Title: Characterizations of the Beta Distributions via Some Regression Assumptions

Author(s): Wen-Jang Huang and Yan-Hau
Issue: Volume 70 Series A Part 1 Year 2008
Pages: 73 -- 87
Let $X$ and $Y$ be two independent non-degenerate random variables. Also let $(U, V)$ be a bijective map of $(X, Y)$. It is desired to use certain regression assumptions between $U$ and $V$ to characterize the distributions of $X$ and $Y$, and consequently, the distribution of $(U, V)$. In most of the previous investigations, $U$ and $V$ turn out to be independent too. Recently, for $X, Y$ valued in $(0, 1)$, Seshadri and Wesolowski (2003) characterize $X$ and $Y$ to be beta distributed based on two constancy of regression assumptions between $U$ and $V$, where $(U, V)$ is a particular bijective map of $(X, Y)$. In this work, first we will generalize the results in Seshadri and Wesolowski (2003). It will be proved that for the bijective map given in Seshadri and Wesolowski (2003), $X$ and$Y$ are beta distributed under some more general regression assumptions. Next we illustrate that for some other special bijective maps $(U, V)$, under certain regression assumptions between $U$ and $V$, $X$ and $Y$ can also be characterized to be beta distributed, yet $U$ and $V$ may not be independent.
AMS (2000) subject classification. Primary 60E05; secondary 62E10.
Keywords and phrases: Beta distribution, characterization, conditional ex- pectation, distribution theory, regression assumption.