Characterization of Mixtures of Weibull and Pareto Distributions Based on Conditional Expectation of Order Statistics

Abstract

The mixture of Weibull random variables X ₁ and X ₂ are identified in terms of relations between the conditional expectation of given X _{1: 2} (or ( given X _{1: 2}, ∀ k ≤ r) and hazard rate function of the distribution, where X _{1: 2} and X _{2: 2} denote the corresponding order statistics, r is a positive integer and α > 0. In addition, by using the natural logarithm transformation, we also characterize and mixture of Pareto distributions based on the conditional expectation of order statistics. Furthermore, when the sample size is n, the above results are also valid, and we also give an application to Multi-Hit models of carcinogenesis (Parallel Systems). Finally, we also characterize mixture of Weibull and Pareto distributions based on the conditional expectation of upper record values.

Keywords:

• Characterization
• Conditional expectation
• Differential equation
• Mixture
• Order statistics
• Pareto distribution
• Record values
• Weibull distribution

Mathematics Subject Classification:

• 60E05
• 62E10
• 62H05

Acknowledgments

The authors are very grateful to the referees for their suggestions and helpful comments which led to improvement in this article. This research was partially supported by the National Science Council, R.O.C. (Plan No. NSC 99-2118-M-415-001).