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A multivariate IFR class

Published online by Cambridge University Press:  14 July 2016

Thomas H. Savits
Thomas H. Savits*
Affiliation:
University of Pittsburgh
*
∗Postal address: Department of Mathematics and Statistics, University of Pittsburgh, PA 15260, USA.
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Abstract

A non-negative random vector T is said to have a multivariate increasing failure rate distribution (MIFR) if and only if E[h(x, T)] is log concave in x for all functions h(x, t) which are log concave in (x, t) and are non-decreasing and continuous in t for each fixed x. This class of distributions is closed under deletion, conjunction, convolution and weak limits. It contains the multivariate exponential distribution of Marshall and Olkin and those distributions having a log concave density. Also, it follows that if T is MIFR and ψ is non-decreasing, non-negative and concave then ψ (T) is IFR.

Keywords

LOG CONCAVEINEQUALITIES
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 197 - 204
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

Research supported by ONR Contract N00014–76–C–0839.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Block, H. W. and Savits, T. H. (1980) Multivariate IFRA distributions. Ann. Prob. 8, 793–801.Google Scholar
Block, H. W. and Savits, T. H. (1981) Multivariate classes in reliability theory. Math. Operat. Res. 6, 453–461.CrossRefGoogle Scholar
Brascamp, H. J. and Lieb, E. H. (1975) Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In Functional Integration and Its Application, ed. Arthurs, A. M. Clarendon Press, Oxford.Google Scholar
El-Neweihi, E. (1981) Stochastic ordering and a class of multivariate new better than used distributions. Commun. Statist. Theory Methods A 10, 1655–1672.Google Scholar
Marshall, A. W. (1975) Multivariate distributions with monotone hazard rate. In Reliability and Fault Tree Analysis, ed. Barlow, R. E., Fussell, J. and Singpurwalla, N. D., SIAM, Philadelphia, 259–284.Google Scholar
Marshall, A. W. and Olkin, I. (1967) A multivariate exponential distribution. J. Amer. Statist. Assoc. 62, 30–44.Google Scholar
Marshall, A. W. and Shaked, M. (1982) A class of multivariate new better than used distributions. Ann. Prob. 10, 259–264.Google Scholar
Ohi, F. and Nishida, T. (1982) A definition of NBU probability measures. J. Japan Statist. Soc. 12, 141–151.Google Scholar
Prékopa, A. (1971) Logarithmic concave measures with application to stochastic programming. Acta Math. Szeged 32, 301–315.Google Scholar
Savits, T. H. (1984) Multivariate life classes and inequalities. In Inequalities in Probability and Statistics, ed. Jong, Y. L. IMS Lecture Notes Monograph Series Vol. 5, IMS, Hayward, Ca. Google Scholar