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Crossing Properties of Mixture Distributions

Published online by Cambridge University Press:  27 July 2009

Philip J. Boland ,
Frank Proschan  and
Y. L. Tong
Philip J. Boland
Affiliation:
Department of Statistics University College, Dublin Belfield, Dublin 4, Ireland
Frank Proschan
Affiliation:
Department of StatisticsThe Florida State University Tallahassee, Florida 32306-3033
Y. L. Tong
Affiliation:
School of MathematicsGeorgia Institute of Technology, Atlanta, Georgia 30332
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Abstract

Mixture distributions are a frequently used tool in modelling random phenomena. We consider mixtures of densities from a one-parameter exponenvial family of distributions. Using the tools of totally positive functions and the variation-diminishing property of such, we study the effect of sign-crossing properties of two mixing densities μ1 and μ2 on the resulting mixture distributions f1 and f2. The results enable us to make stochastic and variability cornparisons for binomial-beta, mixed Weibull, and mixed gamma distributions.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 3 , July 1989 , pp. 355 - 366
Copyright
Copyright © Cambridge University Press 1989

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References

Birnbaum, Z.W. (1948). On random variables with comparable peakedness. Annals of Mathematical Statistics 19: 76–81.CrossRefGoogle Scholar
Everitt, B.S. (1985). Mixture distributions. In Kotz, S. & Johnson, N.L., (eds.), Encyclopedia of statistical sciences, Vol. 5. New York: Wiley, pp. 559–569.Google Scholar
Greenwood, M. & Yule, G.U. (1920). An inquiry into the nature of frequency distributions of multiple happenings. Journal of the Royal Statistical Society A 83: 255–279.CrossRefGoogle Scholar
Ishii, G. & Hayakawa, R. (1960). On the compound binomial distribution. Annals of the Institute of Statistical Mathematics 12: 69–80 (Errata, 12: 208).CrossRefGoogle Scholar
Johnson, N. & Kotz, S. (1969). Distributions in statistics: discrete distributions. Houghton Mifflin Co.Google Scholar
Kao, J.H.K. (1959). A graphical estimation of mixed Weibull parameters in life testing of electron tubes. Technometrics 7: 639–643.Google Scholar
Karlin, S. (1968). Total positivity. Stanford, California: Stanford University Press.Google Scholar
Lindley, D.V. & Singpurwalla, N.D. (1986). Multivariate distributions for the life lengths of components of a system sharing a common environment. Journal of Applied Probability 23: 418–431.CrossRefGoogle Scholar
Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics 5: 375–384.CrossRefGoogle Scholar
Rider, P.R. (1961). The method of moments applied to a mixture of two exponential distributions. Annals of Mathematical Statistics 32: 142–147.CrossRefGoogle Scholar
Ross, S. (1983). Stochastic processes. New York: John Wiley and Sons.Google Scholar
Seal, H.L. (1969). Stochastic theory of a risk business. New York: Wiley.Google Scholar
Shaked, M. (1980). On mixtures from exponential families. Journal of the Royal Statistical Society B 42: 192–198.Google Scholar
Shaked, M. (1985). Ordering distributions in dispersion. In Johnson, N.L. & Kotz, S. (eds.), Encyclopedia of statistical sciences, Vol. 6. New York: Wiley, pp. 485–490.Google Scholar
Stoyan, D. (1983). Comparison methods for queues and other stochastic models. New York: John Wiley and Sons.Google Scholar
Whitt, W. (1985). Uniform conditional variability ordering of probability distributions. Journal of Applied Probability 22: 619–633.CrossRefGoogle Scholar