An improved Hoeffding’s inequality of closed form using refinements of the arithmetic mean-geometric mean inequality

In this note, we present an improvement of the probability inequalities of Hoeffding (1963 Hoeffding, W. 1963. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association. 58:13–30. doi: 10.1080/01621459.1963.10500830.Taylor & Francis Online, Web of Science ® , Google Scholar) for sums of independent-bounded random variables. Various refinements of the arithmetic mean-geometric mean inequality were considered to construct the improved bounds. The refinement of Cartwright and Field (1978 Cartwright, D. I., Field, M. J. 1978. A refinement of the arithmetic mean-geometric mean inequality. Proceedings of the American Mathematical Society 71(1):36–38.Crossref, Web of Science ® , Google Scholar), although not as good as some other refinements, provided the best improvement ofHoeffding’s bounds when accuracy and ease of computation (providing bounds of closed form) are both important. Some numerical examples are also presented to demonstrate that significant improvement in tail probability bounds are found when the means of the random variables are at least somewhat diverse.