Enhancing Markov and Chebyshev's inequalities

2023/08/08 by Joan del Castillo, del Castillo, Joan
Computer Science · Mathematics · #Bayesian Modeling and Causal Inference #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2308.04053

openalex publication_date 2023/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Related

• Randomized and Exchangeable Improvements of Markov's, Chebyshev's and Chernoff's Inequalities
• Halving the bounds for the Markov, Chebyshev, and Chernoff Inequalities using smoothing
• Improved Chebyshev inequality: new probability bounds with known supremum of PDF
• A New Generalization of Chebyshev Inequality for Random Vectors
• Tight tail probability bounds for distribution-free decision making