On Pearson–Kotz Dirichlet distributions

Hashorva, Enkelejd

doi:10.1016/j.jmva.2011.01.012

Cited by 8 publications

(11 citation statements)

References 32 publications

(52 reference statements)

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“…Suppose first that S = 1 almost surely. If A is non-singular and A JI has all its elements as 0, then we have the stochastic representation (see [4])…”

Section: The Kotz Approximationmentioning

confidence: 99%

“…If U is a k-dimensional random vector with stochastic representation (11), then the Pearson-Kotz Dirichlet random vector X considered in Balakrishnan and Hashorva [4] has the stochastic representation…”

Section: Pearson-kotz Dirichlet Distributionsmentioning

confidence: 99%

“…Indeed, the t-distributions are both tractable and flexible forming a prominent subclass of the elliptically symmetric distributions; see [24]. The same role is played by Pearson-Kotz Dirichlet distributions within the class of the Dirichlet distributions; see [4]. For simplicity, we discuss below the case when A = I k , with I k being the identity matrix in R k×k .…”

Section: Pearson-kotz Dirichlet Distributionsmentioning

confidence: 99%

“…Balakrishnan and Hashorva [4] proved the conditional approximation of regularly varying Dirichlet random vectors by the Pearson-Kotz Dirichlet random vectors, and so we do not focus on this topic here.…”

Section: Pearson-kotz Dirichlet Distributionsmentioning

confidence: 99%

“…A similar role in the approximation of the conditional distributions of regularly varying Dirichlet random vectors is played by the Pearson-Kotz Dirichlet distribution which is defined via (1), where R p has the beta distribution of second kind with parameters α 1 + α 2 and θ > 0; see [4]. The Pearson-Kotz Dirichlet random vectors are another important subclass of the Dirichlet class including two prominent cases: the Student t random vectors (see e.g., [24] or [19] and the Pearson Type VII L p random vectors introduced by Gupta and Song [14].…”

Section: Introductionmentioning

confidence: 99%

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Scale mixtures of Kotz–Dirichlet distributions

Hashorva¹

2013

Journal of Multivariate Analysis

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a b s t r a c tIn this paper, we first show that a k-dimensional Dirichlet random vector has independent components if and only if it is a Kotz Type I Dirichlet random vector. We then consider in detail the class of k-dimensional scale mixtures of Kotz-Dirichlet random vectors, which is a natural extension of the class of Kotz Type I random vectors. An interesting member of the Kotz-Dirichlet class of multivariate distributions is the family of Pearson-Kotz Dirichlet distributions, for which we present a new distributional property. In an asymptotic framework, we show that the Kotz Type I Dirichlet distributions approximate the conditional distributions of scale mixtures of Kotz-Dirichlet random vectors. Furthermore, we show that the tail indices of regularly varying Dirichlet random vectors can be expressed in terms of the Kotz Type I Dirichlet random vectors.

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“…Suppose first that S = 1 almost surely. If A is non-singular and A JI has all its elements as 0, then we have the stochastic representation (see [4])…”

Section: The Kotz Approximationmentioning

confidence: 99%

Section: Pearson-kotz Dirichlet Distributionsmentioning

confidence: 99%

Section: Pearson-kotz Dirichlet Distributionsmentioning

confidence: 99%

Section: Pearson-kotz Dirichlet Distributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Scale mixtures of Kotz–Dirichlet distributions

Hashorva¹

2013

Journal of Multivariate Analysis

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Archimedean copulas in finite and infinite dimensions—with application to ruin problems

Constantinescu

Hashorva

2011

Insurance: Mathematics and Economics

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Second-order tail asymptotics of deflated risks

Hashorva¹,

Ling²,

Peng³

2014

Insurance: Mathematics and Economics

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Random deflated risk models have been considered in recent literatures. In this paper, we investigate second-order tail behavior of the deflated risk X = RS under the assumptions of second-order regular variation on the survival functions of the risk R and the deflator S. Our findings are applied to approximation of Value at Risk, estimation of small tail probability under random deflation and tail asymptotics of aggregated deflated risk.The main contributions of this paper concern the second-order expansions of the tail probability of the deflated risk X = RS which are then illustrated by several examples. Our main findings are utilized for the formulations of three applications, namely approximation of Value-at-Risk, estimation of small tail probability of the deflated risk, and the derivation of the tail asymptotics of aggregated risk under deflation.The rest of this paper is organized as follows. Section 2 gives our main results under second-order regular variation conditions. Section 3 shows the efficiency of our second-order asymptotics through some illustrating examples. Section 4 is dedicated to three applications. The proofs of all results are relegated to Section 5. We conclude the paper with a short Appendix. Main resultsWe start with the definitions and some properties of regular variation followed by our principal findings. A measurable function f : [0, ∞) → IR with constant sign near infinity is said to be of second-order regular variation with parameters α ∈ IR and ρ ≤ 0, denoted by f ∈ 2RV α,ρ , if there exists some function A with constant sign near infinity satisfying lim t→∞ A(t) = 0 such that for all x > 0 (cf. Bingham et al. (1987) and Resnick (2007))Here, A is referred to as the auxiliary function of f . Noting that (2.1) implies lim t→∞ f (tx)/f (t) = x α , i.e., f is regularly varying at infinity with index α ∈ IR, denoted by f ∈ RV α ; RV 0 is the class of slowly varying functions. When f is eventually positive, it is of second-order Π-variation with the second-order parameter ρ ≤ 0, denoted by f ∈ 2ERV 0,ρ , if there exist some functions a and A with constant sign near infinity and lim t→∞ A(t) = 0 such that for all x positive lim t→∞ f (tx)−f (t) a(t)

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On Pearson–Kotz Dirichlet distributions

Cited by 8 publications

References 32 publications

Scale mixtures of Kotz–Dirichlet distributions

Scale mixtures of Kotz–Dirichlet distributions

Archimedean copulas in finite and infinite dimensions—with application to ruin problems

Second-order tail asymptotics of deflated risks

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