In their fundamental paper on cubic variance functions (VFs), Letac and Mora (The Annals of Statistics, 1990) presented a systematic, rigorous and comprehensive study of natural exponential families (NEFs) on the real line, their characterization through their VFs and mean value parameterization. They presented a section that for some reason has been left unnoticed. This section deals with the construction of VFs associated with NEFs of counting distributions on the set of nonnegative integers and allows to find the corresponding generating measures. As EDMs are based on NEFs, we introduce in this paper two new classes of EDMs based on their results. For these classes, which are associated with simple VFs, we derive their mean value parameterization and their associated generating measures. We also prove that they have some desirable properties. Both classes are shown to be overdispersed and zero inflated in ascending order, making them as competitive statistical models for those in use in both, statistical and actuarial modeling. To our best knowledge, the classes of counting distributions we present in this paper, have not been introduced or discussed before in the literature. To show that our classes can serve as competitive statistical models for those in use (e.g., Poisson, Negative binomial), we include a numerical example of real data. In this example, we compare the performance of our classes with relevant competitive models.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/ps/2021001
Keywords: Exponential dispersion model, natural exponential family, overdispersion, variance function, zero-inflated distribution
@article{PS_2021__25_1_31_0,
author = {Bar-Lev, Shaul K. and Ridder, Ad},
title = {New exponential dispersion models for count data: the {ABM} and {LM} classes},
journal = {ESAIM: Probability and Statistics},
pages = {31--52},
year = {2021},
publisher = {EDP-Sciences},
volume = {25},
doi = {10.1051/ps/2021001},
mrnumber = {4224734},
zbl = {1481.60018},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2021001/}
}
TY - JOUR AU - Bar-Lev, Shaul K. AU - Ridder, Ad TI - New exponential dispersion models for count data: the ABM and LM classes JO - ESAIM: Probability and Statistics PY - 2021 SP - 31 EP - 52 VL - 25 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2021001/ DO - 10.1051/ps/2021001 LA - en ID - PS_2021__25_1_31_0 ER -
%0 Journal Article %A Bar-Lev, Shaul K. %A Ridder, Ad %T New exponential dispersion models for count data: the ABM and LM classes %J ESAIM: Probability and Statistics %D 2021 %P 31-52 %V 25 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2021001/ %R 10.1051/ps/2021001 %G en %F PS_2021__25_1_31_0
Bar-Lev, Shaul K.; Ridder, Ad. New exponential dispersion models for count data: the ABM and LM classes. ESAIM: Probability and Statistics, Tome 25 (2021), pp. 31-52. doi: 10.1051/ps/2021001
[1] , and , On Poisson-exponential-Tweedie models for ultra-overdispersed count data. To appear in: AStA Adv. Stat. Anal. (2020). | DOI | MR | Zbl
[2] , and , A new class counting distributions embedded in the Lee-Carter model for mortality projections: a Bayesian approach. A technical report No. 146. Actuarial Research Center, University of Haifa, Israel (2016).
[3] , Discussion on paper by B. Jørgensen, “Exponential dispersion models”. J. Roy. Stat. Soc. Ser. B 49 (1987) 153–154. | MR
[4] and , Reproducibility and natural exponential families with power variance functions. Ann. Stat. 14 (1986) 1507–1522. | MR | Zbl
[5] and , On the mean value parameterization of natural exponential families – a revisited review. Math. Methods Stat. 26 (2017) 159–175. | MR | Zbl | DOI
[6] and , Monte Carlo methods for insurance risk computation. Int. J. Stat. Probab. 8 (2019) 54–74. | DOI
[7] and , Exponential dispersion models for overdispersed zero-inflated count data. Preprint (2020). | arXiv
[8] , Information and Exponential Families in Statistical Theory. Wiley, New York (1978). | MR | Zbl
[9] and , A new infinitely divisible discrete distribution with applications to count data modelling. Commun. Stat. Theory Methods 48 (2019) 1401–1416. | MR | Zbl | DOI
[10] , , , and , Extended Poisson–Tweedie: Properties and regression models for count data. Stat. Model. 18 (2018) 24–49. | MR | Zbl | DOI
[11] and , Approximation operators, q-exponential, and free exponential families. Preprint (2005). | arXiv
[12] , and , On the two-parameter Bell–Touchard discrete distribution. Commun. Stat. Theory Methods 49 (2020) 4834–4852. | MR | Zbl | DOI
[13] , Generalized Poisson Distributions: Properties and Applications. Marcel Dekker, New York (1989). | MR | Zbl
[14] and , Lagrangian Probability Distributions. Birkhäuser Boston, Basel, Berlin (2006). | MR | Zbl
[15] , A new distribution of Poisson-type for the number of claims. ASTIN Bull. 27 (1997) 229–242. | DOI
[16] , and , Modelling species abundance using the Poisson-Tweedie family. Environmetrics 22 (2011) 152–164. | MR | DOI
[17] and , Modelling claim number using a new mixture model: negative binomial gamma distribution. J. Stat. Comput. Simul. 86 (2016) 1829–1839. | MR | Zbl | DOI
[18] and , The discrete Lindley distribution: properties and applications. J. Stat. Comput. Simul. 81 (2011) 1405–1416. | MR | Zbl | DOI
[19] , and , A new discrete distribution with actuarial applications. Insurance: Math. Econ. 48 (2011) 406–412. | MR | Zbl
[20] , and , A suitable discrete distribution for modelling automobile claim frequencies. Bull. Malays. Math. Sci. Soc. 39 (2016) 633–647. | Zbl | MR | DOI
[21] and , Methodes d’ajustement de distributions de sinistres. Bull. Assoc. Swiss Actuar. 81 (1981) 87–95.
[22] and , Overdispersion: models and estimation. Comput. Stat. Data Anal. 27 (1998) 151–170. | Zbl | DOI
[23] , Exponential dispersion models (with discussion). J. Roy. Stat. Soc. B 49 (1987) 127–162. | MR | Zbl
[24] , The Theory of Exponential Dispersion Models. Vol. 76 of Monographs on Statistics and Probability. Chapman and Hall, London (1997). | MR | Zbl
[25] and , Discrete dispersion models and their Tweedie asymptotics. AStA Adv. Stat. Anal. 100 (2016) 43–78. | MR | Zbl | DOI
[26] , and , Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demétrio classes. Stat. Oper. Res. Trans. 28 (2004) 201–214. | MR | Zbl
[27] , and , On Hinde–Demétrio regression models for overdispered count data. Stat. Methodol. 4 (2007) 277–291. | MR | Zbl | DOI
[28] and , On strict arcsine distribution. Commun. Stat. - Theory Methods 33 (2004) 993–1006. | MR | Zbl | DOI
[29] and , A property of count distributions in the Hinde-Demétrio family. Commun. Stat. Theory Methods 37 (2008) 1823–1834. | Zbl | MR | DOI
[30] and , Modelling and forecasting the time series of US mortality. J. Am. Stat. Assoc. 87 (1992) 659–671. | Zbl
[31] and , Natural real exponential families with cubic variance functions. Ann. Stat. 18 (1990) 1–37. | MR | Zbl | DOI
[32] , Natural exponential families with quadratic variance functions. Ann. Stat. 10 (1982) 65–80. | MR | Zbl | DOI
[33] , Special Functions. The Macmillan Company, New York (1960). | MR | Zbl
[34] and , A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Math. Econ. 38 (2006) 556–570. | Zbl
[35] , On a model for the claim number process. ASTIN Bull. 18 (1988) 57–68. | DOI
[36] , An index which distinguishes between some important exponential families. Statistics: Applications and New Directions. Edited by and . Proc. Indian Institute Golden Jubilee lnternat Conf. Indian Statist. Inst. Calcutta (1984) 579–604. | MR
[37] , The Poisson-inverse Gaussian distribution as an alternative to the negative binomial. Scand. Actuar. J. 1987 (1987) 113–127. | MR | DOI
Cité par Sources :
