Unified relationship between mean, variance, and an arbitrary number of quantiles

Kensho Kobayashi () and Hidekazu Tanaka ()
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Kensho Kobayashi: Osaka Prefecture University
Hidekazu Tanaka: Osaka Prefecture University

Metrika: International Journal for Theoretical and Applied Statistics, 2025, vol. 88, issue 6, No 18, 1187-1201

Abstract: Abstract In mathematical statistics, quantities such as mean, variance and quantile are often used to summarize a probability distribution. A number of relationships between them have thus been developed. However, all the preceding studies on the relationships have treated only the case for single quantile (one-sided tail probability) or linear combination of two quantiles (two-sided tail probabilities). Also, they have constraints, making them technically inflexible. In this paper, a unified relationship between mean, variance, and an arbitrary number of individual quantiles is derived without restriction on probability distribution. The existence domains of quantiles and tail probabilities are specified. An application of them in the special case where the number of quantiles is two is investigated. As the examples, Rényi’s inequality, Selberg’s inequality, and a bound of threshold for outlier detection are easily obtained. In addition, the probability distribution that achieves the equality in the derived relationship is also clearly identified.

Keywords: Existence domain; Quantiles; Rényi’s inequality; Selberg’s inequality; Tail probabilities; Threshold for outlier detection; 60E05; 62E10; 60E15 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s00184-025-01001-6

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