Estimating Skewness and Kurtosis for Asymmetric Heavy-Tailed Data: A Regression Approach

Joseph H. T. Kim () and Heejin Kim
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Joseph H. T. Kim: Department of Applied Statistics, College of Business and Economics, Yonsei University, Seoul 03722, Republic of Korea
Heejin Kim: Department of Applied Statistics, College of Business and Economics, Yonsei University, Seoul 03722, Republic of Korea

Mathematics, 2025, vol. 13, issue 16, 1-21

Abstract: Estimating skewness and kurtosis from real-world data remains a long-standing challenge in actuarial science and financial risk management, where these higher-order moments are critical for capturing asymmetry and tail risk. Traditional moment-based estimators are known to be highly sensitive to outliers and often fail when the assumption of normality is violated. Despite numerous extensions—from robust moment-based methods to quantile-based measures—being proposed over the decades, no universally satisfactory solution has been reported, and many existing methods exhibit limited effectiveness, particularly under challenging distributional shapes. In this paper we propose a novel method that jointly estimates skewness and kurtosis based on a regression adaptation of the Cornish–Fisher expansion. By modeling the empirical quantiles as a cubic polynomial of the standard normal variable, the proposed approach produces a reliable and efficient estimator that better captures distributional shape without strong parametric assumptions. Our comprehensive simulation studies show that the proposed method performs much better than existing estimators across a wide range of distributions, especially when the data are skewed or heavy-tailed, as is typical in actuarial and financial applications.

Keywords: skewness; kurtosis; Cornish–Fisher expansion; median regression; heavy-tailed data (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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