Some characterizations of unimodal distribution functions

Tailweight with respect to the mode for unimodal distributions

“…F is said to be unimodal whenever there is an a, called a mode of F, such that F is convex on (-c~,a) and concave on (a,~). Further let x' = inf{x: F(x) > 0} and x" = sup{x: F(x) < 1}; F is said to be strictly unimodal (Bertin et al, 1981) if it is unimodal with mode a and if it is strictly convex on (x',a) and strictly concave on (a,x"); here SF = [x',x"] is the support of F.…”

“…Let o// be the set of all unimodal continuous distribution functions; then F E q/ if and only if QF E ~ and AF(2) is an interval for each 2 E N+ (Bertin et al, 1981). IfF is strictly unimodal, then AF(2) is a singleton, say {x~}.…”

“…(ii) is an immediate consequence of continuity properties of 2 ~ inf AF(2) and 2 H sup AF(2) (Bertin et al, 1981). [] In the following, we restrict ourselves to unimodal distribution functions with continuous and bounded probability density function.…”

Some characterizations of discrete unimodality

“…In BHT (= Bertin, Hengartner and Theodorescu (1981)), we gave a characterization of unimodal distribution functions and a representation theorem for the class of unimodal distribution functions, both in terms of their L6vy concentration functions. Discrete analogs of these results are given in the present paper.…”