Ordinary Exponential Dispersion Models

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B. Multivariate Additive Exponential Dispersion Models

The construction of multivariate additive exponential dispersion model is based on an extended convolution method, which interpolates between the set of fully correlated pairs of variables and the set of independent mar- gins of the prescribed form so that the marginal distributions follow a given univariate exponential dispersion model. This method explores the con- volution property of conventional additive exponential dispersion models in order to generate the desired number of parameters, namely k means and kk + 1 2 variance and covariance parameters. Multivariate additive exponential dispersion models are particularly suitable for discrete data, and include multivariate versions of the Poisson, binomial and negative binomial distributions. We start by the extended convolution method in the bivariate additive case as described by Jørgensen and Martínez 2013 and Jørgensen 2013. Suppose that we are given an ordinary bivariate additive ED ∗ µ, λ with CGF s 1 , s 2 ⊤ 7→ λκs 1 , s 2 ; θ 1 , θ 2 = λ[κs 1 + θ 1 , s 2 + θ 2 − κθ 1 , θ 2 ] . From this model we define a multivariate additive exponential dispersion bivariate case by means of the following stochastic representation for the random vector X: X 1 X 2 = U 11 U 12 + U 1 + U 2 , 2.15 where the three vectors on the right-hand side of 2.15 are assumed inde- pendent. More precisely, with θ = θ 1 , θ 2 ⊤ and the weight matrix: Λ = λ 11 λ 12 λ 12 λ 22 , where λ ii = λ 12 + λ i for i = 1 , 2; the joint CGF for X is defined to be Ks 1 , s 2 ; θ, Λ = λ 12 κs 1 , s 2 ; θ 1 , θ 2 + λ 1 κs 1 , 0; θ 1 , θ 2 + λ 2 κ0, s 2 ; θ 1 , θ 2 = λ 12 κs 1 + θ 1 , s 2 + θ 2 + λ 1 κs 1 + θ 1 , θ 2 + λ 2 κθ 1 , s 2 + θ 2 − λ 12 + λ 1 + λ 2 κθ 1 , θ 2 2.16 The equation 2.16 above is interpreted as interpolating between inde- pendence λ 12 = 0 and the maximally correlated case λ 1 = λ 2 = 0. We note that the margins have the same form as for λκs 1 , s 2 ; θ 1 , θ 2 , as seen from the two marginal CGFs as follow Ks 1 , 0; θ, Λ = λ 12 + λ 1 κs 1 , 0; θ 1 , θ 2 = λ 11 s 1 , 0; θ 1 , θ 2 K0 , s 2 ; θ, Λ = λ 12 + λ 2 κ0, s 2 ; θ 1 , θ 2 = λ 22 , s 2 ; θ 1 , θ 2 The construction hence preserves the form of the univariate margins, 14 while replacing the single parameter λ by the three parameters of Λ, for a total of five parameters. Using the notation 2.11, the mean vector for X is EX = diagΛµ = λ 11 κ ′ 1 θ 1 , θ 2 λ 22 κ ′ 2 θ 1 , θ 2 , where κ ′ i denotes the ith partial derivative of κ for i = 1, 2, and where diagΛ is a 2 ×2 diagonal matrix. We use the notation X ∼ ED ∗ 2 µ, Λ for the bivariate additive exponential dispersion defined by 2.16, parametrized by the rate vector µ and weight matrix Σ. The covariance matrix for X is VarX = λ 11 V 11 µ λ 12 V 21 µ λ 12 V 12 µ λ 22 V 22 µ = Λ ⊙ Vµ where the V ij are elements of the unit variance function defined by 2.12. The correlation between X 1 and X 2 is CorrX 1 , X 2 = λ 12 √ λ 11 λ 22 V 12 µ p V 11 µV 22 µ , which for given µ varies between zero and V 12 µ p V 11 µV 22 µ, which may be either positive or negative, depending on the sign of V 12 µ. Now let us consider for the trivariate case, following Jørgensen 2013 we define the trivariate random vector X = X 1 , X 2 , X 3 ⊤ as the sum of six independent vectors    X 1 X 2 X 3    =    U 11 U 12   +    U 21 U 23   +    U 32 U 33   +    U 1   +    U 2   +    U 3   , 2.17 of which three terms are bivariate and three are univariate. However, rather than starting from a trivariate CGF κs 1 , s 2 , s 3 ; θ as was done by Jørgensen, 2013, Jørgensen and Martínez 2013 started from the three bivariate distri- butions corresponding to the first three terms 2.17. For this construction to work, it is crucial that the univariate margins of the three bivariate terms are consistent, so that for example U 11 and U 21 have distributions that belong to the same class. In order to avoid the intricacies of such a construction in the general case, we concentrate here on the homogeneous case, where all three margins belong to the same class, for example a multivariate gamma distribution with gamma margins. Using µ parameterization, the trivariate exponential dispersion model 15 ED ∗ 3 µ, Λ is defined via the joint CGF for the vector X as follows: Ks 1 , s 2 , s 3 ; µ, Λ = λ 12 κs 1 , s 2 ; µ 1 , µ 2 + λ 13 κs 1 , s 3 ; µ 1 , µ 3 + λ 23 κs 2 , s 3 ; µ 2 , µ 3 + λ 1 κs 1 , µ 1 + λ 2 κs 2 , µ 2 + λ 3 κs 3 , µ 3 This definition satisfies the requirement that each marginal distribution belongs to the univariate model ED ∗ µ, λ. For example, the CGF of the first margin is Ks 1 , 0, 0; µ, Λ = λ 12 κs 1 , 0; µ 1 , µ 2 + λ 13 κs 1 , 0; µ 1 , µ 3 + λ 23 κ0, 0; µ 2 , µ 3 + λ 1 κs 1 , µ 1 + λ 2 κ0, µ 2 + λ 3 κ0, µ 3 = λ 12 κs 1 , µ 1 + λ 13 κs 1 , µ 1 + λ 1 κs 1 , µ 1 While all three bivariate marginal distributions are of the form 2.16. For example, the CGF of the joint distribution of X 1 and X 2 is Ks 1 , s 2 , 0; µ, Λ = λ 12 κs 1 , s 2 ; µ 1 , µ 2 + λ 13 κs 1 , 0; µ 1 , µ 3 + λ 23 κs 2 , 0; µ 2 , µ 3 + λ 1 κs 1 , µ 1 + λ 2 κs 2 , µ 2 + λ 3 κ0, µ 3 = λ 12 κs 1 , s 2 ; µ 1 , µ 2 + λ 1 + λ 13 κs 1 , µ 1 + λ 2 + λ 23 κs 2 , µ 2 , which has rate vector µ 1 , µ 2 ⊤ and weight matrix given by the upper left 2 × 2 block of Λ. Based on these considerations the construction for bivariate and trivari- ate cases, the multivariate exponential dispersion model ED ∗ k µ, Λ for gen- eral k is defined as the following. Definition 2.2.2. An additive k-variate exponential dispersion model ED ∗ k µ, Λ with k × 1 rate vector µ = µ 1 , . . . , µ k ⊤ and k × k weignt matrix Λ = {λ ij } k i ,j=1 , is a family distribution with a joint cumulant generating function of the form Ks; µ, Λ = X i j λ ij κs i , s j ; µ i , µ j + k X i=1 λ i κs i , µ i , 2.18 where all weights λ ij and λ i are positive, with λ ii = P j;j,i λ ij + λ i , The mean vector of 2.18 is EX = diagΛµ, where diag{Λ} is a k × k diagonal matrix. By arguments similar to those given in the trivariate case above it can be shown that each univariate margin belongs to the univariate model ED ∗ µ, λ, and that all marginal distributions for a subset of the k variables are again of the form 2.18. ED ∗ k µ, Λ of 2.18 also satisfies the following generalized additive prop-