Modern Stochastics: Theory and Applications

The stochastic literature contains several extensions of the exponential distribution which increase its applicability and flexibility. In the present article, some properties of a new power modified exponential family with an original Kies correction are discussed. This family is defined as a Kies distribution which domain is transformed by another Kies distribution. Its probabilistic properties are investigated and some limitations for the saturation in the Hausdorff sense are derived. Moreover, a formula of a semiclosed form is obtained for this saturation. Also the tail behavior of these distributions is examined considering three different criteria inspired by the financial markets, namely, the VaR, AVaR, and expectile based VaR. Some numerical experiments are provided, too.

Let $\{\xi _{1},\xi _{2},\dots \}$ be a sequence of independent random variables, and η be a counting random variable independent of this sequence. We consider conditions for $\{\xi _{1},\xi _{2},\dots \}$ and η under which the distribution function of the random sum $S_{\eta }=\xi _{1}+\xi _{2}+\cdots +\xi _{\eta }$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi _{1},\xi _{2},\dots \}$ are not necessarily identically distributed.