Abstract
The convex transform order is one way to make precise comparison between the skewness of probability distributions on the real line. We establish a simple and complete characterization of when one Beta distribution is smaller than another according to the convex transform order. As an application, we derive monotonicity properties for the probability of Beta distributed random variables exceeding the mean or mode of their distribution. Moreover, we obtain a simple alternative proof of the mode‐median‐mean inequality for unimodal distributions that are skewed in a sense made precise by the convex transform order. This new proof also gives an analogous inequality for the anti‐mode of distributions that have a unique anti‐mode. Such inequalities for Beta distributions follow as special cases. Finally, some consequences for the values of distribution functions of binomial distributions near to their means are mentioned.
Highlights
How to order probability distributions according to criteria that have consequences with probabilistic interpretations is a common question in probability theory
See Cortes, Mansour, and Mohri (2010) for more general questions. Such an inequality for the binomial random variables was used by Wiklund (2018) when studying the amount of information lost when resampling. These properties allow one to compare the relative location of the mode, median, and mean of certain distributions that are skewed in a sense made precise by the convex transform order
Convex transform order ≤c: Let P and Q be two probability distributions on the real line supported by the intervals I and J that have strictly increasing distribution functions F : I → [0, 1] and G : J → [0, 1], respectively
Summary
How to order probability distributions according to criteria that have consequences with probabilistic interpretations is a common question in probability theory. The main contribution of this paper is to characterize when one Beta distribution is smaller than another according to the convex- and star-shaped transform orders This characterization implies various monotonicity properties for the probabilities of Beta distributed random variables exceeding the mean or mode of their distribution. Such an inequality for the binomial random variables was used by Wiklund (2018) when studying the amount of information lost when resampling These properties allow one to compare the relative location of the mode, median, and mean of certain distributions that are skewed in a sense made precise by the convex transform order. Such mode–median–mean inequalities are a classical subject in probability theory. Some auxiliary results concerning the main tools of analysis are given later in Appendix A1
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