Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values

Abstract

Let XU(1) < XU(2) < ... < XU(n) < ... be the sequence of the upper record values from a population with common distribution function F. In this paper, we first give a theorem to characterize the generalized mixtures of geometric distribution by the relation between E[(XU(n+1)-XU(n)2¦XU(n)=x and the function of the failure rate of the distribution, for any positive integer n. Secondly, we also use the same relation to characterize the generalized mixtures of exponential distribution. The characterizing relations were motivated by the work of Balakrishnan and Balasubramanian (1995).