Ordering the smallest claim amounts from two sets of interdependent heterogeneous portfolios

TL;DR

This paper compares the smallest claim amounts across two interdependent portfolios using stochastic orders, providing bounds for their survival functions and illustrating results with applicable models.

Contribution

It introduces new stochastic ordering results for smallest claim amounts in interdependent portfolios with heterogeneous parameters.

Findings

Establishes conditions for usual and likelihood ratio orderings.

Provides bounds for the survival function of the smallest claim.

Validates results with practical actuarial models.

Abstract

Let $ X_{\lambda_1},\ldots,X_{\lambda_n}$ be a set of dependent and non-negative random variables share a survival copula and let $Y_i= I_{p_i}X_{\lambda_i}$, $i=1,\ldots,n$, where $I_{p_1},\ldots,I_{p_n}$ be independent Bernoulli random variables independent of $X_{\lambda_i}$'s, with ${\rm E}[I_{p_i}]=p_i$, $i=1,\ldots,n$. In actuarial sciences, $Y_i$ corresponds to the claim amount in a portfolio of risks. This paper considers comparing the smallest claim amounts from two sets of interdependent portfolios, in the sense of usual and likelihood ratio orders, when the variables in one set have the parameters $\lambda_1,\ldots,\lambda_n$ and $p_1,\ldots,p_n$ and the variables in the other set have the parameters $\lambda^{*}_1,\ldots,\lambda^{*}_n$ and $p^*_1,\ldots,p^*_n$. Also, we present some bounds for survival function of the smallest claim amount in a portfolio. To illustrate validity of the results, we serve some applicable models.