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Stochastic ordering of order statistics. (English) Zbl 0729.62048

The main result of this paper is the following:
Theorem 1. Let \(X_ 1,...,X_ n\), \(Y_ 1,...,Y_ n\) be independent, non-negative random variables with distributions \[ P(X_ j>t)=\bar F(\lambda_ jt)\equiv \exp (-\int^{\lambda_ jt}_{0}r(s)ds),\quad P(Y_ j>t)=\bar F({\hat \lambda}t), \] where \(\lambda_ 1,...,\lambda_ n\) are positive constants with average \({\hat \lambda}\) and \(\bar F\) is the tail of the distribution function \(F\equiv 1-\bar F\), with failure-rate function r. Assume \( x\mapsto r(x)\) is decreasing; \(x\mapsto xr(x)\) is increasing. Then it is possible to construct on some probability space random variables \[ (X'_ 1,...,X'_ n)=^{{\mathcal D}}(X_ 1,...,X_ n)\text{ and } (Y'_ 1,...,Y'_ n)=^{{\mathcal D}}(Y_ 1,...,Y_ n) \] such that \(Y'_{(j)}\leq X'_{(j)}\) for \(j=1,...,n\) a.s.

MSC:

62G30 Order statistics; empirical distribution functions
62P10 Applications of statistics to biology and medical sciences; meta analysis
92D30 Epidemiology
60E99 Distribution theory
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