Abstract
We begin this paper by introducing the Linnik distributions in both the univariate and multivariate case. An overview of simulation methods and two estimation procedures for the multivariate Linnik distribution are presented. Experiments demonstrating the accuracy of the procedures are also included. Then a novel multivariate Linnik copula is derived. The primary focus of this part of work is on simulation and estimation procedures for this copula, applying existing algorithms for simulation and estimation procedures for the multivariate Linnik distribution derived in the prior section. Several theoretical properties of the copula in relation to different dependence metrics are derived.
Highlights
Linnik Distributions we recall the basic classical concepts related to Levy processes, and infinitely divisible distributions
The primary focus of this part of work is on simulation and estimation procedures for this copula, applying existing algorithms for simulation and estimation procedures for the multivariate Linnik distribution derived in the prior section
We recall the basic classical concepts related to Levy processes, and infinitely divisible distributions
Summary
We recall the basic classical concepts related to Levy processes, and infinitely divisible distributions. 1.1 Preliminaries on General One-Dimensional Levy Processes and Infinitely Divisible Distributions. The 1-D distributions of the Levy processes are infinitely divisible (ID) which, in terms of the characteristics functions, can be expressed as the obvious equality: φX(ξ, t) = E exp(iξX(t)) = [φX(ξ, t/n)]n, ξ ∈ R. for every n ≥ 1, t ≥ 0. A more detailed description of the structure of characteristic functions of infinitely divisible distributions is given by the following classical Levy-Khinchine Representation Theorem: THEOREM 2.1. A random variable X has an infinitely divisi∫ble distribution if, and only if, there exist μ ∈ R, σ ∈ R+, and a nonegative measure Λ on R\{0} satisfying the condition R(1 ∧ |x|2)Λ(dx) < ∞, such that its characteristic function is of the form, φX(ξ) := E[eiξX] = eψ(ξ),. The Levy measure describes the “intensity” of jumps of a certain size of a Levy process
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