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STOCHASTIC MONOTONICITY OF CONDITIONAL ORDER STATISTICS IN MULTIPLE-OUTLIER SCALE POPULATION

  • Ebrahim Amini-Seresht (a1) and Yiying Zhang (a2)

Abstract

This paper discusses the stochastic monotonicity property of the conditional order statistics from independent multiple-outlier scale variables in terms of the likelihood ratio order. Let X 1, …, X n be a set of non-negative independent random variables with X i , i=1, …, p, having common distribution function F1 x), and X j , j=p+1, …, n, having common distribution function F2 x), where F(·) denotes the baseline distribution. Let X i:n (p, q) be the ith smallest order statistics from this sample. Denote by $X_{i,n}^{s}(p,q)\doteq [X_{i:n}(p,q)|X_{i-1:n}(p,q)=s]$ . Under the assumptions that xf′(x)/f(x) is decreasing in x∈ℛ+, λ1≤λ2 and s 1s 2, it is shown that $X_{i:n}^{s_{1}}(p+k,q-k)$ is larger than $X_{i:n}^{s_{2}}(p,q)$ according to the likelihood ratio order for any 2≤in and k=1, 2, …, q. Some parametric families of distributions are also provided to illustrate the theoretical results.

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