Series evaluation of Tweedie exponential dispersion model densities

Exponential dispersion models, which are linear exponential families with a dispersion parameter, are the prototype response distributions for generalized linear models. The Tweedie family comprises those exponential dispersion models with power mean-variance relationships. The normal, Poisson, gamma and inverse Gaussian distributions belong to theTweedie family. Apart from these special cases, Tweedie distributions do not have density functions which can be written in closed form. Instead, the densities can be represented as infinite summations derived from series expansions. This article describes how the series expansions can be summed in an numerically efficient fashion. The usefulness of the approach is demonstrated, but full machine accuracy is shown not to be obtainable using the series expansion method for all parameter values. Derivatives of the density with respect to the dispersion parameter are also derived to facilitate maximum likelihood estimation. The methods are demonstrated on two data examples and compared with with Box-Cox transformations and extended quasi-likelihoood.

[1]  A. Siegel.  The noncentral chi-squared distribution with zero degrees of freedom and testing for uniformity , 1979 .

[2]  P. Hougaard.  Survival models for heterogeneous populations derived from stable distributions , 1986 .

[3]  Karl P. Murphy,et al.  Using Generalized Linear Models to Build Dynamic Pricing Systems , 2000 .

[4]  J. N. Perry,et al.  Taylor's Power Law for Dependence of Variance on Mean in Animal Populations , 1981 .

[5]  Ananda Sen,et al.  The Theory of Dispersion Models , 1997, Technometrics.

[6]  Peter K. Dunn,et al.  Randomized Quantile Residuals , 1996 .

[7]  Odd O. Aalen,et al.  Modelling Heterogeneity in Survival Analysis by the Compound Poisson Distribution , 1992 .

[8]  D. Cox.  Tests of Separate Families of Hypotheses , 1961 .

[9]  E. Wright.  On the Coefficients of Power Series Having Exponential Singularities , 1933 .

[10]  A. E. Renshaw,et al.  Modelling the Claims Process in the Presence of Covariates , 1994 .

[11]  Gordon K. Smyth,et al.  Generalized linear models with varying dispersion , 1989 .

[12]  P. McCullagh,et al.  Generalized Linear Models , 1992 .

[13]  Philip Hougaard,et al.  Measuring the Similarities between the Lifetimes of Adult Danish Twins Born between 1881–1930 , 1992 .

[14]  J. Ghosh,et al.  Statistics : applications and new directions , 1984 .

[15]  Shaul K. Bar-Lev,et al.  Reproducibility and natural exponential families with power variance functions , 1986 .

[16]  Eric R. Ziegel,et al.  Generalized Linear Models , 2002, Technometrics.

[17]  Bent Jørgensen,et al.  Fitting Tweedie's compound poisson model to insurance claims data , 1994 .

[18]  John A. Nelder.  An Alternative View of the Splicing Data , 1994 .

[19]  Andrew F. Siegel,et al.  Modelling Data Containing Exact Zeroes Using Zero Degrees of Freedom , 1985 .

[20]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[21]  J. Nelder,et al.  An extended quasi-likelihood function , 1987 .

[22]  Shaul K. Bar-Lev,et al.  Characterizations of natural exponential families with power variance functions by zero regression properties , 1987 .

[23]  Bent Jørgensen,et al.  Fitting Tweedie's Compound Poisson Model to Insurance Claims Data: Dispersion Modelling , 2002, ASTIN Bulletin.

[24]  B. Jørgensen.  Exponential Dispersion Models , 1987 .

[25]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[26]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[27]  Marie Davidian,et al.  Estimation of variance functions in assays with possibly unequal replication and nonnormal data , 1990 .

[28]  Gordon K. Smyth,et al.  Adjusted likelihood methods for modelling dispersion in generalized linear models , 1999 .

[29]  P. W. Gandar,et al.  Analysis of Distribution of Root Length Density of Apple Trees on Different Dwarfing Rootstocks , 1999 .

[30]  S. Mildenhall.  A SYSTEMATIC RELATIONSHIP BETWEEN MINIMUM BIAS AND GENERALIZED LINEAR MODELS , 1999 .

[31]  Steven Haberman,et al.  Generalized linear models and actuarial science , 1996 .

[32]  Marie Davidian,et al.  Variance functions and the minimum detectable concentration in assays. Technical report, August 1985-August 1986 , 1988 .

[33]  John A. Nelder,et al.  Likelihood, Quasi-likelihood and Pseudolikelihood: Some Comparisons , 1992 .

[34]  Jorge Nocedal,et al.  A Limited Memory Algorithm for Bound Constrained Optimization , 1995, SIAM J. Sci. Comput..

[35]  G. Smyth.  Regression Analysis of Quantity Data with Exact Zeroes ∗ , 2007 .

[36]  P. McCullagh,et al.  Generalized Linear Models, 2nd Edn. , 1990 .

[37]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .