Enhancing Markov and Chebyshev's inequalities

Abstract

The idea of the restricted mean has been used to establish a significantly improved version of Markov's inequality that does not require any new assumptions. The result immediately extends on Chebyshev's inequalities and Chernoff's bound. The improved Markov inequality yields a bound that is hundreds or thousands of times more accurate than the original Markov bound for high quantiles in the most prevalent and diverse situations. The Markov inequality benefits from being model-independent, and the long-standing issue of its imprecision is solved. Practically speaking, avoidance of model risk is decisive when multiple competing models are present in a real-world situation.