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ON STOCHASTIC COMPARISONS OF ORDER STATISTICS FROM HETEROGENEOUS EXPONENTIAL SAMPLES

Published online by Cambridge University Press:  29 October 2019

Yaming Yu Open the ORCID record for Yaming Yu [Opens in a new window]
Yaming Yu*
Affiliation:
Department of Statistics, University of California, Irvine, CA 92697, USA E-mail: yamingy@uci.edu
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Abstract

We show that the kth order statistic from a heterogeneous sample of n ≥ k exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan [14]. As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea [11]. Extensions to general spacings are also presented.

Keywords

convolutiondispersive orderingexponential distributionmajorizationstochastic order
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 532 - 537
Copyright
© Cambridge University Press 2019

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References

1Balakrishnan, N. & Zhao, P. (2013). Ordering properties of order statistics from heterogeneous populations: A review with an emphasis on some recent developments. Probability in the Engineering and Informational Sciences 27: 403–443.CrossRefGoogle Scholar
2Bon, J.-L. & Pǎltǎnea, E. (2006). Comparison of order statistics in a random sequence to the same statistics with iid variables. ESAIM: Probability and Statistics 10: 1–10.CrossRefGoogle Scholar
3Da, G., Xu, M., & Balakrishnan, N. (2014). On the Lorenz ordering of order statistics from exponential populations and some applications. Journal of Multivariate Analysis 127: 88–97.CrossRefGoogle Scholar
4Diaconis, P. & Perlman, M.D. (1990). Bounds for tail probabilities of weighted sums of independent gamma random variables. IMS Lecture Notes–Monograph Series 16: 147–166.CrossRefGoogle Scholar
5Genest, C., Kochar, S.C., & Xu, M. (2009). On the range of heterogeneous samples. Journal of Multivariate Analysis 100: 1587–1592.CrossRefGoogle Scholar
6Karlin, S. (1968). Total positivity. Stanford: Stanford University Press.Google Scholar
7Khaledi, B.-E. & Kochar, S. (2000). Some new results on stochastic comparisons on parallel systems. Journal of Applied Probability 37: 1123–1128.CrossRefGoogle Scholar
8Kochar, S.C. & Xu, M. (2009). Comparisons of parallel systems according to the convex transform order. Journal of Applied Probability 46: 342–352.CrossRefGoogle Scholar
9Kochar, S.C. & Xu, M. (2011). On the skewness of order statistics in the multiple-outlier models. Journal of Applied Probability 48: 271–284.CrossRefGoogle Scholar
10Marshall, A.W., Olkin, I., & Arnold, B. (2009). Inequalities: Theory of majorization and its applications (2nd ed.). New York: Springer.Google Scholar
11Pǎltǎnea, E. (2008). On the comparison in hazard rate ordering of fail-safe systems. Journal of Statistical Planning and Inference 138: 1993–1997.CrossRefGoogle Scholar
12Pǎltǎnea, E. (2011). Bounds for mixtures of order statistics from exponentials and applications. Journal of Multivariate Analysis 102: 896–907.CrossRefGoogle Scholar
13Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
14Xu, M. & Balakrishnan, N. (2012). On the sample ranges from heterogeneous exponential variables. Journal of Multivariate Analysis 109: 1–9.CrossRefGoogle Scholar
15Yu, Y. (2009). Stochastic ordering of exponential family distributions and their mixtures. Journal of Applied Probability 46: 244–254.CrossRefGoogle Scholar
16Yu, Y. (2011). Some stochastic inequalities for weighted sums. Bernoulli 17: 1044–1053.CrossRefGoogle Scholar
17Yu, Y. (2017). On the unique crossing conjecture of Diaconis and Perlman on convolutions of gamma random variables. Annals of Applied Probability 27: 3893–3910.CrossRefGoogle Scholar
18Zhao, P., Li, X., & Balakrishnan, N. (2009). Likelihood ratio order of the second order statistic from independent heterogeneous exponential random variables. Journal of Multivariate Analysis 100: 952–962.CrossRefGoogle Scholar