Parameter estimation of Tukey-type distributions: A comparative analysis

Abstract

Tukey-type distributions (a generalization of Tukey’s gh(normal)-distribution) are attractive for a broad spectrum of applications since they can explicitly model flexible skewness and kurtosis features of practical data. However, since the likelihood function is not available in closed-form, the derivation of an accurate estimation method plays a crucial role and henceforth various procedures to overcome this issue were introduced, especially for Tukey’s gh(normal)-distribution. After extending these approaches to all Tukey-type distributions, our main contribution is to provide a comprehensive simulation study to compare the methods against each other. Moreover, we illustrate the practical usage for the Nikkei225 stock index data.

Keywords:

• Tukey-type distributions
• Parameter estimation
• Simulation study
• Market risk

Acknowledgments

The authors would like to thank the anonymous referee for the comments and suggestions, which contributed to improving the quality of this publication.

Disclosure statement

The views presented in this manuscript reflect those of the authors and do not necessarily coincide with the views of Bayerische Landesbank.

Notes

1 Other kurtosis transformations are e.g., $T_e(z)=z\hspace{0.17em}\text{exp}\hspace{0.17em}{(\cosh (z)-1)}^e, e\in \mathbb{R}$ (Fischer and Klein 2004) and $T_l(z)=z\sinh {(z)}^l, l\in \mathbb{R}$ (Fischer 2008)

2 See the Supplementary Material of Xu and Genton (2015) for the derivation of these formulas and we thank G. Xu for the provision of R code in the case of a gh(normal)-distribution on his website.

3 For each Tukey-type distribution 10000 times 1000 random numbers are drawn and the mean of the empirical skewness and kurtosis are taken.

4 For reasons of comparability, GARCH effects are not taken into account, although they have to be considered in a forecast model.

5 Once an estimate could not be found.