In Search of the Perfect Fit: Interpretation, Flexible Modelling, and the Existing Generalisations of the Normal Distribution

Andriette Bekker (), Matthias Wagener () and Mohammad Arashi ()
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Andriette Bekker: University of Pretoria, Department of Statistics
Matthias Wagener: University of Pretoria, Department of Statistics
Mohammad Arashi: University of Pretoria, Department of Statistics

A chapter in Directional and Multivariate Statistics, 2025, pp 291-316 from Springer

Abstract: Abstract Many generalised distributions exist for modelling data with vastly diverse characteristics. However, very few of these generalisations of the normal distribution have shape parameters with clear roles that determine, for instance, skewness and tail shape. In this chapter, we review existing skewing mechanisms and their properties in detail. Using the knowledge acquired, we add a skewness parameter to the body-tail generalised normal distribution (Wagener et al. in Mathematics 9(21):2648, 2021) that yields the flexible and interpretable normal distribution (FIN) with parameters for location, scale, body shape, skewness, and tail weight. Basic statistical properties of the FIN are provided, such as the probability density function (PDF), cumulative distribution function, moments, and likelihood equations. Additionally, the FIN PDF is extended to a multivariate setting using a Student t-copula, yielding the multivariate flexible and interpretable normal distribution (MFIN). The MFIN is applied to stock returns data, where it outperforms the t-copula multivariate generalised hyperbolic, Azzalini skew-t, hyperbolic, and normal inverse Gaussian distributions.

Keywords: Body tail; Kurtosis; Maximum likelihood; Multivariate; Skewness; Skew normal; Stock returns (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-96-2004-3_15

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DOI: 10.1007/978-981-96-2004-3_15

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