## Variability of total claim amounts under dependence between claims severity and number of events.(English)Zbl 1100.60005

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.

### MSC:

 60E15 Inequalities; stochastic orderings 62P05 Applications of statistics to actuarial sciences and financial mathematics 62E10 Characterization and structure theory of statistical distributions
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