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Convex transform order of Beta distributions with some consequences

Author

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  • Idir Arab
  • Paulo Eduardo Oliveira
  • Tilo Wiklund

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.

Suggested Citation

  • Idir Arab & Paulo Eduardo Oliveira & Tilo Wiklund, 2021. "Convex transform order of Beta distributions with some consequences," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(3), pages 238-256, August.
    • Handle: RePEc:bla:stanee:v:75:y:2021:i:3:p:238-256
    DOI: 10.1111/stan.12233
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    References listed on IDEAS

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