Tailoring the Gaussian Law for Excess Kurtosis and Skewness by Hermite Polynomials

Zoia, Maria Grazia

doi:10.1080/03610920802696596

Cited by 22 publications

(19 citation statements)

References 7 publications

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“…This paper extends recent insights (see [3,5,6]) on the issue of tailoring distributions in order to account for over-kurtosis. Starting from a given spherical distribution, the orthogonal polynomials of its related modular variable are provided and used to design the intended distribution shape adapter.…”

Section: Discussionmentioning

confidence: 68%

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The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

Bagnato¹,

Faliva²,

Zoia³

2016

Symmetry

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This paper carries out an investigation of the orthogonal-polynomial approach to reshaping symmetric distributions to fit in with data requirements so as to cover the multivariate case. With this objective in mind, reference is made to the class of spherical distributions, given that they provide a natural multivariate generalization of univariate even densities. After showing how to tailor a spherical distribution via orthogonal polynomials to better comply with kurtosis requirements, we provide operational conditions for the positiveness of the resulting multivariate Gram-Charlier-like expansion, together with its kurtosis range. Finally, the approach proposed here is applied to some selected spherical distributions.

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Section: Discussionmentioning

confidence: 68%

“…This clears the way for the tailoring of the shape of a given distribution from "inside" through a polynomial shape-adapter, built on the orthogonal polynomials engendered by the same distribution (e.g., [1][2][3][4]). …”

Section: Introductionmentioning

confidence: 99%

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

Bagnato¹,

Faliva²,

Zoia³

2016

Symmetry

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“…Motivated by this consideration we propose the multivariate leptokurtic‐normal (MLN) distribution which is the multivariate Gram–Charlier expansion of the MN distribution. In detail, the MLN distribution is obtained by reshaping the generating variate of its elliptical representation (Cambanis, Huang, & Simons, ) with the associated orthogonal polynomials (Zoia, ) which, unlike the Hermite ones, are not common in the literature. The result is a distribution characterized by one additional parameter corresponding to the excess kurtosis with respect to the original MN distribution.…”

Section: Introductionmentioning

confidence: 99%

“…The article is organized as follows. Some preliminary results are given in Section about elliptical distributions and the polynomial reshaping method proposed by Zoia (). Section presents two of the main contributions of the work namely the MLN distribution and its genesis; moreover maximum likelihood is described to estimate the parameters of the proposed distribution.…”

Section: Introductionmentioning

confidence: 99%

The multivariate leptokurtic‐normal distribution and its application in model‐based clustering

2016

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This article proposes the elliptical multivariate leptokurtic‐normal (MLN) distribution to fit data with excess kurtosis. The MLN distribution is a multivariate Gram–Charlier expansion of the multivariate normal (MN) distribution and has a closed‐form representation characterized by one additional parameter denoting the excess kurtosis. It is obtained from the elliptical representation of the MN distribution, by reshaping its generating variate with the associated orthogonal polynomials. The strength of this approach for obtaining the MLN distribution lies in its general applicability as it can be applied to any multivariate elliptical law to get a suitable distribution to fit data. Maximum likelihood is discussed as a parameter estimation technique for the MLN distribution. Mixtures of MLN distributions are also proposed for robust model‐based clustering. An EM algorithm is presented to obtain estimates of the mixture parameters. Benchmark real data are used to show the usefulness of mixtures of MLN distributions. The Canadian Journal of Statistics 45: 95–119; 2017 © 2016 Statistical Society of Canada

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“…However, as the paper shows, this hurdle can, nonetheless, be largely overcome. This is achieved by moving from PHS distributions to GCT expansions, using an argument similar to that used to alter the moments of a Gaussian via a Gram-Charlier series (see, e.g., [5] and more recently [6]). A first approach to the issue, with reference to logistic and hyperbolic secant laws, can be found in [7].…”

Section: Introductionmentioning

confidence: 99%

A Distribution Family Bridging the Gaussian and the Laplace Laws, Gram–Charlier Expansions, Kurtosis Behaviour, and Entropy Features

Faliva

Zoia

2017

Entropy

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Abstract:The paper devises a family of leptokurtic bell-shaped distributions which is based on the hyperbolic secant raised to a positive power, and bridges the Laplace and Gaussian laws on asymptotic arguments. Moment and cumulant generating functions are then derived and represented in terms of polygamma functions. The behaviour of shape parameters, namely kurtosis and entropy, is investigated. In addition, Gram-Charlier-type (GCT) expansions, based on the aforementioned distributions and their orthogonal polynomials, are specified, and an operational criterion is provided to meet modelling requirements in a possibly severe kurtosis and skewness environment. The role played by entropy within the kurtosis ranges of GCT expansions is also examined.

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Tailoring the Gaussian Law for Excess Kurtosis and Skewness by Hermite Polynomials

Cited by 22 publications

References 7 publications

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

The Role of Orthogonal Polynomials in Tailoring Spherical Distributions to Kurtosis Requirements

The multivariate leptokurtic‐normal distribution and its application in model‐based clustering

A Distribution Family Bridging the Gaussian and the Laplace Laws, Gram–Charlier Expansions, Kurtosis Behaviour, and Entropy Features

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