# Stochastic comparisons of the largest claim amounts from two sets of interdependent heterogeneous portfolios

## TL;DR

This paper compares the largest claim amounts from two interdependent portfolios using stochastic orders, providing theoretical results and applications in actuarial models to understand risk differences.

## Contribution

It introduces new stochastic comparison results for the maximum claim amounts of interdependent portfolios with different parameters, extending existing actuarial risk models.

## Key findings

- Established stochastic orderings for largest claim amounts between portfolios.
- Provided conditions under which one portfolio's maximum claim exceeds another's.
- Applied theoretical results to practical actuarial models.

## Abstract

Let $ X_{\lambda_1},\ldots,X_{\lambda_n}$ be dependent non-negative random variables and $Y_i=I_{p_i} X_{\lambda_i}$, $i=1,\ldots,n$, where $I_{p_1},\ldots,I_{p_n}$ are 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. In this paper, we compare the largest claim amounts of two sets of interdependent portfolios, in the sense of usual stochastic order, 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 $p^*_1,\ldots,p^*_n$. For illustration, we apply the results to some important models in actuary.