The Beta-Hyperbolic Secant Distribution

Authors

  • Matthias J. Fischer University of Erlangen-Nürnberg, Nürnberg, Germany
  • David Vaughan Wilfrid Laurier University, Waterloo, Ontario, Canada

DOI:

https://doi.org/10.17713/ajs.v39i3.247

Abstract

The shape of a probability distribution is often summarized by the distribution’s skewness and kurtosis. Starting from a symmetric “parent” density f on the real line, we can modify its shape (i.e. introduce skewness and in-/decrease kurtosis) if f is appropriately weighted. In particular, every density w on the interval (0; 1) is a specific weighting function. Within this work, we follow up a proposal of Jones (2004) and choose the Beta distribution as
underlying weighting function w. “Parent” distributions like the Student-t, the logistic and the normal distribution have already been investigated in the literature. Based on the assumption that f is the density of a hyperbolic secant distribution, we introduce the Beta-hyperbolic secant (BHS) distribution. In contrast to the Beta-normal distribution and to the Beta-Student-t distribution, BHS densities are always unimodal and all moments exist. In contrast to the Beta-logistic distribution, the BHS distribution is more flexible
regarding the range of skewness and leptokurtosis combinations. Moreover,
we propose a generalization which nests both the Beta-logistic and the BHS distribution. Finally, the goodness-of-fit between all above-mentioned distributions is compared for glass fibre data and aluminium returns.

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Published

2016-02-24

How to Cite

Fischer, M. J., & Vaughan, D. (2016). The Beta-Hyperbolic Secant Distribution. Austrian Journal of Statistics, 39(3), 245–258. https://doi.org/10.17713/ajs.v39i3.247

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