Enhancing Markov and Chebyshev's inequalities

TL;DR

This paper introduces a significantly improved, model-independent version of Markov's inequality using the restricted mean, greatly enhancing accuracy for high quantiles and extending to Chebyshev's and Chernoff's bounds.

Contribution

The paper presents a novel, assumption-free enhancement of Markov's inequality that substantially improves its precision and extends its applicability to related inequalities.

Findings

Improved Markov inequality is hundreds to thousands times more accurate for high quantiles.

Extension of the improved inequality to Chebyshev's and Chernoff's bounds.

The new bounds are model-independent and reduce model risk in practical applications.

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.