Real natural exponential families and generalized orthogonality

References

• Abid, R., C. C. Kokonendji, and A. Masmoudi. 2020. Geometric Tweedie regression models for continuous and semicontinuous data with variation phenomenon. AStA Advances in Statistical Analysis 104 (1):33–58. doi: 10.1007/s10182-019-00350-8.
   Web of Science ®Google Scholar
• Abid, R., C. C. Kokonendji, and A. Masmoudi. 2021. On Poisson-exponential-Tweedie models for ultra-overdispersed count data. AStA Advances in Statistical Analysis 105 (1):1–23. doi: 10.1007/s10182-020-00375-4.
   Web of Science ®Google Scholar
• Bar-Lev, S. K. 1987. Discussion on paper by B. Jørgensen, “Exponential dispersion models. Journal of the Royal Statistical Society Series B 49 (2):153–4.
   Google Scholar
• Bar-Lev, S. K., and P. Enis. 1986. Reproducibility and natural exponential families with power variance functions. The Annals of Statistics 14 (4):1507–22. doi: 10.1214/aos/1176350173.
   Web of Science ®Google Scholar
• Bar-Lev, S. K., and C. C. Kokonendji. 2017. On the mean value parametrization of natural exponential families, a revisited review. Mathematical Methods of Statistics 26 (3):159–75. doi: 10.3103/S1066530717030012.
   Google Scholar
• Boubacar Maïnassara, Y., and C. C. Kokonendji. 2014. On normal stable Tweedie models and power-generalized variance functions of only one component. TEST 23 (3):585–606. doi: 10.1007/s11749-014-0363-9.
   Web of Science ®Google Scholar
• Bryc, W., R. Fakhfakh, and W. Mlotkowski. 2019. Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality. Probability and Mathematical Statistics 39 (2):237–58. doi: 10.19195/0208-4147.39.2.1.
   Web of Science ®Google Scholar
• Casalis, M. 1996. The 2d + 4 simple quadratic natural exponential families on Rd. The Annals of Statistics 24 (4):1828–54.
   Web of Science ®Google Scholar
• Consonni, G., and P. Veronese. 1992. Conjugate priors for exponential families having quadratic variance functions. Journal of the American Statistical Association 87 (420):1123–7. doi: 10.1080/01621459.1992.10476268.
   Web of Science ®Google Scholar
• Feinsilver, P. 1986. Some classes of orthogonal polynomials associated with martingales. Proceedings of the American Mathematical Society 98 (2):298–302. doi: 10.1090/S0002-9939-1986-0854037-1.
   Web of Science ®Google Scholar
• Hamza, M., and A. Hassairi. 2013. Bayesian approach to cubic natural exponential families. Statistics & Probability Letters 83 (9):1946–55. doi: 10.1016/j.spl.2013.04.020.
   Web of Science ®Google Scholar
• Hassairi, A. 1992. La classification des familles exponentielles naturelles sur Rn par laction du groupe linaire de Rn+1. Comptes-rendus de l’Académie des Sciences de Paris 315:207–10.
   Google Scholar
• Hassairi, A., and M. Zarai. 2004. Characterization of the cubic exponential families by orthogonality of polynomials. The Annals of Probability 32 (3B):2463–76. doi: 10.1214/009117904000000522.
   Web of Science ®Google Scholar
• Hassairi, A., and M. Zarai. 2005. Bhattacharyya matrices and cubic exponential families. Statistical Methodology 2 (3):226–32. doi: 10.1016/j.stamet.2005.04.003.
   Google Scholar
• Hassairi, A., and M. Zarai. 2006. Characterization of the simple cubic multivariate exponential families. Journal of Functional Analysis 235 (1):69–89. doi: 10.1016/j.jfa.2005.09.005.
   Web of Science ®Google Scholar
• Hinde, J., and C. G. Demétrio. 1998. Overdispersion: Models and estimation. Computational Statistics & Data Analysis 27 (2):151–70. doi: 10.1016/S0167-9473(98)00007-3.
   Web of Science ®Google Scholar
• Jørgensen, B. 1987. Exponential dispersion models. Journal of the Royal Statistical Society: Series B (Methodological) 49 (2):127–45. doi: 10.1111/j.2517-6161.1987.tb01685.x.
   Web of Science ®Google Scholar
• Jørgensen, B. 1997. The theory of exponential dispersion models. In Monographs on statistics and probability, vol. 76. London: Chapman and Hall, 1997.
   Google Scholar
• Jørgensen, B., and C. C. Kokonendji. 2016. Discrete dispersion models and their Tweedie asymptotics. AStA Advances in Statistical Analysis 100 (1):43–78. doi: 10.1007/s10182-015-0250-z.
   Web of Science ®Google Scholar
• Kokonendji, C. C. 2005a. Characterizations of some polynomial variance functions by d-pseudo-orthogonality. Journal of Applied Mathematics and Computing 19:1–2. doi: 10.1007/BF02935816.
   Google Scholar
• Kokonendji, C. C. 2005b. On d-orthogonality of the Sheffer systems associated to a convolution semigroup. Journal of Computational and Applied Mathematics 181 (1):83–91. doi: 10.1016/j.cam.2004.11.019.
   Web of Science ®Google Scholar
• Kokonendji, C. C., C. G. Demétrio, and S. S. Zocchi. 2007. On Hinde–Demétrio regression models for overdispersed count data. Statistical Methodology 4 (3):277–91. doi: 10.1016/j.stamet.2006.10.001.
   Google Scholar
• Kokonendji, C. C., S. Dossou-Gbété, and C. G. B. Demétrio. 2004. Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demetrio classes. Statistics and Operations Research Transactions 28 (2):201–14.
   Google Scholar
• Kokonendji, C. C., and D. Pommeret. 2005. Characterization of multivariate exponential families with polynomial variance function. Advances in Mathematics- African Diaspora Journal of Mathematics 1:78–84.
   Google Scholar
• Kokonendji, C. C., and M. Zarai. 2007. Transorthogonal polynomials and simple cubic multivariate distributions. Far East Journal of Theoretical Statistics 21:171–201.
   Google Scholar
• Labeye-Voisin, E., and D. Pommeret. 1995. Polynômes orthogonaux associées aux familles exponentielles. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics 320:79–84.
   Google Scholar
• Letac, G., and M. Mora. 1990. Natural real exponential families with cubic variance functions. The Annals of Statistics 18:1–37. doi: 10.1214/aos/1176347491.
   Web of Science ®Google Scholar
• Meixner, J. 1934. Orthogonale polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden funktion. Journal of the London Mathematical Society s1-9 (1):6–13. doi: 10.1112/jlms/s1-9.1.6.
   Google Scholar
• Morris, C. N. 1982. Natural exponentials families with quadratic variance function. The Annals of Statistics 10:65–80. doi: 10.1214/aos/1176345690.
   Web of Science ®Google Scholar
• Pommeret, D. 2000. Orthogonality of the Sheffer system associated to a Levy process. Journal of Statistical Planning and Inference 86 (1):1–10. doi: 10.1016/S0378-3758(99)00158-5.
   Web of Science ®Google Scholar
• Schoutens, W., and J. L. Teugels. 1998. Lévy processes, polynomials and martingales. Communications in Statistics. Stochastic Models 14 (1-2):335–49. doi: 10.1080/15326349808807475.
   Google Scholar
• Shanbhag, D. N. 1972. Some characterizations based on the Bhattacharya matrix. Journal of Applied Probability 9 (03):580–7. doi: 10.1017/S0021900200035889.
   Google Scholar
• Sheffer, I. M. 1937. Concerning Appell sets and associated linear functional equations. Duke Mathematical Journal 3:593–609. doi: 10.1215/S0012-7094-37-00347-8.
   Google Scholar
• Tweedie, M. C. K. 1984. An index which distinguishes between some important exponential families. In Statistics: Applications and new directions. Proc. Indian Institute Golden Jubilee lnternat. Conf, ed. J. K. Ghosh and J. Roy, 579–604. Calcutta: Indian Statistical Institute.
   Google Scholar
• Zarai, M. 2007. Heisenberg-Weyl Lie algebra and natural exponential families. Infinite Dimensional Analysis. Quantum Probability and Related Topics 10:293–303.
   Web of Science ®Google Scholar