On a conjecture of Kakosyan, Klebanov and Melamed

Abstract

Suppose X₁, X₂, ..., X_m is a random sample of size m from a population with probability density function f(x), x>0 and let X_1,m<...<X_m,m be the corresponding order statistics. We assume m as an integer valued random variable with P(m=k)=p(1−p)^k−1, k=1, 2, ... and 0<p<1. Kakosyan, Klebanov and Melamed (1984) conjectured that the identical distribution of

and n X_1,n for fixed n characterizes the exponential distribution. In this paper we prove that under the assumption of monotone hazard rate the identical distribution of

and (n−r+1) (X_r,n−X_r−1,n) for some fixed r and n with 1≤r≤n, n≥2, X_0,n=0, characterizes the exponential distribution. Under the assumption of monotone hazard rate the conjecture of Kakosyan, Klebanov and Melamed follows from the above result with r=1.