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Zeitschrift für Operations Research

, Volume 31, Issue 1, pp A1–A13 | Cite as

Ordering distributions by scaled order statistics

  • M. Scarsini
  • M. Shaked
Article

Abstract

Motivated by applications in reliability theory, we define a preordering (X1, ...,Xn)\(\mathop \lesssim \limits^{{\text{(}}k)}\) (Y1 ...,Yn) of nonnegative random vectors by requiring thek-th order statistic ofa1X1,..., anXn to be stochastically smaller than thek-th order statistic ofa1Y1, ...,anYn for all choices ofai>0,i=1, 2, ...,n. We identify a class of functionsMk, n such that\(X\mathop \lesssim \limits^{{\text{(}}k)} Y\) if and only ifEφ(X)⩽Eφ(Y) for allφεMk,n. Some preservation results related to the ordering\(\mathop \lesssim \limits^{{\text{(}}k)}\) are obtained. Some applications of the results in reliability theory are given.

Key words and phrases

Stochastic orderings order statistics 

Zusammenfassung

Motiviert durch Anwendungen in der Zuverlässigkeitstheorie wird eine Prä Ordnung (X1, ...,Xn)\(\mathop \lesssim \limits^{{\text{(}}k)}\) (Y1, ...,Yn) auf nichtnegativen Zufallsvektoren dadurch definiert, daß gefordert wird, daß für jede Wahl vonai>0,i=1, 2, ...,n diek-te Ranggröße vona1X1, ...,anXn stochastisch kleiner als diek- te Ranggröße vona1Y1, ...,anYn ist. Es wird eine Klasse von FunktionenMk,n beschrieben, so daß\(X\mathop \lesssim \limits^{{\text{(}}k)} Y\) genau dann gilt, wennEφ(X)⩽Eφ(Y) für alleφ εMk,n.

Für die Ordnung\(\mathop \lesssim \limits^{{\text{(}}k)}\) werden einige Erhaltungsgesetze hergeleitet. Ferner werden einige Anwendungen der Resultate in der Zuverlässigkeitstheorie angegeben.

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Copyright information

© Physica-Verlag 1987

Authors and Affiliations

  • M. Scarsini
    • 1
  • M. Shaked
    • 2
  1. 1.Istituto di Matematica FinanzieriaUniversità di ParmaItaly
  2. 2.Department of MathematicsUniversity of ArizonaUSA

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