Stochastic orderings between distributions and their sample spacings  II
Let X1:n[lessthanorequals, slant]X2:n[lessthanorequals, slant]...[lessthanorequals, slant]Xn:n denote the order statistics of a random sample of size n from a probability distribution with distribution function F. Similarly, let Y1:m[lessthanorequals, slant]Y2:m[lessthanorequals, slant]...[lessthanorequals, slant]Ym:m denote the order statistics of an independent random sample of size m from another distribution with distribution function G. We assume that F and G are absolutely continuous with common support (0,[infinity]). The corresponding normalized spacings are defined by Ui:n[reverse not equivalent](ni+1)(Xi:nXi1:n) and Vj:m[reverse not equivalent](mj+1)(Yj:mYj1:m), for i=1,...,n and j=1,...,m, where X0:n=Y0:n[reverse not equivalent]0. It is proved that if X is smaller than Y in the hazard rate order sense and if either F or G is a decreasing failure rate (DFR) distribution, then Ui:n is stochastically smaller than Vj:m for i[lessthanorequals, slant]j and ni[greaterorequal, slanted]mj. If instead, we assume that X is smaller than Y in the likelihood ratio order and if either F or G is DFR, then this result can be strengthened from stochastic ordering to hazard rate ordering. Finally, under a stronger assumption on the shapes of the distributions that either F or G has logconvex density, it is proved that X being smaller than Y in the likelihood ratio order implies that Ui:n is smaller than Vj:m in the sense of likelihood ratio ordering for i[lessthanorequals, slant]j and ni=mj.
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Volume (Year): 44 (1999)
Issue (Month): 2 (August)
Pages: 161166
Handle:  RePEc:eee:stapro:v:44:y:1999:i:2:p:161166 
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