Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T12:51:58.168Z Has data issue: false hasContentIssue false
  • English
  • Français

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: sangitadas118@gmail.com, suchandan.kayal@gmail.com, kayals@nitrkl.ac.in
Suchandan Kayal
Affiliation:
Department of Mathematics, National Institute of Technology Rourkela, Rourkela 769008, India. E-mails: sangitadas118@gmail.com, suchandan.kayal@gmail.com, kayals@nitrkl.ac.in
N. Balakrishnan
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1. E-mail: bala@mcmaster.ca
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Balakrishnan, N., Zhang, Y., & Zhao, P. (2017). Ordering the largest claim amounts and ranges from two sets of heterogeneous portfolios. Scandinavian Actuarial Journal 2018(1): 23–41.CrossRefGoogle Scholar
Barmalzan, G., Akrami, A., & Balakrishnan, N. (2020). Stochastic comparisons of the smallest and largest claim amounts with location-scale claim severities. Insurance: Mathematics and Economics 93: 341–352.Google Scholar
Barmalzan, G. & Payandeh Najafabadi, A.T. (2015). On the convex transform and right-spread orders of smallest claim amounts. Insurance: Mathematics and Economics 64: 380–384.Google Scholar
Barmalzan, G., Payandeh Najafabadi, A.T., & Balakrishnan, N. (2016). Likelihood ratio and dispersive orders for smallest order statistics and smallest claim amounts from heterogeneous Weibull sample. Statistics & Probability Letters 110: 1–7.CrossRefGoogle Scholar
Barmalzan, G., Payandeh Najafabadi, A.T., & Balakrishnan, N. (2017). Ordering properties of the smallest and largest claim amounts in a general scale model. Scandinavian Actuarial Journal 2017(2): 105–124.CrossRefGoogle Scholar
Castillo, E., Hadi, A.S., Balakrishnan, N., & Sarabia, J.M. (2005). Extreme value and related models with applications in engineering and science. Hoboken, NJ: John Wiley & Sons.Google Scholar
Das, S. & Kayal, S. (2021). Ordering extremes of exponentiated location-scale models with dependent and heterogeneous random samples. Metrika 83(8): 869–893.CrossRefGoogle Scholar
Das, S., Kayal, S., & Balakrishnan, N. (2020). Orderings of the smallest claim amounts from exponentiated location-scale models. Methodology and Computing in Applied Probability. 1–29. doi:10.1007/s11009-020-09793-y.Google Scholar
Das, S., Kayal, S., & Choudhuri, D. (2021). Ordering results on extremes of exponentiated location-scale models. Probability in the Engineering and Informational Sciences 35(2): 331–354.CrossRefGoogle Scholar
Dolati, A. & Dehgan Nezhad, A. (2014). Some results on convexity and concavity of multivariate copulas. Iranian Journal of Mathematical Sciences and Informatics 9(2): 87–100, 129.Google Scholar
Kundu, A., Chowdhury, S., Nanda, A.K., & Hazra, N.K. (2016). Some results on majorization and their applications. Journal of Computational and Applied Mathematics 301: 161–177.CrossRefGoogle Scholar
Li, H. & Li, X. (eds) (2013). Stochastic orders in reliability and risk. New York: Springer.CrossRefGoogle Scholar
Li, X. & Fang, R. (2015). Ordering properties of order statistics from random variables of Archimedean copulas with applications. Journal of Multivariate Analysis 133: 304–320.CrossRefGoogle Scholar
Marshall, A.W., Olkin, I., & Arnold, B.C. (2011). Inequalities: theory of majorization and its applications, 2nd ed. New York: Springer.CrossRefGoogle Scholar
McNeil, A.J. & Nešlehová, J. (2009). Multivariate Archimedean copulas, $d$-monotone functions and $\ell ^{1}$-norm symmetric distributions. The Annals of Statistics 37: 3059–3097.CrossRefGoogle Scholar
Müller, A. & Stoyan, D. (2002). Comparison methods for stochastic models and risks. Chichester, UK: John Wiley & Sons.Google Scholar
Nadeb, H., Torabi, H., & Dolati, A. (2018). Ordering the smallest claim amounts from two sets of interdependent heterogeneous portfolios. preprint arXiv:1812.06166.Google Scholar
Nadeb, H., Torabi, H., & Dolati, A. (2020). Stochastic comparisons between the extreme claim amounts from two heterogeneous portfolios in the case of transmuted-G model. North American Actuarial Journal 24(3): 475–487.CrossRefGoogle Scholar
Nadeb, H., Torabi, H., & Dolati, A. (2020). Stochastic comparisons of the largest claim amounts from two sets of interdependent heterogeneous portfolios. Mathematical Inequalities & Applications 23(1): 35–56.CrossRefGoogle Scholar
Nelsen, R. (2006). An introduction to copulas, 2nd ed. New York: Springer.Google Scholar
Shaked, M. & Shanthikumar, J.G. (2007). Stochastic orders. New York: Springer.CrossRefGoogle Scholar
Torrado, N. & Navarro, J. (2020). Ranking the extreme claim amounts in dependent individual risk models. Scandinavian Actuarial Journal. 1–30. doi:10.1080/03461238.2020.1830845.Google Scholar
Zhang, Y., Cai, X., & Zhao, P. (2019). Ordering properties of extreme claim amounts from heterogeneous portfolios. ASTIN Bulletin 49(2): 525–554.CrossRefGoogle Scholar