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

Sangita Das

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.

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

Abstract

Consider two sequences of heterogeneous and independent portfolios of risks and and, let and be two positive integer-valued random variables, independent of and , respectively. In this article, we investigate different stochastic inequalities involving and and and 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.

Paper Structure

This paper contains 7 sections, 18 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Ordering results
  4. Application
  5. Reliability theory
  6. Auction Theory
  7. Conclusion

Key Result

Lemma 2.1

(Marshall2011) Consider the real-valued continuously differentiable function $\phi$ on $J^n,$ where $J\subseteq\mathcal{R}$ is an open interval. Then, $\phi$ is Schur-convex $(\text{Schur-concave})$ on $J^n$ if and only if $\phi$ is symmetric on $J^n$, and for all $i\neq j$ and all $\boldsymbol{u}\i where $\frac{\partial\phi(\boldsymbol{u})}{\partial u_i}$ denotes the partial derivative of $\phi(\

Figures (2)

  • Figure 1: (a) Plot of $\bar{F}_{T_{{N_1}:{N_{1}}}}(x)-\bar{F}_{T^{*}_{{N_2}:{N_{2}}}}(x)$ as in Example \ref{['ex3.1']}. (b) Plot of ${F}_{X_{{N_1}:{N_{1}}}}(x)-{F}_{Y_{{N_2}:{N_{2}}}}(x)$ as in Counterexample \ref{['cex3.1']}.
  • Figure 2: (a) Plot of ${F}_{T_{{N_1}:{N_{1}}}}(x)/{F}_{T^{*}_{{N_2}:{N_{2}}}}(x)$ as in Counterexample \ref{['ex3.3']}. (b) Plot of $\bar{F}_{X_{1:{N_{1}}}}(x)-\bar{F}_{Y_{1:{N_{2}}}}(x)$ as in Counterexample \ref{['cex3.2']}.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.1
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 30 more