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On normal stable Tweedie models and power-generalized variance functions of only one component. (English) Zbl 1308.62111
Summary: As an extension to normal gamma and normal inverse Gaussian models, all normal stable Tweedie (NST) models are introduced for getting a simple form of the determinant of the covariance matrix, so-called generalized variance. As alternatives to the standard normal model, multivariate 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. Within the framework of exponential dispersion models, a new form of variance functions is firstly established. Then, their generalized variance functions are shown to be powers of only the fixed mean component. Their modified Lévy measures are generally of the normal gamma type, which is connected to NST models through some Monge-Ampère equations. Two kinds of generalized variance estimators are discussed and variance modelling under only observations of normal terms is evoked. Finally, reasonable extensions of NST to multiple stable Tweedie models and corresponding problems are presented.

62H05 Characterization and structure theory for multivariate probability distributions; copulas
60E07 Infinitely divisible distributions; stable distributions
62F10 Point estimation
62H99 Multivariate analysis
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