2015
DOI: 10.1007/s00009-015-0562-y
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Multivariate Normal α-Stable Exponential Families

Abstract: The normal inverse Gaussian distributions are used to introduce the class of multivariate normal α-stable distributions. Some fundamental properties of these new distributions are established. We give the expression of the variance function of the generated natural exponential family and we use the Lévy-Khintchine representation to determine the associated Lévy measure. We also study the relationship between these distributions and the multivariate inverse Gaussian ones.Mathematics Subject Classification. 60E0… Show more

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Cited by 7 publications
(5 citation statements)
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“…Its importance comes from the fact that it identifies the underling NEF and many works focused on the classifications of these families according to the form of their variance functions (see Casalis, 1996;Letac, 1992;Letac & Mora, 1990;Louati, Masmoudi, & Mselmi, 2015a;Morris, 1982;Mselmi, Kokonendji, Louati, & Masmoudi, 2018). Moreover, it is useful in the determination of the quasi-likelihood function (…”
Section: Natural Exponential Familymentioning
confidence: 99%
See 1 more Smart Citation
“…Its importance comes from the fact that it identifies the underling NEF and many works focused on the classifications of these families according to the form of their variance functions (see Casalis, 1996;Letac, 1992;Letac & Mora, 1990;Louati, Masmoudi, & Mselmi, 2015a;Morris, 1982;Mselmi, Kokonendji, Louati, & Masmoudi, 2018). Moreover, it is useful in the determination of the quasi-likelihood function (…”
Section: Natural Exponential Familymentioning
confidence: 99%
“…The second derivative K μ ′′ defines the covariance operator of P ( θ , μ ) and is given by the following expression Kμfalse(θfalse)=boldRdboldxboldxPfalse(θ,μfalse)false(dboldxfalse)Kμfalse(θfalse)Kμfalse(θfalse), where x ⊤ and K μ ′ ( θ ) ⊤ represent the transpose of the vectors x and K μ ′ ( θ ). The variance function of the NEF F ( μ ) is defined on M F ( μ ) by mVF(μ)(m)=Kμ(ψμ(m))=ψμ(m)1. Its importance comes from the fact that it identifies the underling NEF and many works focused on the classifications of these families according to the form of their variance functions (see Casalis, 1996; Letac, 1992; Letac & Mora, 1990; Louati, Masmoudi, & Mselmi, 2015a; Morris, 1982; Mselmi, Kokonendji, Louati, & Masmoudi, 2018). Moreover, it is useful in the determination of the quasi‐likelihood function (or the quasi‐log‐likelihood function; see (McCullagh & Nelder, 1989), p. 325) which is defined by scriptQfalse(z,mfalse)=-0.5emzmzsVFfalse(μfalse)false(sfalse)ds.5emwhere.5emzsupportfalse(μfalse). …”
Section: Preliminariesmentioning
confidence: 99%
“…Remark Corollary 1 (1) allows us to determine the link and the variance functions of the subordinated α‐stable process by the Poisson subordinator (S‐P), with α(0,2]{1} (for more details about stable distributions and processes see Hassairi & Louati, 2009, Louati et al, 2015a, 2017, 2020, and Mselmi, 2018b, 2018c). We have the following results ψμ2,t(m)=(α1)mtc1α1exp1α𝒲mtcαα1 and VF(μ2,t)(m)=tmtα2α1c1α1exp1α𝒲mtcαα1+mt…”
Section: The Lambert Classmentioning
confidence: 99%
“…This variance function does not have an explicit form and it depends on the unknown link function. Note that Louati et al (2015a) gave the explicit form of the variance function of the normal-inverse Gaussian distribution. The work of Mselmi et al (2018) was extended by Mselmi (2018a) who determined the form of the variance function of the class of Lévy processes time-changed by the first-exit time of the inverse Gaussian subordinator.…”
Section: Introductionmentioning
confidence: 99%
“…Within this framework, [6] have characterized the exponential families of the Markov processes using an additive functional model. Besides, [11,12] have investigated on the characterization of the class of normal tempered stable distributions, which can be interpreted as a Brownian motion time-changed by a stable subordinator. They have also established a characterization of a multivariate Lévy process based on the notion of cut in natural exponential family.…”
Section: Introductionmentioning
confidence: 99%