On the powers of polynomial logistic distributions

Abstract

Let $P(x)$ be a polynomial monotone increasing on $(-\infty,+\infty)$. The probability distribution possessing the distribution function

\[F(x)=\frac{1}{1+\exp\{-P(x)\}}\] is called the polynomial logistic distribution associated with polynomial $P$ and denoted by PL($P$). It has recently been introduced, as a generalization of the logistic distribution, by V. M. Koutras, K. Drakos, and M. V. Koutras who have also demonstrated the importance of this distribution in modeling financial data. In the present paper, for a random variable $X\sim\mathrm{PL}(P)$, the analytical properties of its characteristic function are examined, the moment-(in)determinacy for the powers $X^{m},\;m\in\mathbb{N}$ and $|X|^{p},\;p\in(0,+\infty)$ depending on the values of $m$ and $p$ is investigated, and exemplary Stieltjes classes for the moment-indeterminate powers of $X$ are constructed.