The mean, median, and mode of unimodal distributions: a characterization

For a unimodal distribution on the real line, the celebrated mean-median-mode inequality states that they often occur in an alphabetical (or its reverse) order. Various sufficient conditions for the validity of the inequality are known. This article explicitly characterizes the three dimensional set of means, medians, and modes of unimodal distributions. It is found that the set is pathwise connected but not convex. Some fundamental inequalities among the mean, the median and mode of unimodal distributions are also derived. These inequalities are used: (i) to prove nonunimodality of certain distributions, and (ii) for obtaining bounds on the median of a unimodal distribution. In a multivariate setting, the generalized notion of $\alpha$-unimodality is used, and characterizations are given for the set of mean vectors, when the mode is fixed, or when it varies in a sphere. In particular, it is found that the set of mean vectors for generalized unimodal distributions with a specified mode and covariance matrix is an exact ellipsoid and this ellipsoid is explicitly described.