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On normal stable Tweedie models and power-generalized variance functions of only one component

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Abstract

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.

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Notes

  1. Steepness can be characterized as follows: the mean domain (\(M_{p})\) is equal to the interior of the closed convex hull of the distribution support (\(S_{p}\)).

    Table 1 Summary of univariate stable Tweedie models with unit mean domain \(M_{p}\) of (2.4) and support \(S_{p}\) of distributions.

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Acknowledgments

The authors would like to thank the co-editor-in-chief, the associate editor and three anonymous referees for their constructive comments and suggestions. We are grateful to Khoirin Nisa for fruitful discussions and her attentive reading. The second author dedicates this work to Professor Gaston M. N’guérékata for his 60th birthday.

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Correspondence to Célestin C. Kokonendji.

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Boubacar Maïnassara, Y., Kokonendji, C.C. On normal stable Tweedie models and power-generalized variance functions of only one component. TEST 23, 585–606 (2014). https://doi.org/10.1007/s11749-014-0363-9

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