, Volume 23, Issue 3, pp 585–606 | Cite as

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

  • Yacouba Boubacar Maïnassara
  • Célestin C. KokonendjiEmail author
Original Paper


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.


Covariance matrix Determinant Generalized variance estimator Lévy measure Monge–Ampère equation Multivariate exponential dispersion model 

Mathematics Subject Classification (2010)

62H05 60E07 62F10 62H99 



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.


  1. Alfaro JL, Ortega JF (2012) Robust Hotelling’s \(T^{2}\) control charts under non-normality: the case of \(t\)-Student distribution. J Stat Comput Simul 82:1437–1447CrossRefzbMATHMathSciNetGoogle Scholar
  2. Antoniadis A, Besbeas P, Sapatinas T (2001) Wavelet shrinkage for natural exponential families with cubic variance functions. Sankhya A 63:309–327zbMATHMathSciNetGoogle Scholar
  3. Bar-Lev S, Bschouty D, Enis P, Letac G, Lu I, Richard D (1994) The diagonal multivariate natural exponential families and their classification. J Theor Probab 7:883–929CrossRefzbMATHGoogle Scholar
  4. Barndorff-Nielsen OE (1997) Normal inverse Gaussian distribution and stochastic volatility modelling. Scand J Stat 24:1–13CrossRefzbMATHMathSciNetGoogle Scholar
  5. Barndorff-Nielsen OE (1998) Processes of normal inverse Gaussian type. Financ Stoch 2:41–68CrossRefzbMATHMathSciNetGoogle Scholar
  6. Barndorff-Nielsen OE, Shephard N (2002) Normal modified stable processes. Theory Probab Math Stat 65:1–19MathSciNetGoogle Scholar
  7. Barndorff-Nielsen OE, Kent J, Sørensen M (1982) Normal variance-mean mixtures and z distributions. Int Stat Rev 50:145–159CrossRefzbMATHGoogle Scholar
  8. Bernardo JM, Smith AFM (1993) Bayesian theory. Wiley, New YorkGoogle Scholar
  9. Bernardoff P, Kokonendji CC, Puig B (2008) Generalized variance estimators in the multivariate gamma models. Math Methods Stat 17:66–73CrossRefzbMATHMathSciNetGoogle Scholar
  10. Casalis M (1996) The \(2d+4\) simple quadratic natural exponential families on \(\mathbb{R}^{d}\). Ann Stat 24:1828–1854CrossRefzbMATHMathSciNetGoogle Scholar
  11. Chen ZY (2005) The S-system computation of non-central gamma distribution. J Stat Comput Simul 75:813–829CrossRefzbMATHMathSciNetGoogle Scholar
  12. Consonni G, Veronese P, Gutiérrez-Peña E (2004) Reference priors for exponential families with simple quadratic variance function. J Multivar Anal 88:335–364CrossRefzbMATHGoogle Scholar
  13. Dunn PK (2013) The R package ‘tweedie’ Version 2.1.7 of 2013.01.15.
  14. Feller W (1971) An introduction to probability theory and its applications, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  15. Hassairi A (1993) La classification des familles exponentielles naturelles sur \(\mathbb{R}^{n}\) par l’action du groupe linéaire de \( \mathbb{R}^{n+1}\). C R Acad Sci Paris Série I 315:207–210MathSciNetGoogle Scholar
  16. Hassairi A (1999) Generalized variance and exponential families. Ann Stat 27:374–385CrossRefzbMATHMathSciNetGoogle Scholar
  17. Iliopoulos G, Kourouklis S (1998) On improved interval estimation for the generalized variance. J Stat Plan Inference 66:305–320CrossRefzbMATHMathSciNetGoogle Scholar
  18. Jørgensen B (1997) The theory of dipersion models. Chapman & Hall, LondonGoogle Scholar
  19. Jørgensen B (2013) Construction of multivariate dispersion models. Braz J Probab Stat 27:285–309CrossRefMathSciNetGoogle Scholar
  20. Kokonendji CC (1994) Exponential families with variance functions in \(\sqrt{\Delta }P(\sqrt{\Delta })\): Seshadri’s class. Test 3:123–172CrossRefzbMATHMathSciNetGoogle Scholar
  21. Kokonendji CC, Khoudar M (2006) On Lévy measures for infinitely divisible natural exponential families. Stat Probab Lett 76:1364–1368CrossRefzbMATHMathSciNetGoogle Scholar
  22. Kokonendji CC, Masmoudi A (2006) A characterization of Poisson–Gaussian families by generalized variance. Bernoulli 12:371–379CrossRefzbMATHMathSciNetGoogle Scholar
  23. Kokonendji CC, Masmoudi A (2013) On the Monge–Ampère equation for characterizing gamma-Gaussian model. Stat Probab Lett 83:1692–1698CrossRefzbMATHMathSciNetGoogle Scholar
  24. Kokonendji CC, Pommeret D (2007) Comparing UMVU and ML estimators of the generalized variance for natural exponential families. Statistics 41:547–558CrossRefzbMATHMathSciNetGoogle Scholar
  25. Kokonendji CC, Seshadri V (1996) On the determinant of the second derivative of a Laplace transform. Ann Stat 24:1813–1827CrossRefzbMATHMathSciNetGoogle Scholar
  26. Kotz S, Balakrishnan N, Johnson NL (2000) Continuous multivariate distributions, 2nd edn. Models and applicationsWiley, New YorkGoogle Scholar
  27. Koudou AE, Pommeret D (2002) A characterization of Poisson–Gaussian families by convolution-stability. J Multivar Anal 81:120–127CrossRefzbMATHMathSciNetGoogle Scholar
  28. Knüsel L, Bablok B (1996) Computation of the noncentral gamma distribution. SIAM J Sci Comput 17:1224–1231CrossRefzbMATHMathSciNetGoogle Scholar
  29. Lee KJ, Thompson SG (2008) Flexible parametric models for random-effects distributions. Stat Med 27:418–434CrossRefMathSciNetGoogle Scholar
  30. Letac G (1989) Le problème de la classification des familles exponentielles naturelles sur \(\mathbb{R}^{d}\) ayant une fonction variance quadratique. In: Heyer H (ed) Probability measures on groups IX, vol 1306, Lecture Notes in MathSpringer, Berlin, pp 194–215Google Scholar
  31. Loeper G, Rapetti F (2005) Numerical solution of the Monge–Ampère equation by a Newton’s algorithm. C R Acad Sci Paris Série I 340:319–324CrossRefzbMATHMathSciNetGoogle Scholar
  32. Ølgård TA, Hanssen A, Hansen RE, Godtliebsen F (2005) EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution. Signal Process 85:1655–1673CrossRefGoogle Scholar
  33. Patnaik PB (1949) The noncentral chi-square and \(F\)-distributions and their applications. Biometrika 36: 202–232Google Scholar
  34. Roberts GE, Kaufman H (1966) Table of Laplace transforms. Saunders, LondonzbMATHGoogle Scholar
  35. Samorodnitsky G, Taqqu MS (1994) Stable non-Gaussian random processes—stochastic models with infinite variance. Chapman & Hall, New YorkzbMATHGoogle Scholar
  36. Sato K (1999) Lévy processes and infinitely divisible distributions. Cambridge University Press, CambridgezbMATHGoogle Scholar
  37. Seshadri V (1993) The inverse Gaussian distribution: a case study in exponential families. Oxford University Press, New YorkGoogle Scholar
  38. Shorrock RW, Zidek JV (1976) An improved estimator of the generalized variance. Ann Stat 4:629–638CrossRefzbMATHMathSciNetGoogle Scholar
  39. Tweedie MCK (1984) An index which distinguishes between some important exponential families. In: Ghosh JK, Roy J (eds) Statistics: applications and new directions. Proceedings of the Indian Statistical Golden Jubilee International Conference, Calcutta, pp 579–604.Google Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2014

Authors and Affiliations

  • Yacouba Boubacar Maïnassara
    • 1
  • Célestin C. Kokonendji
    • 1
    Email author
  1. 1.Université de Franche-Comté, UFR Sciences et Techniques, Laboratoire de Mathématiques de Besançon, UMR 6623 CNRS-UFCBesançon CedexFrance

Personalised recommendations