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Ordering results for smallest claim amounts from two portfolios of risks with dependent heterogeneous exponentiated location-scale claims

Part of: Distribution theory - Probability Special processes Nonparametric inference Operations research and management science

Published online by Cambridge University Press:  26 July 2021

Sangita Das Open the ORCID record for Sangita Das [Opens in a new window] ,
Suchandan Kayal Open the ORCID record for Suchandan Kayal [Opens in a new window]  and
N. Balakrishnan
Sangita Das
Affiliation:
Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India. E-mails: [email protected], [email protected], [email protected]
Suchandan Kayal
Affiliation:
Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India. E-mails: [email protected], [email protected], [email protected]
N. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. E-mail: [email protected]
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Abstract

Let $\{Y_{1},\ldots ,Y_{n}\}$ be a collection of interdependent nonnegative random variables, with $Y_{i}$ having an exponentiated location-scale model with location parameter $\mu _i$, scale parameter $\delta _i$ and shape (skewness) parameter $\beta _i$, for $i\in \mathbb {I}_{n}=\{1,\ldots ,n\}$. Furthermore, let $\{L_1^{*},\ldots ,L_n^{*}\}$ be a set of independent Bernoulli random variables, independently of $Y_{i}$'s, with $E(L_{i}^{*})=p_{i}^{*}$, for $i\in \mathbb {I}_{n}.$ Under this setup, the portfolio of risks is the collection $\{T_{1}^{*}=L_{1}^{*}Y_{1},\ldots ,T_{n}^{*}=L_{n}^{*}Y_{n}\}$, wherein $T_{i}^{*}=L_{i}^{*}Y_{i}$ represents the $i$th claim amount. This article then presents several sufficient conditions, under which the smallest claim amounts are compared in terms of the usual stochastic and hazard rate orders. The comparison results are obtained when the dependence structure among the claim severities are modeled by (i) an Archimedean survival copula and (ii) a general survival copula. Several examples are also presented to illustrate the established results.

Keywords

Archimedean survival copulaExponentiated location-scale modelGeneral survival copulaHazard rate orderMajorization ordersSmallest claim amounts

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 62G30: Order statistics; empirical distribution functions 60K10: Applications (reliability, demand theory, etc.) 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 1116 - 1137
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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