Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-18T14:11:14.783Z Has data issue: false hasContentIssue false
  • English
  • Français

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

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.

Information

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

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.)

Article purchase

Temporarily unavailable

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