Variability of total claim amounts under dependence between claims severity and number of events (Q2499826)

Let \(\chi\subseteq\mathbb{R}^2\). For each \((\theta_0,\theta_1)\in\chi\), let \(N_i(\theta_0)\) be a nonnegative integer-valued random variable, and let \(\mathbf{X}_i(\theta_1)=\{X_{ik}(\theta_1),\;k\in\mathbb{N}\}\), be a sequence of nonnegative random variables, \(i=1,2,\ldots,n\). Assume that \(N_1(\theta_0),N_2(\theta_0),\ldots,N_n(\theta_0)\), \(\mathbf{X}_1(\theta_1),\mathbf{X}_2(\theta_1),\ldots,\mathbf{X}_n(\theta_1)\) are mutually independent. Now, let \((\Theta_{1,0},\Theta_{1,1})\) and \((\Theta_{2,0},\Theta_{2,1})\) be two random vectors taking on values in \(\chi\). The authors are interested in the vectors of random sums \[ (Z_1(\Theta_{j,0},\Theta_{j,1}),\;Z_2(\Theta_{j,0},\Theta_{j,1}),\ldots,Z_n(\Theta_{j,0},\Theta_{j,1})),\quad j=1,2, \] where \[ Z_i(\Theta_{j,0},\Theta_{j,1})=\sum_{k=1}^{N_i(\Theta_{j,0})}X_{ik}(\Theta_{j,1}),\quad i=1,2,\ldots,n,\quad j=1,2. \] They show that if \((\Theta_{1,0},\Theta_{1,1})\) and \((\Theta_{2,0},\Theta_{2,1})\) are ordered according to some positive dependence stochastic order, and if the \(X_{ik}(\theta_1)\)'s satisfy some stochastic monotonicity properties, then the vectors \((Z_1(\Theta_{j,0},\Theta_{j,1}),Z_2(\Theta_{j,0},\Theta_{j,1}),\ldots,Z_n(\Theta_{j,0},\Theta_{j,1}))\), \(j=1,2\), are ordered with respect to some stochastic orders. Some applications in insurance are described.