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Stochastic Comparisons of Order Statistics

In: Stochastic Comparisons with Applications

Author

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  • Subhash C. Kochar

    (Portland State University, Fariborz Maseeh Department of Mathematics and Statistics)

Abstract

Order statistics and record values are compared according to various stochastic orders which are discussed in the previous chapters. It is shown that whereas order statistics are always increasing according to the stochastic and the hazard rate orders, that may not be the case with the likelihood ratio order when the observations are not identically distributed. Sufficient conditions for such a result to hold are identified. The case of order statistics when sampling without replacement (SRSWOR) from a finite population is also considered. Order statistics are also compared according to the dispersive as well as the right spread orders. It is shown that in the case of a random sample from a decreasing failure rate (DFR) distribution, the successive order statistics are ordered according to the dispersive order. They are also compared according to convex transform, star order, Lorenz, and other skewness orders. Conditions on the shapes of the parent distributions are obtained under which order statistics are ordered according to the star and the Lorenz orders. We also consider the properties of the order statistics case when the observations follows the proportional hazard rate (PHR) model and examine the effect of changes in the proportionality parameters on their stochastic properties. The theory of majorization and Schur convexity is used for making these comparisons. Bounds on the moments, the hazard rate and the survival functions of order statistics are obtained. Stochastic comparisons of order statistics from two distributions differing in their scale parameters is discussed in detail. The topic of dependence among order statistics is discussed in Chap. 8 .

Suggested Citation

  • Subhash C. Kochar, 2022. "Stochastic Comparisons of Order Statistics," Springer Books, in: Stochastic Comparisons with Applications, chapter 0, pages 139-174, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-12104-3_6
    DOI: 10.1007/978-3-031-12104-3_6
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