Ordering results for extreme claim amounts based on random number of claims

TL;DR

This paper establishes new stochastic inequalities for the minimum and maximum of random claim amounts in portfolios with random sizes, extending existing results and applying to reliability and auction theories.

Contribution

It introduces generalized stochastic inequalities involving minima and maxima of random claim sizes with random portfolio sizes, broadening prior theoretical frameworks.

Findings

Derived inequalities under stochastic and hazard rate orders

Generalized previous results on claim size orderings

Provided numerical examples and applications in reliability and auction theory

Abstract

Consider two sequences of heterogeneous and independent portfolios of risks $T_1,T_2,\ldots$ and $T^*_{1}, T^*_{2},\ldots$ and, let $N_1$ and $N_2$ be two positive integer-valued random variables, independent of $T_i'$ and $T^*_i$, respectively. In this article, we investigate different stochastic inequalities involving $\min\{T_1,\ldots,T_{N_1}\}$ and $\min\{T^*_1,\ldots,T^*_{N_2}\},$ and $\max\{T_1,\ldots,T_{N_1}\}$ and $\max\{T^*_1,\ldots,T^*_{N_2}\}$ in the sense of usual stochastic order and reversed hazard rate order concerning maltivariate chain majorization order. These new results strengthen and generalize some of the well known results in the literature, including \cite{barmalzan2017ordering}, \cite{balakrishnan2018} and \cite{kundu2021_shock} for the case of random claim sizes. Different numerical examples are provided to highlight the applicability of this work. Finally, some interesting applications of our results in reliability theory and auction theory are presented.