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Stochastic comparisons of series systems with heterogeneous Weibull components

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  • Fang, Longxiang
  • Zhang, Xinsheng

Abstract

Let X1,…,Xn be independent random variables with Xi∼W(α,λi), where W(α,λi) denotes a Weibull distribution with shape parameter α and scale parameter λi, i=1,…,n. Let Y1,…,Yn be a random sample of size n from a Weibull distribution with shape parameter α and a common scale parameter λ. Firstly, we prove that the smallest order statistic X1:n is greater than the smallest order statistic Y1:n according to the convex transform order. Secondly, we prove that λ≥(1n∑i=1nλiα)1α implies Y1:n≤dispX1:n; and λ=(∏i=1nλi)1n implies X1:n≤rhY1:n. Let X1∗,…,Xn∗ be independent random variables with Xi∗∼W(α,λi∗),i=1,…,n. Then (λ1∗,…,λn∗)≤m(λ1,…,λn) implies that X1:n≤rhX1:n∗ for α>1 and X1:n∗≤rhX1:n for 0<α≤1.

Suggested Citation

  • Fang, Longxiang & Zhang, Xinsheng, 2013. "Stochastic comparisons of series systems with heterogeneous Weibull components," Statistics & Probability Letters, Elsevier, vol. 83(7), pages 1649-1653.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:7:p:1649-1653
    DOI: 10.1016/j.spl.2013.03.012
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    1. Longxiang Fang & N. Balakrishnan, 2016. "Likelihood ratio order of parallel systems with heterogeneous Weibull components," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(6), pages 693-703, August.
    2. Barmalzan, Ghobad & Payandeh Najafabadi, Amir T. & Balakrishnan, Narayanaswamy, 2016. "Likelihood ratio and dispersive orders for smallest order statistics and smallest claim amounts from heterogeneous Weibull sample," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 1-7.
    3. Mesfioui Mhamed & Trufin Julien, 2021. "Dispersive order comparisons on extreme order statistics from homogeneous dependent random vectors," Dependence Modeling, De Gruyter, vol. 9(1), pages 385-393, January.
    4. Li, Chen & Li, Xiaohu, 2015. "Likelihood ratio order of sample minimum from heterogeneous Weibull random variables," Statistics & Probability Letters, Elsevier, vol. 97(C), pages 46-53.

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