Modelling overdispersion with the normalized tempered stable distribution
Kolossiatis, Michalis, Griffin, Jim E. and Steel, Mark F. J. (2010) Modelling overdispersion with the normalized tempered stable distribution. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers, Vol.2010).
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This paper discusses a multivariate distribution which generalizes the Dirichlet distribution and demonstrates its usefulness for modelling overdispersion in count data. The distribution is constructed by normalizing a vector of independent Tempered Stable random variables. General formulae for all moments and cross-moments of the distribution are derived and they are found to have similar forms to those for the Dirichlet distribution. The univariate version of the distribution can be used as a mixing distribution for the success probability of a Binomial distribution to define an alternative to the well-studied Beta-Binomial distribution. Examples of fitting this model to simulated and real data are presented.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Multivariate analysis, Distribution (Probability theory)|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||25|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|Version or Related Resource:||Kolossiatis, M., et al. (2011). Modeling overdispersion with the normalized tempered stable distribution. Computational Statistics & Data Analysis, 55(7), pp. 2288-2301. http://wrap.warwick.ac.uk/id/eprint/41084|
|References:||Abramowitz, M. and Stegun, I. A. (1964): “Handbook of mathematical functions with formulas, graphs, and mathematical tables, Volume 55 of National Bureau of Standards Applied Mathematics Series,” Aeschbacher, H. U., Vuataz, L., Sotek, J., and Stalder, R. (1977): “The use of the Beta- Binomial distribution in dominant-lethal testing for weak mutagenic activity (Part 1),” Mutation Research, 44, 369-390. Altham, P. M. E. (1978): “Two generalizations of the Binomial distribution,” Applied Statistics, 27, 162-167. Billingsley, P. (1995): “Probability and measure,” John Wiley & Sons: New York. Bohning, D., Schlattmann, P. and Lindsay, B. (1992): “Computer-Assisted Analysis of Mixtures (C.A.MAN): Statistical Algorithms,” Biometrics, 48, 283-303. Brix, A. (1999): “Generalized Gamma measures and shot-noise Cox processes,” Advances in Applied Probability, 31, 929-953. Brooks, S. P. (2001): “On Bayesian analyses and finite mixtures for proportions,” Statistics and Computing, 11, 179-190. Brooks, S. P., Morgan, B. J. T., Ridout, M. S. and Pack, S. E. (1997): “Finite Mixture Models for Proportions,” Biometrics, 53, 1097-1115. Charalambides, Ch. A. (2005): “Combinatorial methods in discrete distributions” John Wiley & Sons: Hoboken, New Jersey. Charalambides, Ch. A. and Singh, J. (1988): “A review of the Stirling numbers, their generalizations and statistical applications,” Communications in Statistics-Theory and Methods, 17, 2533-2595. Feller, W. (1971): “An introduction to probability theory and it applications,” John Wiley & Sons: New York. Garren, S. T., Smith, R. L. and Piegorsch, W. (2001): “Bootstrap goodness-of-fit test for the Beta-Binomial model,” Journal of Applied Statistics, 28, 561-571. George, E. O. and Bowman, D. (1995): “A full likelihood procedure for analysing exchangeable binary data,” Biometrics, 51, 512-523. Gradshteyn, I. S. and Ryzhik, I. M. (1994): “Table of integrals, series and products,” Academic Press: Boston. Haseman, J.K . and Soares, E. R. (1976): “The distribution of fetal death in control mice and its implications on statistical tests for dominant lethal effects,” Mutation Research, 41, 272-288. Hougaard, P. (1986): “Survival models for heterogeneous populations derived from stable distributions,” Biometrika, 73, 387-396. James, D. A. and Smith, D. M. (1982): “Analysis of results from a collaborative study of the dominant lethal assay,” Mutation Research, 97, 303-314 James, L. F., Lijoi, A. and Pr¨unster, I. (2006): “Conjugacy as a distinctive feature of the Dirichlet process,” Scandinavian Journal of Statistics, 33, 105-120. Johnson, W. P. (2002): “The curious history of Fa´a di Bruno’s formula,” American Mathmetical Monthly, 109, 217-234. Jorgensen, B. (1987): “Exponential dispersion models (with discussion),” Journal of the Royal Statistical Society B, 49, 127-162. Kuk, A. Y. C. (2004): “A litter-based approach to risk assessment in developmental toxicity studies via a power family of completely monotone functions,” Journal of the Royal Statistical Society. Series C., 53, 369-386. Kupper, L. L. and Haseman, J. K. (1978): “The use of a correlated Binomial model for the analysis of certain toxicological experiments,” Biometrics, 35, 281-293. Lijoi, A., Mena, R. H. and Pr¨unster, I. (2005): “Hierarchical mixture modeling with normalized inverse-Gaussian priors,” Journal of the American Statistical Association, 100, 1278-1291. Ochi, Y. and Prentice, R. L. (1984): “Likelihood inference in a correlated probit regression models,” Biometrika, 71, 531-543. Palmer, K. J., Ridout, M. S. and Morgan, B. J. T. (2008a): “Modelling cell generation times using the Tempered Stable distribution,” Journal of the Royal Statistical Society Series C, 57, 379-397. Palmer, K. J., Ridout, M. S. and Morgan, B. J. T (2008b): R functions for the Tempered Stable distribution (R code). Available from http://www.kent.ac.uk/IMS/personal/msr/TemperedStable.html Pang, Z. and Kuk, A. Y. C. (2005): “A shared response model for clustered binary data in developmental toxicity studies,” Biometrics, 61, 1076-1084. Paul, S. R. (1985): “A three-parameter generalisation of the Binomial distribution,” Communications in Statistics Theory and Methods, 14(6), 1497-1506. Paul, S. R. (1987): “On the Beta-Correlated Binomial (BCB) distribution in a three parameter generalization of the Binomial distribution,” Communications in Statistics Theory and Methods, 16(5), 1473-1478. Tweedie, M. (1984): “An index which distinguishes between some important exponential families,” in Statistics: Applications and New Directions: Proceedings of the Indian statistical Institute Golden Jubilee International Conference, Eds: J. Ghosh and J. Roy, 579-604. Williams, D. A. (1982): “Extra-Binomial variation in logistic linear models,” Journal of the Royal Statistical Society C, 31, 144-148. Zhang, S. and Jin, J. (1996): “Computation of Special Functions,” John Wiley & Sons: New York.|
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