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

In: Directional and Multivariate Statistics

Author

Listed:
  • Andriette Bekker

    (University of Pretoria, Department of Statistics
    Centre of Excellence in Mathematical and Statistical Sciences)

  • Matthias Wagener

    (University of Pretoria, Department of Statistics)

  • Mohammad Arashi

    (University of Pretoria, Department of Statistics
    Ferdowsi University of Mashhad, Department of Statistics, Faculty of Mathematical Sciences)

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.

Suggested Citation

  • Andriette Bekker & Matthias Wagener & Mohammad Arashi, 2025. "In Search of the Perfect Fit: Interpretation, Flexible Modelling, and the Existing Generalisations of the Normal Distribution," Springer Books, in: Somesh Kumar & Barry C. Arnold & Kunio Shimizu & Arnab Kumar Laha (ed.), Directional and Multivariate Statistics, pages 291-316, Springer.
    • Handle: RePEc:spr:sprchp:978-981-96-2004-3_15
    DOI: 10.1007/978-981-96-2004-3_15
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