On the distribution theory of over-dispersion. (English) Zbl 1329.62069
Summary: An overview of the evolution of probability models for over-dispersion is given looking at their origins, motivation, first main contributions, important milestones and applications. A specific class of models called the Waring and generalized Waring models will be a focal point. Their advantages relative to other classes of models and how they can be adapted to handle multivariate data and temporally evolving data will be highlighted.
MSC:
| 62E10 | Characterization and structure theory of statistical distributions |
Keywords:
heterogeneity; contagion; clustering; spells model; accident proneness; mixtures; zero-adjusted models; biased sampling; generalized Waring distribution; generalized Waring processReferences:
| [1] | Ajiferuke, I, Wolfram, D, Xie, H: Modelling website visitation and resource usage characteristics by IP address data. In: Julien, H, Thompson, S (eds.). Proceedings of the 32nd Annual Conference of the Canadian Association for Information Science, Manitoba, Canada (2004). Available at: www.cais-acsi.ca/proceedings/2004/ajiferuke_2004.pdf · Zbl 0606.62053 |
| [2] | Andersen, P, Borgan, Ø, Gill, R, Keiding, N: Statistical Models based on Counting Processes. Springer, New York (1993) · Zbl 0769.62061 |
| [3] | Andrews, DF, Mallows, CL: Scale mixtures of normal distributions. J. R. Stat. Soc. B 36, 99-102 (1974) · Zbl 0282.62017 |
| [4] | Arbous, AG, Kerrich, JE: Accident statistics and the concept of accident proneness. Biometrics 7, 340-432 (1951) · doi:10.2307/3001656 |
| [5] | Bates, GE: Joint distributions of time intervals for the occurrence of successive accidents in a generalized Polya sheme. Ann. Math. Stat. 26, 705-720 (1955) · Zbl 0067.37501 · doi:10.1214/aoms/1177728429 |
| [6] | Bates, GE, Neyman, J: Contributions to the theory of accident proneness. U. Calif. Publ. Stat. 1, 215-275 (1952) |
| [7] | Breslow, N: Tests of hypotheses in overdispersed poisson regression and other quasi-likelihood models. J. Am. Stat. Assoc. 85, 565-571 (1990) · doi:10.1080/01621459.1990.10476236 |
| [8] | Brillinger, D.; Krishnaiah, PR (ed.), Comparative Aspects of the Study of Ordinary Time Series and of Point Processes, 33-133 (1978), New York · Zbl 0415.62066 |
| [9] | Cane, VR: The concept of accident proneness. B. Inst. Math. Bulgarian Acad. Sci. 15, 183-188 (1974) · Zbl 0337.62072 |
| [10] | Cane, VR: A class of non-identifiable stochastic models. J. Appl. Probab. 14, 475-782 (1977) · Zbl 0375.60083 · doi:10.2307/3213450 |
| [11] | Carriere, J: Nonparametric tests for mixed poisson distributions. Insur. Math. Econ. 12, 3-8 (1993) · Zbl 0768.62031 · doi:10.1016/0167-6687(93)90994-Z |
| [12] | Chetwynd, AG, Diggle, PJ: On estimating the reduced second moment measure of a stationary spatial point process. Aust. N.Z. J. Stat. 40(1), 11-15 (1998) · Zbl 0924.62098 · doi:10.1111/1467-842X.00002 |
| [13] | Cox, DR: Renewal Theory. Barnes & Noble, New York (1962) · Zbl 0103.11504 |
| [14] | Cox, DR: Some remarks on overdispersion. Biometrika 70, 269-274 (1983) · Zbl 0511.62007 · doi:10.1093/biomet/70.1.269 |
| [15] | Cresswell, WL, Froggatt, P: The Causation of Bus Driver Accidents. Oxford University Press, London (1963) |
| [16] | Davidian, M, Carroll, RJ: A note on extended quasi-likelihood. J. R. Stat. Soc. B 50, 74-82 (1988) |
| [17] | Dean, CB, Lawless, J, Willmot, GE: A mixed poisson-inverse Gaussian regression model. Can. J. Stat. 17, 171-182 (1989) · Zbl 0679.62051 · doi:10.2307/3314846 |
| [18] | Diggle, PJ, Chetwynd, AG: Second- order analysis of spatial clustering for inhomogeneous populations. Biometrics 47, 1155-1163 (1991) · doi:10.2307/2532668 |
| [19] | Dimaki, C, Xekalaki, E: Identifiability of income distributions in the context of damage and generating models. Commun. Stat. A-Theor. 19(8), 2757-2766 (1990) · Zbl 0728.62019 · doi:10.1080/03610929008830346 |
| [20] | Dimaki, C, Xekalaki, E: Additive and multiplicative distortion of observations: some characteristic properties. J. Appl. Stat. Sc. 5(2/3), 113-127 (1996) · Zbl 0874.62006 |
| [21] | Dubourdieu, J: Remarques Relatives sur la Théorie Mathématique de l’Assurance-Accidents. Bull. Trim. Inst. Actuaries Fran. 44, 79-146 (1938) · JFM 64.1233.02 |
| [22] | Faddy, MJ: Extending poisson process modelling and analysis of count data. Biometrical J. 39(4), 431-440 (1997) · Zbl 0890.62074 · doi:10.1002/bimj.4710390405 |
| [23] | Famoye, F, Singh, KP: Zero-inflated generalized poisson regression model with an application to domestic violence. Data. J. Data Sci. 4, 117-130 (2006) |
| [24] | Fisher, RA: The effects of methods of ascertainment upon the estimation of frequencies. Ann. Eugenic. 6, 13-25 (1934) · JFM 62.1373.02 · doi:10.1111/j.1469-1809.1934.tb02105.x |
| [25] | Fisher, RA: The significance of deviations from expectation in a poisson series. Biometrics 6, 17-24 (1950) · doi:10.2307/3001420 |
| [26] | Greenwood, M.; Woods, HM, On the Incidence of Industrial Accidents upon Individuals with Special Reference to Multiple Accidents, 1-28 (1919), London |
| [27] | Greenwood, M, Yule, GU: An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attack of disease or repeated accidents. J. R. Stat. Soc. 83, 255-279 (1920) · doi:10.2307/2341080 |
| [28] | Gschlößl, S, Czado, C: Modelling count data with overdispersion and spatial effects. Stat. Pap. 49, 531-552 (2008) · Zbl 1310.62083 · doi:10.1007/s00362-006-0031-6 |
| [29] | Gupta, PL, Gupta, RC, Tripathi, RC: Analysis of zero adjusted count data. Comput. Stat. Data An. 23, 207-218 (1996) · Zbl 0875.62096 · doi:10.1016/S0167-9473(96)00032-1 |
| [30] | Gupta, PL, Gupta, RC, Tripathi, RC: Score test for zero inflated generalized poisson regression model. Commun. Stat. A-Theor. 33, 47-64 (2004) · Zbl 1185.62041 · doi:10.1081/STA-120026576 |
| [31] | Gurland, J: Some applications of the negative binomial and other contagious distributions. Am. J. Public Health 49, 1388-1399 (1959) · doi:10.2105/AJPH.49.10.1388 |
| [32] | Hinde, J, Demetrio, CGB: Overdispersion: models and estimation. Comput. Stat. Data An. 27, 151-170 (1998) · Zbl 1042.62578 · doi:10.1016/S0167-9473(98)00007-3 |
| [33] | Irwin, JO: Discussion on chambers and Yule’s paper. J. R. Stat. Soc. Supplement 7, 101-109 (1941) · Zbl 0063.00771 |
| [34] | Irwin, JO: The place of mathematics in medical and biological statistics. J. R. Stat. Soc. A 126, 1-44 (1963) · doi:10.2307/2982445 |
| [35] | Irwin, JO: The generalized Waring distribution applied to accident theory. J. R. Stat. Soc. A 131, 205-225 (1968) · doi:10.2307/2343842 |
| [36] | Irwin, JO: The Generalized Waring Distribution. J. R. Stat. Soc. A 138, 18-31 (1975). Part I), 204-227 (Part II), 374-384 (Part III · doi:10.2307/2345247 |
| [37] | Jewell, N: Mixtures of exponential distributions. Ann. Stat. 10, 479-484 (1982) · Zbl 0495.62042 · doi:10.1214/aos/1176345789 |
| [38] | Karlis, D, Xekalaki, E: On testing for the number of components in a mixed poisson model. Ann. I. Stat. Math. 51, 149-162 (1999) · Zbl 0928.62061 · doi:10.1023/A:1003839420071 |
| [39] | Karlis, D.; Xekalaki, E.; Panaretos, J. (ed.), Mixtures Everywhere, 78-95 (2003), USA |
| [40] | Karlis, D, Xekalaki, E: Mixed poisson distributions. Int. Stat. Rev. 73(1), 35-58 (2005) · Zbl 1104.62010 · doi:10.1111/j.1751-5823.2005.tb00250.x |
| [41] | Kemp, CD: On a contagious distribution suggested for accident data. Biometrics 23, 241-255 (1967) · doi:10.2307/2528159 |
| [42] | Kemp, CD; Patil, GP (ed.), “Accident proneness” and discrete distribution theory, 41-65 (1970), University Park, USA |
| [43] | Krishnaji, N: Characterization of the Pareto distribution through a model of under-reported incomes. Econometrica 38, 251-255 (1970) · doi:10.2307/1913007 |
| [44] | Kulinskaya, E, Olkin, I: An overdispersion model in meta-analysis. Stat. Model. 14(1), 49-76 (2014) · Zbl 07257895 · doi:10.1177/1471082X13494616 |
| [45] | Lawless, JF: Negative binomial and mixed poisson regression. Can. J. Stat. 15, 209-225 (1987) · Zbl 0632.62060 · doi:10.2307/3314912 |
| [46] | Lewis, PAW; Lewis, PAW (ed.), Recent results in the statistical analysis of univariate point processes, 1-54 (1972), New York · Zbl 0263.62056 |
| [47] | Lexis, W: Über die Theorie der Statilität Statistischer Reichen. Jahrb. Nationalökon. u. Statist. 32, 60-98 (1879) |
| [48] | McCullagh, P, Nelder, JA: Generalized Linear Models, 2nd edn. Chapman & Hall, London (1989) · Zbl 0744.62098 · doi:10.1007/978-1-4899-3242-6 |
| [49] | McKendrick, AG: The applications of mathematics to medical problems. Proc. Edinb. Math. Soc. 44, 98-130 (1925). doi:10.1017/S0013091500034428 · JFM 52.0542.04 · doi:10.1017/S0013091500034428 |
| [50] | McLachlan, JA, Peel, D: Finite Mixture Models. Wiley, New York (2001) |
| [51] | Moore, DF: Asymptotic properties of moment estimators for overdispersed counts and proportions. Biometrika 23, 583-588 (1986) · Zbl 0614.62031 · doi:10.1093/biomet/73.3.583 |
| [52] | Nelder, JA, Pregibon, D: An extended quasi-likelihood function. Biometrika 74, 221-232 (1987) · Zbl 0621.62078 · doi:10.1093/biomet/74.2.221 |
| [53] | Ngatchou-Wandji, J, Paris, C: On the zero-inflated count models with application to modelling annual trends in incidences of some occupational allergic diseases in France. J. Data Sci. 9, 639-659 (2011) |
| [54] | Panaretos, J: An extension of the damage model. Metrika 29, 189-194 (1982) · Zbl 0496.62012 · doi:10.1007/BF01893378 |
| [55] | Panaretos, J: A generating model involving Pascal and logarithmic series distributions. Commun. Stat. A-Theor. 12(7), 841-848 (1983) · Zbl 0512.62025 · doi:10.1080/03610928308828499 |
| [56] | Panaretos, J: On the evolution of surnames. Int. Stat. Rev. 57(2), 161-167 (1989) · Zbl 0715.62299 · doi:10.2307/1403384 |
| [57] | Patil, GP, Ord, JK: On size-biased sapling and related form-invariant weighted distributions. Sankhya 38, 48-61 (1976) · Zbl 0414.62015 |
| [58] | Rao, CR: On discrete distributions arising out of methods of ascertainment. Sankhya A25, 311-324 (1963) · Zbl 0212.21903 |
| [59] | Rao, CR; Atkinson, AC (ed.); Fienberg, SE (ed.), Weighted Distributions Arising Out of Methods of Ascertainment, 543-569 (1985), New York · Zbl 0603.62013 · doi:10.1007/978-1-4613-8560-8_24 |
| [60] | Ridout, M, Demetrio, CGB, Hinde, J: Models for count data with many zeros, pp. 179-190. Invited paper presented at the Nineteenth International Biometric Conference, Cape Town, South Africa (1998) |
| [61] | Ridout, M, Hinde, J, Demetrio, CGB: A score test for testing zero inflated Poisson regression model against zero inflated negative binomial alternatives. Biometrics 57, 219-223 (2001) · Zbl 1209.62079 · doi:10.1111/j.0006-341X.2001.00219.x |
| [62] | Ripley, BD: The second-order analysis of stationary point processes. J. Appl. Probab. 13, 255-266 (1976) · Zbl 0364.60087 · doi:10.2307/3212829 |
| [63] | Ripley, BD: Modelling spatial patterns (with Discussion). J. R. Stat. Soc. B 39, 172-212 (1977) · Zbl 0369.60061 |
| [64] | Shaw, L, Sichel, HS: Accidents Proneness. Pergamon Press, Oxford (1971) |
| [65] | Steutel, FW, van Harn, K: Discrete analogues of self-decomposability and stability. Ann. Prob. 7, 893-899 (1979) · Zbl 0418.60020 · doi:10.1214/aop/1176994950 |
| [66] | Student: An explanation of deviations from poisson’s law in practice. Biometrika 12, 211-215 (1919) · doi:10.2307/2331767 |
| [67] | Thyrion, P: Extension of the collective risk theory. Skand. Aktuaritidskrift 52(Supplement), 84-98 (1969) · Zbl 0223.62118 |
| [68] | Titterington, DM: Some recent research in the analysis of mixture distributions. Statistics 21, 619-641 (1990) · Zbl 0714.62023 · doi:10.1080/02331889008802274 |
| [69] | Tripathi, R, Gupta, R, Gurland, J: Estimation of parameters in the beta binomial model. Ann. I. Stat. Math. 46, 317-331 (1994) · Zbl 0816.62024 · doi:10.1007/BF01720588 |
| [70] | Wang, P, Puterman, M, Cokburn, I, Le, N: Mixed poisson regression models with covariate dependent rates. Biometrics 52, 381-400 (1996) · Zbl 0875.62407 · doi:10.2307/2532881 |
| [71] | Winkelmann, R: Duration dependence and dispersion in count-data models. J. Bus. Econ. Stat. 13(4), 467-474 (1995) |
| [72] | Xekalaki, E.; Taillie, C. (ed.); Patil, GP (ed.); Baldessari, B. (ed.), Chance mechanisms for the univariate generalized Waring distribution and related characterizations, 157-171 (1981), Dordrecht · Zbl 0482.62003 · doi:10.1007/978-94-009-8549-0_12 |
| [73] | Xekalaki, E: The univariate generalized Waring distribution in relation to accident theory: proneness, spells or contagion? Biometrics 39(3), 887-895 (1983a) · Zbl 0536.62089 · doi:10.2307/2531324 |
| [74] | Xekalaki, E: Infinite divisibility, completeness and regression properties of the univariate generalized Waring distribution. Ann. I. Stat. Math. A 35, 279-289 (1983b) · Zbl 0525.60018 · doi:10.1007/BF02480983 |
| [75] | Xekalaki, E: A property of the Yule distribution and its applications. Commun. Stat. A-Theor. 12(10), 1181-1189 (1983c) · Zbl 0527.62016 · doi:10.1080/03610928308828523 |
| [76] | Xekalaki, E: The bivariate generalized Waring distribution and its application to accident theory. J. R. Stat. Soc. A 147(3), 488-498 (1984a) · Zbl 0562.62047 · doi:10.2307/2981580 |
| [77] | Xekalaki, E: Linear regression and the Yule distribution. J. Econometrics 24(1), 397-403 (1984b) · Zbl 0559.62009 · doi:10.1016/0304-4076(84)90061-7 |
| [78] | Xekalaki, E: Models leading to the bivariate generalized Waring distribution. Utilitas Math. 25, 263-290 (1984c) · Zbl 0525.62018 |
| [79] | Xekalaki, E: The multivariate generalized Waring distribution. Commun. Stat. A-Theor. 15(3), 1047-1064 (1986) · Zbl 0606.62053 · doi:10.1080/03610928608829168 |
| [80] | Xekalaki, E: Under- and Overdispersion. Enc. Act. Sci. 3, (2006) [http://onlinelibrary.wiley.com/doi/10.1002/9780470012505.tau003/abstract] · Zbl 0525.60018 |
| [81] | Xekalaki, E, Panaretos, J: Identifiability of compound poisson distributions. Scand. Actuar. J. 66, 39-45 (1983) · Zbl 0507.62010 · doi:10.1080/03461238.1983.10408688 |
| [82] | Xekalaki, E, Zografi, M: The generalized Waring process and its application. Commun. Stat. A-Theor. 37(12), 1835-1854 (2008) · Zbl 1162.60020 · doi:10.1080/03610920801893707 |
| [83] | Xue, D, Deddens, J: Overdispersed negative binomial models. Commun. Stat. A-Theor. 21, 2215-2226 (1992) · Zbl 0775.62044 · doi:10.1080/03610929208830908 |
| [84] | Yule, GW: A mathematical theory of evolution based on the conclusions of J.C. Willis, F.R.S. Philos. T. R. Soc. B 213, 21-87 (1924) · doi:10.1098/rstb.1925.0002 |
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