TY - JOUR
T1 - Stochastic order of sample range from heterogeneous exponential random variables
AU - Zhao, Peng
AU - Li, Xiaohu
PY - 2009/1
Y1 - 2009/1
N2 - Let X1,...,Xn be independent exponential random variables with their respective hazard rates 1,...,n, and let Y1,...,Yn be independent exponential random variables with common hazard rate . Denote by Xn:n, Yn:n and X 1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n-X1:n is proved to be larger than Y n:n-Y1:n according to the usual stochastic order if and only if λ ≥ (λ̄-1||n i=1λi)1/(n-1) with λ̄ = ∑ni=1λi}/n. Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.
AB - Let X1,...,Xn be independent exponential random variables with their respective hazard rates 1,...,n, and let Y1,...,Yn be independent exponential random variables with common hazard rate . Denote by Xn:n, Yn:n and X 1:n, Y1:n the corresponding maximum and minimum order statistics. Xn:n-X1:n is proved to be larger than Y n:n-Y1:n according to the usual stochastic order if and only if λ ≥ (λ̄-1||n i=1λi)1/(n-1) with λ̄ = ∑ni=1λi}/n. Further, this usual stochastic order is strengthened to the hazard rate order for n=2. However, a counterexample reveals that this can be strengthened neither to the hazard rate order nor to the reversed hazard rate order in the general case. The main result substantially improves those related ones obtained in Kochar and Rojo and Khaledi and Kochar.
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U2 - 10.1017/S0269964809000023
DO - 10.1017/S0269964809000023
M3 - Article
AN - SCOPUS:56149126653
SN - 0269-9648
VL - 23
SP - 17
EP - 29
JO - Probability in the Engineering and Informational Sciences
JF - Probability in the Engineering and Informational Sciences
IS - 1
ER -