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Inequalities and bounds for expected order statistics from transform-ordered families

Part of: Distribution theory - Probability Nonparametric inference

Published online by Cambridge University Press:  08 April 2025

Idir Arab ,
Tommaso Lando  and
Paulo Eduardo Oliveira Open the ORCID record for Paulo Eduardo Oliveira [Opens in a new window]
Idir Arab*
Affiliation:
University of Coimbra
Tommaso Lando*
Affiliation:
University Bergamo
Paulo Eduardo Oliveira*
Affiliation:
University of Coimbra
*
*Postal address: Centro de Matemática da Universidade de Coimbra, Department of Mathematics, University of Coimbra, Portugal.
***Postal address: Department of Economics, University of Bergamo, Italy. Email: [email protected]
*Postal address: Centro de Matemática da Universidade de Coimbra, Department of Mathematics, University of Coimbra, Portugal.
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Abstract

We introduce a comprehensive method for establishing stochastic orders among order statistics in the independent and identically distributed case. This approach relies on the assumption that the underlying distribution is linked to a reference distribution through a transform order. Notably, this method exhibits broad applicability, particularly since several well-known nonparametric distribution families can be defined using relevant transform orders, including the convex and the star transform orders. Moreover, for convex-ordered families, we show that an application of Jensen’s inequality gives bounds for the probability that a random variable exceeds the expected value of its corresponding order statistic.

Keywords

Stochastic ordershazard rateoddsconvexstar-shapedexceedance probability

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62G30: Order statistics; empirical distribution functions 62G15: Tolerance and confidence regions
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 19
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Applied Probability Trust

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