Multivariate Poisson Tweedie models

28

3. NORMAL STABLE TWEEDIE MODELS

In this chapter, we present the family of normal stable Tweedie models which is the extension of normal gamma Bernardo and Smith, 1993 and normal inverse Gaussian Bandorff-Nielsen et al., 1982 models. Normal Stable Tweedie NST models are composed by a fixed univariate stable Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with the same variance equal to the fixed component. A random variable X is said to be stable or to have a stable distribution if it has the property that a linear combination of two independent copies of the variable, e.g. aX 1 + bX 2 where X 1 and X 2 are two independent copies of X, has the same distribution with cX + d for some positive c and d ∈ R Nolan, 2017. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution. While Tweedie distributed variable is a variable that belongs to the exponential dispersion family of distribution with specific power mean-variance relationships, i.e, V µ = µ p see Chapter 2, Equation 2.21 for some p . Then Tweedie distribution is to the subclass of the exponential dispersion models that admits a power variance function V µ = µ p Jørgensen, 1997. Some details on univariate stable Tweedie models are given in Appendix A.

3.1. Definition and Properties

Motivated by normal gamma and normal inverse Gaussian models, Boubacar Maïnassara and Kokonendji 2014 introduced a new form of gen- eralized variance functions which are generated by the so-called normal stable Tweedie NST models of k-variate distributions k 1. The generating σ-finite positive measure µ α,t on R k of NST models is composed by the well- known probability measure ξ α,t of univariate positive σ-stable distribution generating Lévy process X α t t which was introduced by Feller 1971 as follows ξ α,t dx = 1 πx ∞ X r=0 t r Γ1 + αrsin−rπα r α r α − 1 −r [1 − αx] αr 1 x dx = ξ α,t xdx , 3.1 where α ∈ 0,1 is the index parameter, Γ. is the classical gamma function, and I A denotes the indicator function of any given event A that takes the value 1 if the event occurs and 0 otherwise. Parameter α can be extended into α ∈ −∞,2] see Tweedie, 1984. For α = 2, we obtain the normal distribution with density ξ 2 ,t dx = 1 √ 2 πt exp −x 2 2t dx . For a k-dimensional NST random vector X = X 1 , . . . , X k ⊤ , the generating 29 30 σ-finite positive measure ν α,t is given by ν α,t dx = ξ α,t dx 1 k Y j=2 ξ 2 ,x 1 dx j , 3.2 where X 1 is a univariate non-negative stable Tweedie variable and all other variables X 2 , . . . , , X k ⊤ =: X c 1 given X 1 are k − 1 real independent Gaussian variables with variance X 1 . By introducing power variance parameter p defined by p − 11 − α = 1 and equivalent to p = p α = α − 2 α − 1 or α = αp = p − 2 p − 1 , now we consider X j as the univariate stable component instead of X 1 for j ∈ {1,2,...,k}, t 0 the dispersion parameter of the associated Lévy process and α = αp the index parameter which is connected to the power variance parameter, the generating σ-finite positive measure ν α,t on R k of NST models is ν α,t;j dx = ξ α,t dx j Y ℓ,j ξ 2 ,x j dx ℓ 3.3 with α = αp ∈ [−∞,0. Since p − 11 − α = 1 then 3.1 can be expressed in term of p namely ξ p ,t with ξ p ,t = ξ p α,t , then equation 3.3 can be written as follows ν p ,t:j dx = ξ p ,t dx j Y ℓ,j ξ ,x j dx ℓ . 3.4 For suitable univariate NEF F p ,t = F ξ αp,t of stable Tweedie types, we can interpret the multivariate NEFs G p ,t = G ν αp,t as composed by the dis- tribution 3.4 of the random vector X = X 1 , . . . , X k ⊤ where X 1 is a univariate stable Tweedie variable generated by xi ,x 1 with mean 0 and variance x 1 . So from Table A.1 in Appendix with S p ⊆ [0,∞, we must retain α in [−∞,1 and the associated univariate model may be called the non-negative stable Tweedie, which include normal Poisson models appearing as new multi- variate distribution having one discrete component. Definition 3.1.1. For X = X 1 , . . . , X k ⊤ a k-dimensional normal stable Tweedie random vector, it must hold that 1. X j is a univariate stable Tweedie random variable, and

2. X

c j |X j := X 1 , . . . , X j−1 , X j+1 , . . . , X k given X j follows the k − 1-variate inde- pendent normal N k−1 , X j I k−1 distribution, where I k−1 = diag k−1 1 , . . . , 1 denotes the k − 1 × k − 1 unit matrix. By equation 3.4, one can obtain the cumulant function K ν p ,t θ = log R R k exp 31 θ T x ν p ,t dx: K ν p ,t;j θ = log    Z R exp θ j x j    Y ℓ,j Z R exp θ ℓ x ℓ ξ ,x j dx ℓ   ξ p ,t dx j    = log    Z R exp θ j x j    Y ℓ,æ exp x j θ 2 ℓ 2   ξ p ,t dx j    = log    Z R exp   θ j x j + 1

2 X

ℓ,j x j θ 2 ℓ   ξ p ,t dx j    = log    Z R exp    x j   θ j + 1

2 X

ℓ,j θ 2 ℓ       ξ p ,t dx j    = tK ξ p ,t   θ j + 1

2 X

ℓ,j θ 2 ℓ    Here K ξ p ,t is the cumulant function of univariate stable Tweedie NEF F ξ p ,t generated by σ-finite positive measures ξ p ,t as follow: K ξ p ,t = tK ξ p ,1 with K ξ p ,1 θ =    exp θ for p = 1 −log−θ for p = 2 1 2 − p 1 − pθ p−2p−1 for 1 , 1 , 2 3.5 for all θ in their respective canonical domains Θ ξ p ,1 =      R for p = 0 , 1 [0 , ∞ for p 0 or 0 p 1 −∞,0 for 1 p 6 2 or p = ∞ −∞,0] for 2 p ∞. The function K ν p ,t;j θ is finite for all θ = θ 1 , . . . , θ k ⊤ in the canonical domain Θ ν p ,t;j =   θ ∈ R k ; θ ⊤ ˜θ c j :=   θ j + 1

2 X

ℓ,j θ 2 ℓ    ∈ Θ p    3.6 where ˜θ c j = θ 1 , . . . , θ j−1 , 1, θ j+1 , . . . , θ k and Θ p =      R for p = 0 , 1 [0 , ∞ for p 0 or 0 p 1 −∞,0 for 1 p 6 2 or p = ∞ −∞,0] for 2 p ∞.