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CHARACTERIZATIONS OF THE EXPONENTIAL DISTRIBUTION BY ORDER STATISTICS AND CONDITIONAL

  • Lee, Min-Young (Department of Mathematics Dankook University) ;
  • Chang, Se-Kyung (Department of Mathematics Dankook University) ;
  • Jung, Kap-Hun (Department of Mathematics Dankook University)
  • Published : 2002.07.01

Abstract

Let X$_1$, X$_2$‥‥,X$\_$n/ be n independent and identically distributed random variables with continuous cumulative distribution function F(x). Let us rearrange the X's in the increasing order X$\_$1:n/ $\leq$ X$\_$2:n/ $\leq$ ‥‥ $\leq$ X$\_$n:n/. We call X$\_$k:n/ the k-th order statistic. Then X$\_$n:n/ - X$\_$n-1:n/ and X$\_$n-1:n/ are independent if and only if f(x) = 1-e(equation omitted) with some c > 0. And X$\_$j/ is an upper record value of this sequence lf X$\_$j/ > max(X$_1$, X$_2$,¨¨ ,X$\_$j-1/). We define u(n) = min(j|j > u(n-1),X$\_$j/ > X$\_$u(n-1)/, n $\geq$ 2) with u(1) = 1. Then F(x) = 1 - e(equation omitted), x > 0 if and only if E[X$\_$u(n+3)/ - X$\_$u(n)/ | X$\_$u(m)/ = y] = 3c, or E[X$\_$u(n+4)/ - X$\_$u(n)/|X$\_$u(m)/ = y] = 4c, n m+1.

Keywords

absolutely continuous distribution;characterization;conditional expectation;order statistic record value

References

  1. Commun. Korean Math. Soc. v.16 no.2 Characterizations of the exponential distribution by conditional expectations of record values Lee, M. Y.
  2. Commack NY Record Statistics, Nova Science publishers, Inc M. Ahsanuallah
  3. Berlin Heidelberg NY Characterization of probability distributions, Springer-verlag J. Galambos
  4. Characterization of the exponential distribution by some properties of the record values