Tauber Theory for Infinitely Divisible Variance Functions

Jørgensen, Bent; Martínez, José Raúl

doi:10.2307/3318587

Cited by 10 publications

(3 citation statements)

References 19 publications

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“…Remark 8.2 Jørgensen and Martínez (1997) and have developed Tauberian methods for variance functions, where power asymptotics for V is replaced by regular variation. Similar methods may be used for characterizing the domains of attraction for the generalized extreme value distributions under the assumptions of Theorem 8.2 or similar, see also de Haan (1970) and de Haan and Ferreira (2006).…”

Section: Extreme Convergencementioning

confidence: 99%

Dispersion models for extremes

2009

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We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. A class of extreme generalized linear regression models for analysis of extremes and lifetime data is introduced. The set of quadratic and power slope functions characterize well-known families such as the Rayleigh, Gumbel, power, Pareto, logistic, negative exponential, Weibull and Fréchet. We show a convergence theorem for slope functions, by which we may express the classical extreme value convergence results in terms of asymptotics for extreme dispersion models. The key idea is to explore the parallels between location families and natural exponential families, and between the convolution and minimum operations.

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Section: Extreme Convergencementioning

confidence: 99%

Dispersion models for extremes

2009

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“…See e.g. [2] for a description of distribution functions with polynomial variance function and [15] for tail equivalence for the variance function in infinitely divisible distributions. In the Regularly varying case, i.e.…”

Section: A Local Resultsmentioning

confidence: 99%

A Gibbs Conditional theorem under extreme deviation

Biret,

Broniatowski,

Cao

2016

Preprint

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We explore some properties of the conditional distribution of an i.i.d. sample under large exceedances of its sum. Thresholds for the asymptotic independance of the summands are observed, in contrast with the classical case when the conditioning event is in the range of a large deviation. This paper is an extension to [7]. Tools include a new Edgeworth expansion adapted to specific triangular arrays where the rows are generated by tilted distribution with diverging parameters, together with some Abelian type results.

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“…The Tweedie model corresponding to the exponent p is denoted Tw * p (θ, λ). Many exponential dispersion models have variance functions that are asymptotically of the Tweedie form (2.3), and there is a corresponding general convergence theorem with the Tweedie models as limiting distributions Martínez 1997), providing a kind of central limit theory for exponential dispersion models. For this reason, Tweedie models occupy a central position among exponential dispersion models.…”

Section: Tweedie Modelsmentioning

confidence: 99%

Stationary Time Series Models with Exponential Dispersion Model Margins

Jørgensen

Song

1998

Journal of Applied Probability

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We consider a class of stationary infinite-order moving average processes with margins in the class of infinitely divisible exponential dispersion models. The processes are constructed by means of the thinning operation of Joe (1996), generalizing the binomial thinning used by McKenzie (1986, 1988) and Al-Osh and Alzaid (1987) for integer-valued time series. As a special case we obtain a class of autoregressive moving average processes that are different from the ARMA models proposed by Joe (1996). The range of possible marginal distributions for the new models is extensive and includes all infinitely divisible distributions with finite moment generating functions, hereunder many known discrete, continuous and mixed distributions.

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Tauber Theory for Infinitely Divisible Variance Functions

Cited by 10 publications

References 19 publications

Dispersion models for extremes

Dispersion models for extremes

A Gibbs Conditional theorem under extreme deviation

Stationary Time Series Models with Exponential Dispersion Model Margins

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