Partially Schur-constant models

Castañer, Anna; Bielsa, M. Mercè Claramunt; Lefèvre, Claude; Loisel, Stéphane

doi:10.1016/j.jmva.2019.01.007

Cited by 7 publications

(2 citation statements)

References 20 publications

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“…, for some α, d > 0, are applied in many different areas of mathematics including actuarial science (see e.g. [8,32]) and dependence modeling by copulas [15,31,34,35].…”

Section: Definition 4 We Say That the Borel Functionmentioning

confidence: 99%

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Jasiulis-Gołdyn,

Misiewicz,

Omey

et al. 2023

Results Math

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This paper deals with the generalized convolutions connected with the Williamson transform and the maximum operation. We focus on such convolutions which can define transition probabilities of renewal processes. They should be monotonic since the described time or destruction does not go back, it should admit existence of a distribution with a lack of memory property because the analog of the Poisson process shall exist. Another valuable property is the simplicity of calculating and inverting the corresponding generalized characteristic function (in particular Williamson transform) so that the technique of generalized characteristic function can be used in description of our processes. The convex linear combination property (the generalized convolution of two point measures is the convex combination of several fixed measures), or representability (which means that the generalized convolution can be easily written in the language of independent random variables)—they also facilitate the modeling of real processes in that language. We describe examples of generalized convolutions having the required properties ranging from the maximum convolution and its simplest generalization—the Kendall convolution (associated with the Williamson transform), up to the most complicated here—Kingman convolution. It is novel approach to apply in the extreme value theory. Stochastic representation of the Kucharczak-Urbanik in the order statistics terms is proved, which open new paths to investigate Archimedean copulas. This paper open the door to solve an old open problem of the relationship between copulas and generalized convolutions mentioned by B. Schweizer and A. Sklar in 1983. This indicates the path of further research towards extremes and dependency modelling.

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“…, for some α, d > 0, are applied in many different areas of mathematics including actuarial science (see e.g. [8,32]) and dependence modeling by copulas [15,31,34,35].…”

Section: Definition 4 We Say That the Borel Functionmentioning

confidence: 99%

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

Jasiulis-Gołdyn,

Misiewicz,

Omey

et al. 2023

Results Math

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“…+ γ(dx), for some α, d > 0, are applied in many different areas of mathematics including actuarial science (see e.g. [7,31]) and dependence modeling by copulas ( [13,30,33,34]). Note that it is easy to retrieve the measure knowing its Williamson transform.…”

Section: The Kendall Convolution By the Corresponding Williamson Tran...mentioning

confidence: 99%

How exceptional is the extremal Kendall and Kendall-type convolution

Jasiulis-Gołdyn¹,

Misiewicz²,

Omey³

et al. 2019

Preprint

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In the paper we describe several important properties of the Kendall convolution at the same time pointing to these generalized convolutions which have the same property. For example the monotonic property is necessary to build a renewal process with respect to generalized convolution, lack of memory property is needed for the construction of the Poisson process with respect to generalized convolution. Another valuable property is the simplicity of inverting the corresponding generalized characteristic function e.g. inverting the Williamson transform in the case of the Kendall convolution. The convex linear combination property makes calculations easier and the representation as a weak convolution with respect to max-convolution allows describing extreme phenomena.In the last section we discuss the representable generalized convolutions in the sense similar to the one defined in [6], i.e. such convolutions ⋄ for which the random variable Z satisfies the condition L(θ 1 ) ⋄ L(θ 1 ) = L(Z) (L(X) denotes the distribution of X) can be explicitly given by some function on variables X, Y and maybe some additional variables. This property enables the computer simulation of stochastic processes with respect to representable generalized convolution and makes possible studying path properties of such processes. Contents 1. Motivations 2 2. A primer on generalized convolutions 5

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Schur-Constant and Related Dependence Models, with Application to Ruin Probabilities

Lefèvre

Simon

2019

Methodol Comput Appl Probab

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Partially Schur-constant models

Cited by 7 publications

References 20 publications

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

How Exceptional is the Extremal Kendall and Kendall-Type Convolution

How exceptional is the extremal Kendall and Kendall-type convolution

Schur-Constant and Related Dependence Models, with Application to Ruin Probabilities

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