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Some characterizations based on the Bhattacharya matrix

Published online by Cambridge University Press:  14 July 2016

D. N. Shanbhag
D. N. Shanbhag*
Affiliation:
University of Sheffield
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Abstract

Laha and Lukacs (1960) have studied distributions with the property that a quadratic statistic has quadratic regression on the sample mean. In doing this, they have arrived at some interesting characterizations for the normal, Poisson, gamma, binomial and negative binomial distributions. Starting with an exponential-type probability density function, the present paper investigates all the distributions for which the 3 × 3 Bhattacharya matrix is diagonal. It is found that the normal, Poisson, gamma, binomial and negative binomial distributions can be characterized by this property. Further, it is observed that for these distributions an s × s Bhattacharya matrix is defined for all s and is also diagonal.

Keywords

BHATTACHARYA MATRIXCHARACTERIZATIONEXPONENTIAL-TYPE DENSITY FUNCTIONNORMAL, POISSON, GAMMA, BINOMIAL AND NEGATIVE BINOMIAL DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 3 , June 1972 , pp. 580 - 587
Copyright
Copyright © Applied Probability Trust 1972 

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References

Gurland, J. (1956) An inequality satisfied by the gamma function. Skand. Aktuarietidsk. 39, 171–172.Google Scholar
Laha, R. G. and Lukacs, E. (1960) On a problem connected with quadratic regression. Biometrika, 47, 335–343.Google Scholar
Raja Rao, B. (1969) An improved inequality satisfied by the gamma function. Skand. Aktuarietidsk. 52, 78–83.Google Scholar
Shanbhag, D. N. (1970) Some remarks on Raja Rao's inequality concerning the gamma function. Submitted for publication.Google Scholar