APPLICATIONS OF THE HAZARD RATE ORDERING IN RELIABILITY AND ORDER-STATISTICS

被引:104
作者
BOLAND, PJ
ELNEWEIHI, E
PROSCHAN, F
机构
[1] UNIV ILLINOIS,DEPT MATH,CHICAGO,IL 60680
[2] FLORIDA STATE UNIV,DEPT STAT,TALLAHASSEE,FL 32306
关键词
STOCHASTIC ORDER; HAZARD RATE ORDER; LIKELIHOOD RATIO ORDER; COHERENT SYSTEMS; K-OUT-OF-N SYSTEMS; PROPORTIONAL HAZARD RATES;
D O I
10.2307/3215245
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T1, ..., T(n)) to the vector (T1', ..., T(n)'), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the caw for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (lambda1r(t), lambda2r(t)), the more diverse (lambda1, lambda2) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if tau(k/n) is the lifetime of a k-out-of-n system, then tau(k/n) is greater than tau(k+1/n) in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T1, ..., T(n), we let T(k:n) represent the kth order statistic (in increasing order). Then it is shown that T(k+1:n) is greater than T(k:n) in the hazard rate ordering for all k = 1, ..., n - 1. The result does not, however, extend to the stronger likelihood ratio order.
引用
收藏
页码:180 / 192
页数:13
相关论文
共 14 条
[1]  
Alzaid AA., 1988, STAT PAP, V29, P35, DOI [947506, DOI 10.1007/BF02924509]
[2]  
[Anonymous], 1996, STOCHASTIC PROCESSES
[3]   STOCHASTIC ORDER FOR REDUNDANCY ALLOCATIONS IN SERIES AND PARALLEL SYSTEMS [J].
BOLAND, PJ ;
ELNEWEIHI, E ;
PROSCHAN, F .
ADVANCES IN APPLIED PROBABILITY, 1992, 24 (01) :161-171
[4]   STATISTICAL-INFERENCE FOR UNIFORM STOCHASTIC ORDERING IN SEVERAL POPULATIONS [J].
DYKSTRA, R ;
KOCHAR, S ;
ROBERTSON, T .
ANNALS OF STATISTICS, 1991, 19 (02) :870-888
[5]   RELATIONSHIP BETWEEN SYSTEM FAILURE RATE AND COMPONENT FAILURE RATES [J].
ESARY, JD ;
PROSCHAN, F .
TECHNOMETRICS, 1963, 5 (02) :183-&
[6]  
Karlin S., 1968, TOTAL POSITIVITY
[7]  
Keilson J, 1982, CAN J STAT, V10, P181, DOI DOI 10.2307/3556181
[8]   UNIFORM STOCHASTIC ORDERINGS AND TOTAL POSITIVITY [J].
LYNCH, J ;
MIMMACK, G ;
PROSCHAN, F .
CANADIAN JOURNAL OF STATISTICS-REVUE CANADIENNE DE STATISTIQUE, 1987, 15 (01) :63-69
[9]  
Marshall A. W., 1979, INEQUALITIES THEORY, V143
[10]   SCHEDULING JOBS SUBJECT TO NON-HOMOGENEOUS POISSON SHOCKS [J].
PINEDO, ML ;
ROSS, SM .
MANAGEMENT SCIENCE, 1980, 26 (12) :1250-1257