Improvement of the Ushakov bound

Abstract

Considering a discrete unimodal distribution, an upper bound on a tail probability about a mode is suggested, which can be shown to be sharper than both the Bienaymé-Chebyshev bound and the Ushakov bound. Furthermore, by using the suggested bound, an upper bound on the probability outside the range of three standard deviations is given when a mean equals a mode.

Keywords:

• Bienaymé-Chebyshev inequality
• Gauss inequality
• three-sigma rule
• unimodal distribution
• Ushakov inequality

AMS 2010 subject classifications:

• 60G05
• 60E15

Acknowledgments

The authors appreciate the anonymous reviewers for their careful reading of the manuscript and comments.

Additional information

Funding

This work was partially supported by JSPS KAKENHI Grant Number [JP19K11864].