Moments of the generalized hyperbolic distribution

Scott, David J.; Würtz, D.; Dong, Christine; Tran, Thanh Tam

doi:10.1007/s00180-010-0219-z

Cited by 33 publications

(21 citation statements)

References 24 publications

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“…In fact, GH distributions possess moments of arbitrary order (except in a limiting case that corresponds to Student's t-distribution), and formulas for these moments are given in [43]. The class is closed under affine transformations [16]: if X ∼ GH(λ, α, β, δ, µ) then aX + b ∼ GH(λ, α/|a|, β/a, δ|a|, aµ + b).…”

Section: The Class Of Generalized Hyperbolic Distributionsmentioning

confidence: 99%

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A Stein characterisation of the generalized hyperbolic distribution

Gaunt

2017

ESAIM: PS

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The generalized hyperbolic (GH) distributions form a five parameter family of probability distributions that includes many standard distributions as special or limiting cases, such as the generalized inverse Gaussian distribution, Student's tdistribution and the variance-gamma distribution, and thus the normal, gamma and Laplace distributions. In this paper, we consider the GH distribution in the context of Stein's method. In particular, we obtain a Stein characterisation of the GH distribution that leads to a Stein equation for the GH distribution. This Stein equation reduces to the Stein equations from the current literature for the aforementioned distributions that arise as limiting cases of the GH superclass.

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Section: The Class Of Generalized Hyperbolic Distributionsmentioning

confidence: 99%

“…Using forward substitution in (3.19), one can then obtain formulas for moments of arbitrary order of the GH(λ, α, β, δ, 0) distribution. The recurrence relation (3.19) appears to be new, although it should be noted that [43] have already established a formula for the moments of general order of the GH distribution.…”

Section: Applications Of Proposition 31mentioning

confidence: 99%

A Stein characterisation of the generalized hyperbolic distribution

Gaunt

2017

ESAIM: PS

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“…This distribution is denoted by GH d (Σ, µ, λ, χ, ψ). Then the third cumulant for B d (Γ) can be obtained from Theorem 3 and from the formula for the moments of Γ ∼ GIG(λ, χ, ψ) (see Lemma 1 of Scott et al (2011)) which are given by…”

Section: The Poisson-normal Inverse Gaussian Claim Modelmentioning

confidence: 99%

Third cumulant for multivariate aggregate claim models

Loperfido

Mazur

Podgórski³

2017

Scandinavian Actuarial Journal

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The third moment cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. It is discussed how this result can be used to obtain a formula for the third cumulant for a classical model of multivariate claims. Two important special cases are considered. In the first one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results.

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“…The one-dimensional normal inverse Gaussian (NIG) pdf contains four parameters (α, β, δ, and μ) and is defined as follows [89][90][91]:…”

Section: Appendix A: Normal Inverse Gaussian Distributionmentioning

confidence: 99%

Estimating higher-order structure functions from geophysical turbulence time series: Confronting the curse of the limited sample size

DeMarco

Basu

2017

Phys. Rev. E

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Utilizing synthetically generated random variates and laboratory measurements, we document the inherent limitations of the conventional structure function approach in limited sample size settings. We demonstrate that an alternative approach, based on the principle of maximum likelihood, can provide nearly unbiased structure function estimates with far less uncertainty under such unfavorable conditions. The superiority of this approach over the conventional approach does not diminish even in the case of strongly correlated samples. Two entirely different types of probability distributions, which have been reported in the turbulence literature, are found to be compatible with the proposed approach.

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Moments of the generalized hyperbolic distribution

Cited by 33 publications

References 24 publications

A Stein characterisation of the generalized hyperbolic distribution

A Stein characterisation of the generalized hyperbolic distribution

Third cumulant for multivariate aggregate claim models

Estimating higher-order structure functions from geophysical turbulence time series: Confronting the curse of the limited sample size

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