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LIKELIHOOD RATIO AND MEAN RESIDUAL LIFE ORDERS FOR ORDER STATISTICS OF HETEROGENEOUS RANDOM VARIABLES

Published online by Cambridge University Press:  10 April 2001

Taizhong Hu ,
Zegang Zhu  and
Ying Wei
Taizhong Hu
Affiliation:
Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China, E-mail: [email protected]
Zegang Zhu
Affiliation:
Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China, E-mail: [email protected]
Ying Wei
Affiliation:
Department of Statistics and Finance, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China, E-mail: [email protected]
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Abstract

Let X1:n ≤ X2:n ≤ ··· ≤ Xn:n denote the order statistics of a set of independent and not necessarily identically distributed random variables X1,..., Xn. Under mild assumptions, it is shown that Xk−1:n−1 ≤lrXk:n for k = 2,..., n if X1 ≤lrX2 ≤lr ··· ≤lrXn and that Xk:n ≤lrXk:n−1 for k = 1,..., n − 1 if X1 ≥lrX2 ≥lr ··· ≥lrXn, where ≤lr denotes the likelihood ratio order. Concerning the mean residual life order (≤mrl), it is shown that Xn−1:n−1 ≤mrlXn:n if Xj ≤mrlXn for j = 1,..., n − 1. Two counterexamples are also given to illustrate that Xk−1:n−1 ≤mrlXk:n in this case is, in general, not true for k < n.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 15 , Issue 2 , April 2001 , pp. 259 - 272
Copyright
© 2001 Cambridge University Press

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