Springer-Verlag, Wien and Innsbruck.Google Scholar
[6] Gradshteyn, I. S. and Ryzhik, I. M. (1965) Table of Integrals, Series and Products Academic Press, New York.Google Scholar