Impact of Kolmogorov–Nagumo’s Average in Statistics and Related Fields

Abstract

The concept of a ‘mean’ is foundational to statistical science, yet the familiar arithmetic mean is but one instance of a much broader and more powerful class of averages. This paper explores the Quasi-Arithmetic Means (QAMs), an axiomatically-grounded generalization that unifies a wide array of specific means, including the arithmetic, geometric, harmonic, and power means, under a single functional form. We provide a comprehensive overview of the QAM framework, detailing its theoretical underpinnings and demonstrating its remarkable ability to connect disparate statistical concepts and inspire novel methodologies. Key axiomatic results are reviewed, and the reach of QAMs is illustrated through a survey of their diverse applications in areas such as economics, information theory, and stochastic modeling. Furthermore, this paper introduces original contributions: novel quasi-arithmetic mean representations for the sample median and data extremes (maximum and minimum as limiting cases), and new QAM-based measures of randomness. These contributions highlight the continued vitality and generative capacity of the QAM concept. Ultimately, we offer a broad perspective on this fundamental statistical tool, underscoring its enduring importance for both theoretical understanding and practical innovation.