Distribution and asymptotics under beta random scaling

TL;DR

This paper explores the distributional properties and tail asymptotics of a random variable scaled by a Beta distribution, providing recursive formulas and applications to extremes and elliptical distributions.

Contribution

It derives a recursive formula for the distribution of Y from the distribution of X scaled by a Beta variable and investigates tail asymptotics relating X and Y.

Findings

Recursive formula for H based on H_{a,b}

Relations between tail behaviors of X and Y

Applications to extremes and elliptical distributions

Abstract

Let X,Y,B be three independent random variables such that $X$ has the same distribution function as Y B. Assume that B is a Beta random variable with positive parameters a,b and Y has distribution function H. Pakes and Navarro (2007) show under some mild conditions that the distribution function H_{a,b} of X determines H. Based on that result we derive in this paper a recursive formula for calculation of H, if H_{a,b} is known. Furthermore, we investigate the relation between the tail asymptotic behaviour of X and Y. We present three applications of our asymptotic results concerning the extremes of two random samples with underlying distribution functions H and H_{a,b}, respectively, and the conditional limiting distribution of bivariate elliptical distributions.