Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions

Abstract. We study for a class of symmetric Lévy processes with state space R n the transition density p t (x) in terms of two one-parameter families of metrics, (d t ) t>0 and (δ t ) t>0 . The first family of metrics describes the diagonal term p t (0); it is induced by the characteristic exponent ψ of the Lévy process by d t (x, y) = tψ(x − y). The second and new family of metrics δ t relates to √ tψ through the formulawhere F denotes the Fourier transform. Thus we obtain the following "Gaussian" representation of the transition density:t (x,0) where p t (0) corresponds to a volume term related to √ tψ and where an "exponential" decay is governed by δ 2 t . This gives a complete and new geometric, intrinsic interpretation of p t (x). MSC 2010: Primary: 60J35. Secondary: 60E07; 60E10; 60G51; 60J45; 47D07; 31E05.

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“…is the density of a generalized hyperbolic distribution on R which is known to be infinitely divisible, see [2] and [3]. Then…”

Section: Examples Of Class N Distributionsmentioning

confidence: 99%

“…From [2] we know that ρ η,κ,λ (s) is an infinitely divisible probability density on (0, ∞). Consider the probability density p ρ as in (6.1) with mixing probability ρ = ρ η,κ,λ as in (6.4).…”

Section: Examples Of Class N Distributionsmentioning

confidence: 99%

A geometric interpretation of the transition density of a symmetric Lévy process

et al. 2012

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“…The distributions (90) and (94) are instances of the so called Generalized Inverse Gaussian (or GIG for short, see [15]) distribution. The GIG distribution was first introduced in relation to hyperbolic distributions in [16]. It can be shown that,…”

Section: Conditional For µmentioning

confidence: 99%

Entropic priors for discrete probabilistic networks and for mixtures of Gaussians models

Rodríguez

2002

AIP Conference Proceedings

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Abstract. The ongoing unprecedented exponential explosion of available computing power, has radically transformed the methods of statistical inference. What used to be a small minority of statisticians advocating for the use of priors and a strict adherence to bayes theorem, it is now becoming the norm across disciplines. The evolutionary direction is now clear. The trend is towards more realistic, flexible and complex likelihoods characterized by an ever increasing number of parameters. This makes the old question of: What should the prior be? to acquire a new central importance in the modern bayesian theory of inference. Entropic priors provide one answer to the problem of prior selection. The general definition of an entropic prior has existed since 1988 [1], but it was not until 1998 [2] that it was found that they provide a new notion of complete ignorance. This paper re-introduces the family of entropic priors as minimizers of mutual information between the data and the parameters, as in [2], but with a small change and a correction. The general formalism is then applied to two large classes of models: Discrete probabilistic networks and univariate finite mixtures of gaussians. It is also shown how to perform inference by efficiently sampling the corresponding posterior distributions.

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“…(i) More generally, the GIG and, hence, (non-skew) hyperbolic distributions are infinitely divisible (see [66]), so methods from [67] can be applied to compute the inverse df and the tail integral. In turn, the iterative procedure Equation (3) can be applied to compute the expectile.…”

Section: Example: Skewed Student T Distributionmentioning

confidence: 99%

Dependence Uncertainty Bounds for the Expectile of a Portfolio

Jakobsons

Vanduffel

2015

Risks

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Abstract:We study upper and lower bounds on the expectile risk measure of risky portfolios when the joint distribution of the risky components is not fully specified. First, we summarize methods for obtaining bounds when only the marginal distributions of the components are known, but not their interdependence (unconstrained bounds). In particular, we provide the best-possible upper bound and the best-possible lower bound (under some conditions), as well as numerical procedures to compute them. We also derive simple analytic bounds that appear adequate in various situations of interest. Second, we study bounds when some information on interdependence is available (constrained bounds). When the variance of the portfolio is known, a simple-to-compute upper bound is provided, and we illustrate that it may significantly improve the unconstrained upper bound. We also show that the unconstrained lower bound cannot be readily improved using variance information. Next, we derive improved bounds when the bivariate distributions of each of the risky components and a risk factor are known. When the factor induces a positive dependence among the components, it is typically possible to improve the unconstrained lower bound. Finally, the unconstrained dependence uncertainty spreads of expected shortfall, value-at-risk and the expectile are compared.

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Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions

Cited by 243 publications

References 5 publications

A geometric interpretation of the transition density of a symmetric Lévy process

A geometric interpretation of the transition density of a symmetric Lévy process

Entropic priors for discrete probabilistic networks and for mixtures of Gaussians models

Dependence Uncertainty Bounds for the Expectile of a Portfolio

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