Abstract
It is well known that any natural exponential family (NEF) is characterized by its variance function on its mean domain, often much simpler than the corresponding generating probability measures. The mean value parametrization appeared to be crucial in some statistical theory, like in generalized linear models, exponential dispersion models and Bayesian framework. The main aim of the paper is to expose the mean value parametrization for possible statistical applications. The paper presents an overview of the mean value parametrization and of the characterization property of the variance function for NEF’s. In particular it introduces the relationships existing between the NEF’s generating measure, Laplace transform and variance function as well as some supplemental results concerning the mean value representation. Some classes of polynomial variance functions are revisited for illustration. The corresponding NEF’s of such classes are generated by counting probabilities on the nonnegative integers and provide Poisson-overdispersed competitors to the homogeneous Poisson distribution.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Y. Awad, S. K. Bar-Lev, and U. Makov, A New Class of Counting Distributions Embedded in the Lee–Carter Model forMortality Projections: A Bayesian Approach, Techn. Rep. (Actuarial Research Center, Univ. of Haifa, Israel, 2016) (in press).
S. K. Bar-Lev, “Discussion on paper by B. Jørgensen ‘Exponential dispersion models’,” J. Roy. Statist. Soc., Ser. B 49, 153–154 (1987).
S. K. Bar-Lev, B. Boukai, and I. Kleiner, “The Kendal–Ressel Exponential Dispersion Model: Some Statistical Aspects and Estimation”, Int. J. Statist. and Probab. 5 (3), 32–41 (2016).
S. K. Bar-Lev and D. Bshouty, “A Class of Infinitely Divisible Variance Functions with an Application to the Polynomial Case”, Probab. and Statist. Lett. 10, 377–379 (1990).
S. K. Bar-Lev, D. Bshouty, and P. Enis, “Variance Functions with Meromorphic Means”, Ann. Probab. 19, 1349–1366 (1991).
S. K. Bar-Lev, D. Bshouty, and P. Enis, “On Polynomial Variance Functions”, Probab. Theory and Rel. Fields 94, 69–82 (1991).
S. K. Bar-Lev, D. Bshouty, P. Enis, and A. Y. Ohayon, “Compositions and Products of Infinitely Divisible Variance Functions”, Scandinavian J. Statist. 19, 83–89 (1992).
S. K. Bar-Lev, D. Bshouty, P. Grünwald, and P. Harremoës, “Jeffreys versus Shtarkov Distributions Associated with Some Natural Exponential Families”, Statist. Methodology 7, 638–643 (2010).
S. K. Bar-Lev, D. Bshouty, and Z. Landsman, “Second Order Minimax Estimation of the Mean”, J. Statist. Planning and Inference 140, 3282–3294 (2010).
S. K. Bar-Lev and P. Enis, “Reproducibility and Natural Exponential Families with Power Variance Functions”, Ann. Statist. 14, 1507–1522 (1986).
S. K. Bar-Lev and G. Letac, “Increasing Hazard Rate of Mixtures for Natural Exponential Families”, Advances in Appl. Probab. 44, 373–390 (2009).
S. K. Bar-Lev and Z. Landsman, “Exponential Dispersion Models: Second-Order Minimax Estimation of the Mean for Unknown Dispersion Parameter”, J. Statist. Planning and Inference 136, 3837–3851 (2006).
O. E. Barndorff-Nielsen, Information and Exponential Families in Statistical Theory (Wiley, New York, 1978).
J. Castillo and M. Pérez-Casany, “Overdispersed and Underdispersed Poisson Generalizations”, J. Statist. Planning and Inference 134, 486–500 (2005).
A. Delwarde, M. Denuit, and C. Partrat, “Negative Binomial Version of the Lee–Carter Model for Mortality Forecasting”, Appl. StochasticModels in Business and Industry 23 (5), 385–401 (2007).
P. K. Dunn and G. K. Smith, “Series Evaluation of Tweedie Exponential Dispersion Densities”, Statistics and Computing 15, 267–280 (2005).
P. K. Dunn and G. K. Smith, “Evaluation of Tweedie Exponential Dispersion Model Densities by Fourier Inversion”, Statistics and Computing 18, 73–86 (2008).
I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (seventh edition) (Elsevier and Academic Press, New York, 2007).
S. Haberman and A. Renshaw, “A Comparative Study of Parametric Mortality Projection Models”, Insurance: Mathematics and Economics 48 (1), 35–55 (2011).
D. B. Hall and K. S. Berenhaut, “Score Tests for Heterogeneity and Overdispersion in Zero-Inflated Poisson and Binomial Regression Models”, Canadian J. Statist. 30, 1–16 (2002).
J. Hinde and C. G. B. Demétrio, “Overdispersion: Models and Estimation”, Comput. Statist. and Data Analysis 27, 151–170 (1998).
P. Hougaard, K.-L. T. Lee, and G. A. Whitmore, “Analysis of Overdispersed Count Data by Mixtures of Poisson Variables and Poisson Processes”, Biometrics 53, 1225–1238 (1997).
N. L. Johnson, S. Kotz, and A. W. Kemp, Univariate Discrete Distributions, 2nd ed. (Wiley, New York, 1992).
B. Jørgensen, “Exponential Dispersion Models (with Discussion)”, J. Roy. Statist. Soc., Ser. B 49, 127–162 (1987).
B. Jørgensen, The Theory of Exponential Dispersion Models, in Monographs on Statistics and Probability (Chapman and Hall, London, 1997), Vol. 76.
C. C. Kokonendji, S. Dossou-Gbété, and C. G. B. Demétrio, “Some Discrete Exponential Dispersion Models: Poisson–Tweedie and Hinde–Demétrio Classes”, Statist. and Operations Research Transactions 28, 201–214 (2004).
C. C. Kokonendji, C. G. B. Demétrio, and S. S. Zocchi, “On Hinde–Demétrio Regression Models for Overdispersed Count Data”, Statist. Methodology 4, 277–291, (2007).
C. C. Kokonendji and M. Khoudar, “On Strict Arcsine Distribution”, Commun. Statist.–Theory and Methods 33, 993–1006 (2004).
C. C. Kokonendji and M. Khoudar, “On LévyMeasures for Infinitely DivisibleNatural Exponential Families”, Statist. and Probab. Lett. 76, 1364–1368 (2006).
C. C. Kokonendji and D. Malouche, “A Property of Count Distributions in the Hinde–Demétrio Family”, Commun. Statist.–Theory and Methods 37, 1823–1834 (2008).
R. D. Lee and L. Carter, “Modeling and Forecasting the Time Series of US Mortality”, J. Amer. Statist. Assoc. 87, 659–671 (1992).
G. Letac, Lectures on Natural Exponential Families and Their Variance Functions, in Monografias de Matemática (Insituto de Matemática Pura Eplicada, Rio de Janeiro, 1992), Vol. 50.
G. Letac, Associated Natural Exponential Families and Elliptic Functions, in The Fascination of Probability and Statistics and their Applications, Ed. by M. Podolskij, R. Stelzer, S. Thorbjornsen, and A. E. D. Veraat (Springer, 2016).
G. Letac and M. Mora, “Natural Real Exponential Families with Cubic Variance Functions”, Ann. Statist. 18, 1–37 (1990).
E. Lukacs, Characteristic Functions, 2nd ed. (Hafner, New York, 1970).
E. Lukacs, Developments in Characteristic Functions Theory (Griffin, London, 1983).
C. N. Morris, “Natural Exponential Families with Quadratic Variance Functions”, Ann. Statist. 10, 65–80 (1982).
C. N. Morris, “Natural Exponential Families with Quadratic Variance Functions: Statistical Theory”, Ann. Statist. 11, 515–529 (1983).
J. A. Nelder and R. W. M. Wedderburn, “Generalized Linear Models”, J. Roy. Statist. Soc., Ser. A 135, 370–384 (1972).
R. S. Pimentel, M. Niewiadomska-Bugaj, and J.-C. Wang, “Association of Zero-Inflated Continuous Variables”, Statist. and Probab. Lett. 96, 61–67 (2015).
N. U. Prabhu, Queues and Inventories: A Study of Basic Stochastic Processes (Wiley, New York, 1965).
A. E. Renshaw and S. Haberman, “A Cohort-Based Extension to the Lee–Carter Model for Mortality Reduction Factors”, Insurance:Mathematics and Economics 38, 556–570 (2006).
M. Ridout, C. G. B. Demétrio, and J. Hinde, “Models for Count Data with Many Zeros”, in Proc. XIXth Intern. Biometrics Conf., Cape Town, 1998, pp. 179–192.
G. K. Smyth and B. Jørgensen, “Fitting Tweedie’s Compound Poisson Model to Insurance Claims Data: Dispersion Modelling”, Astin Bulletin 32, 143–157 (2002).
M. C. K. Tweedie, “An Index Which Distinguishes between Some Important Exponential Families”, in Statistics: Applications and New Directions, Proc. Indian Statist. Inst. Golden Jubilee International Conference, Ed. by J. K. Ghosh and J. Roy (Indian Statist. Inst., Calcutta, 1984), pp. 579–604.
V. Vinogradov, R. B. Paris, and O. Yanushkevichiene, “New Properties for Members of the Power-Variance Family. I”, Lithuanian Math. J. 52, 444–461 (2012).
V. Vinogradov, R. B. Paris, and O. Yanushkevichiene, “New Properties for Members of the Power-Variance Family. II”, Lithuanian Math. J. 53, 103–120 (2013).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Bar-Lev, S.K., Kokonendji, C.C. On the mean value parametrization of natural exponential families — a revisited review. Math. Meth. Stat. 26, 159–175 (2017). https://doi.org/10.3103/S1066530717030012
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530717030012
Keywords
- natural exponential families
- exponential dispersion models
- variance functions
- polynomial variance functions
- Poisson-overdispersed distribution