Definition and Properties Multivariate Discrete Dispersion Models

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B. Multivariate Poisson Tweedie models

Multivariate Poisson Tweedie models Jørgensen and Kokonendji, 2011 are special cases of multivariate discrete dispersion models. The models are considered as a new class of multivariate Poisson-Tweedie mixtures, which is based on the multivariate Tweedie distributions of Jørgensen and Martínez 2013. Consider the k-variate Tweedie distribution Y ∼ Tw p µ, Σ with mean vector µ and covariance matrix CovY = Diagµ p 2 Σ Diagµ p 2 . 2.31 Table 2.3: Summary of Multivariate Poisson Tweedie Models Model p Type Multivariate Neyman Type A p = 1 + Multivariate Poisson-Compound Poisson 1 p 32 + Multivariate Pólya-Aeppli p = 3 2 + Multivariate Poisson-compound Poisson 3 2 p 2 + Multivariate negative binomial p = 2 + Multivariate factorial discrete stable p 2 + Multivariate Poisson-inverse Gaussian p = 3 + The multivariate Poisson-Tweedie model PT p µ, Σ is defined as a Poisson mixture X |Y ∼ independent PoY i for i = 1 , . . . , k, where X 1 , . . . , X k are assumed conditionally independent given Y . The mul- tivariate Poisson-Tweedie model has univariate Poisson-Tweedie margins, X i ∼ PT p µ i , σ ii where σ ij denote the entries of Σ . The mean vector is µ and the dispersion matrix is 2.31 positive-definite making the distribution overdispersed. The covariance matrix for X has the form CovX = Diagµ + Diagµ Σ Diagµ, making it straightforward to fit multivariate Poisson-Tweedie regression models using quasi-likelihood. The multivariate Poisson-Tweedie model satisfies the following dilation property: DiagcPT p µ, Σ = PT p h Diagcµ, Diagc 1−p2 Σ Diagc 1−p2 i , where c is a k-vector with positive elements. 2.2.4. Applications Here we give some examples from literatures which have emphasized the application of a certain dispersion model. The followings are illustra- tive descriptions of the applications of exponential, geometric and discrete dispersion models. 27 Stochastic model for heterogeneities in regional organ blood flow The theory of exponential dispersion models was applied to construct a stochastic model for heterogeneities in regional organ blood flow Kendal, 2001. Regional organ blood flow exhibits a significant degree of spatial heterogeneity when measured by using labeled microspheres or by other means. The related velocity distribution of blood flow has been character- ized by a gamma distribution. To provide a stochastic description for the macroscopic and microscopic heterogeneities in regional organ blood flow, a scale invariant compound Poisson-gamma distribution was employed. Distribution of human single nucleotide polymorphisms A scale invariant Poisson gamma PG exponential dispersion model was used in modeling the distribution of human single nucleotide poly- morphisms NSPs Kendal, 2003. The SNPs appear to be non-uniformly dispersed throughout the human genome, this non-uniformity can be at- tributed to a segmented genealogical substructure within the genome, where older segments may have accumulated greater numbers of SNPs. An analy- sis of 1.42 million human single nucleotide polymorphisms SNPs revealed an apparent power function relationship between the estimated variance and mean number of SNPs per sample bin. By PG-EDM model the sample bins contain random Poisson distributed numbers of identical by descent genomic segments, each with independently gamma distributed numbers of SNPs. Geometric α-Laplace marginals in autoregressive process Geometric Laplace distribution is one of the geometric dispersion model. Lekshmi and Jose 2004 proposed the used of geometric Laplace distribution as stationary marginal in autoregressive process modeling. Some contexts of applications were mentioned in their paper, e.g. for modeling pooled position errors in a large navigation system, for modeling sulphate concen- tration data with ARMA model, and for modeling financial time series data see Hsu 1979, Damsleth and El-Shaarawi 1989, Anderson and Arnold 1993 respectively for details. Integer valued time series modeling A discrete dispersion model was applied in modeling autoregressive time series of count data. The model was developed in terms of a convo- lution of Poisson and negative binomial random variables known as Pois- son–negative binomial PNB distribution Jose and Mariyamma, 2015. The distribution was used as the marginal distribution of the time series model. 28