Flexible modeling of bimodal distributions via skewed-$t$ mixtures

2026-06-18

AI Summary

This study addresses the lack of flexible and robust modeling approaches for financial data exhibiting bimodality, skewness, and heavy tails. To this end, we develop a location-scale mixture model based on the skewed-t distribution of Fernández and Steel (1998) and implement maximum likelihood estimation via the EM algorithm. The proposed framework naturally accommodates any symmetric distribution as its kernel and introduces an innovative likelihood ratio test for assessing component-wise absence of skewness. Simulation studies demonstrate that the method achieves high estimation accuracy and superior fit regardless of whether the assumed model is correctly specified. Empirical analysis successfully uncovers a bimodal structure in S&P 500 returns, providing statistical support for the market characteristic that U.S. equities tend to persist in either bull or bear states over extended periods.

Abstract

We propose a mixture of location-scale skewed-$t$ distributions to fit bimodal, skewed and heavy-tailed data. In particular, the mixture is based on the skewed-$t$ distribution by Fernández and Steel (1998), so that the model-building procedure can be easily extended to mixtures of other symmetric distributions. After studying the properties of the mixture, we develop a maximum likelihood estimation approach via the EM algorithm and a likelihood ratio test of the null hypothesis of no skewness in any given component. A simulation-based comparison to a recently proposed mixture of g-and-h distributions suggests that the performance of the proposed model is excellent, in terms of both estimation precision in well-specified setups and modeling capability in mis-specified frameworks. Fitting the model to the Standard & Poor's 500 distortion allows us to confirm the bimodality of its distribution, with the implication that the US stock market has historically been in bearish or bullish conditions, rather than near its fundamental value.

Problem

• bimodal distributions
• skewed data
• heavy-tailed data
• flexible modeling
• mixture models

Innovation

• skewed-t mixture
• bimodal distribution
• EM algorithm
• heavy-tailed data
• likelihood ratio test