Characterizations of the exponential distribution by order statistics and conditional expectations of record values
Commun. Korean Math. Soc. 2002 Vol. 17, No. 3, 535-540
Printed September 1, 2002
Min-Young Lee, Se-Kyung Chang, Kap-Hun Jung
Dankook University, Dankook University, Dankook University
Abstract : Let $ X_1 , X_2 , \cdots , 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} \le X_{2:n} \le \cdots \le 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^{-\frac{x}{c}} $ with some $ c>0 $. And $ X_j $ is an upper record value of this sequence if $ X_j > \max\{ X_1 , X_2 , \cdots , X_{j-1} \}.$ We define $ u(n) =\min \{j \vert j > u(n-1) , X_j > X_{u(n-1)} , n \geq 2 \} $ with $ u(1) =1.$ Then $ F(x)=1-e^{-\frac{x}{c}}{\hskip-0.05cm}, \, x>0 $ if and only if $ E[X_{u(n+3)} - X_{u(n)} \vert X_{u(m)} =y ] = 3c ,$ or $ E[X_{u(n+4)} - X_{u(n)} \vert X_{u(m)} =y ] = 4c \, , \,\, n \geq m+1 .$
Keywords : absolutely continuous distribution, characterization, conditional expectation, order statistic, record value
MSC numbers : 60E15, 62E10
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