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On ratios of random variables and generalised mortality rates

Part of: Distribution theory

Published online by Cambridge University Press:  14 July 2016

Leigh A. Roberts
Leigh A. Roberts*
Affiliation:
Victoria University, Wellington
*
∗Postal address: Institute of Statistics and Operations Research, P.O. Box 600, Victoria University, Wellington, New Zealand.
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Abstract

We find the distribution function of a ratio of dependent random variables which can represent a generalised mortality rate in a demographic or life insurance context. Each death in the numerator and each unit of exposure in the denominator are weighted by a random sum at risk, which is assumed to follow a gamma distribution. General results on the existence of moments of ratios of random variables are established, and applied to show that the moments of the rates considered here depend in a simple way on the minimum number of entrants into the mortality investigation.

Keywords

DEMOGRAPHYGAMMA DISTRIBUTED SUMS AT RISKINSURANCEMOMENTS OF RATIOS OF RANDOM VARIABLESMORTALITY RATEPRODUCT OF GAMMA AND UNIFORM VARIATES

MSC classification

Primary: 62E20: Asymptotic distribution theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 2 , June 1992 , pp. 268 - 279
Copyright
Copyright © Applied Probability Trust 1992 

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References

Feller, W. (1965) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Marsaglia, G. (1965) Ratios of normal variables and ratios of sums of uniform variables. J. Amer. Statist. Assoc. 60, 193–204.Google Scholar