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On the range of heterogeneous samples

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  • Genest, Christian
  • Kochar, Subhash C.
  • Xu, Maochao

Abstract

Let Rn be the range of a random sample X1,...,Xn of exponential random variables with hazard rate [lambda]. Let Sn be the range of another collection Y1,...,Yn of mutually independent exponential random variables with hazard rates [lambda]1,...,[lambda]n whose average is [lambda]. Finally, let r and s denote the reversed hazard rates of Rn and Sn, respectively. It is shown here that the mapping t|->s(t)/r(t) is increasing on (0,[infinity]) and that as a result, Rn=X(n)-X(1) is smaller than Sn=Y(n)-Y(1) in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X(n) is seen to be more stochastically increasing in X(1) than Y(n) is in Y(1). In other words, the pair (X(1),X(n)) is more dependent than the pair (Y(1),Y(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X1,...,Xn form a random sample from a continuous distribution while Y1,...,Yn are mutually independent lifetimes with proportional hazard rates.

Suggested Citation

  • Genest, Christian & Kochar, Subhash C. & Xu, Maochao, 2009. "On the range of heterogeneous samples," Journal of Multivariate Analysis, Elsevier, vol. 100(8), pages 1587-1592, September.
  • Handle: RePEc:eee:jmvana:v:100:y:2009:i:8:p:1587-1592
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    References listed on IDEAS

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    1. Marco Scarsini, 1984. "On measures of concordance," Post-Print hal-00542380, HAL.
    2. Dolati, Ali & Genest, Christian & Kochar, Subhash C., 2008. "On the dependence between the extreme order statistics in the proportional hazards model," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 777-786, May.
    3. Avérous, Jean & Genest, Christian & C. Kochar, Subhash, 2005. "On the dependence structure of order statistics," Journal of Multivariate Analysis, Elsevier, vol. 94(1), pages 159-171, May.
    4. Kochar, Subhash C & Korwar, Ramesh, 1996. "Stochastic Orders for Spacings of Heterogeneous Exponential Random Variables," Journal of Multivariate Analysis, Elsevier, vol. 57(1), pages 69-83, April.
    5. Kochar, Subhash & Rojo, Javier, 1996. "Some New Results on Stochastic Comparisons of Spacings from Heterogeneous Exponential Distributions," Journal of Multivariate Analysis, Elsevier, vol. 59(2), pages 272-281, November.
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    Cited by:

    1. Zhao, Peng & Zhang, Yiying, 2012. "On sample ranges in multiple-outlier models," Journal of Multivariate Analysis, Elsevier, vol. 111(C), pages 335-349.
    2. Ding, Weiyong & Da, Gaofeng & Zhao, Peng, 2013. "On sample ranges from two sets of heterogenous random variables," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 63-73.
    3. Xu, Maochao & Balakrishnan, N., 2012. "On the sample ranges from heterogeneous exponential variables," Journal of Multivariate Analysis, Elsevier, vol. 109(C), pages 1-9.
    4. Antonia Castaño-Martínez & Gema Pigueiras & Miguel A. Sordo, 2021. "On the Increasing Convex Order of Relative Spacings of Order Statistics," Mathematics, MDPI, vol. 9(6), pages 1-12, March.

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