Characterizations of arcsin and related distributions based on a new generalized unimodality

Abstract

A general characterization for α-unimodal distributions was provided by Alamatsaz (1985) who later introduced a multivariate extension of them (Alamatsaz 1993). Here, by solving the related equations, another generalization for unimodality is presented. As a result of this generalization, a simpler proof of a conjecture, as well as a characterization for generalized arcsin distributions and some generalizations of the author's earlier works, have been obtained. Last, but not the least, it is shown that some elementary methods can be more powerful than some more advanced techniques.

Randomly Weighted Averages: A Multivariate Case

“…The inner product of two random vectors was introduced in Homei (2014) and the exact distribution of this product was investigated for some random vectors with Beta and Dirichlet distributions. In this paper a new generalization for the inner product of two random vectors is introduced.…”

“…) identifies the Stieltjes transformation of the distribution of Z. Alternatively, from Theorem 2.1 (or Corollary 2.2) the distribution of Z is power semicircle (seeHomei (2014) for more details). Since the Stieltjes transformation of the power semicircle distributionis n−2 2 (1−t) (n−3)/2 (z 2 −t) −1/2 dt (see e.g.…”