Definitions

Tweedie distributions

Tweedie distributions

In probability and statistics, the Tweedie distributions are a family of probability distributions which include continuous distributions such as the normal and gamma, the purely discrete scaled Poisson distribution, and the class of mixed compound Poisson-Gamma distributions which have positive mass at zero, but are otherwise continuous. Tweedie distributions belong to the exponential dispersion model family of distributions, a generalization of the exponential family, which are the response distributions for generalized linear models.

Tweedie distributions have a mean mu and a variance phi mu^p, where phi>0 is a dispersion parameter, and p, called the index parameter, (uniquely) determines the distribution in the Tweedie family. Special cases include:

Tweedie distributions exist for all real values of p except for 0. Apart from the four special cases identified above, their probability density function have no closed form. However, software is available that enables the accurate computation of the Tweedie densities (and probability distribution functions).

The Tweedie distributions were so named by Bent Jørgensen after M.C.K. Tweedie, a medical statistician at the University of Liverpool, UK, who presented the first thorough study of these distributions in 1984.

The index parameter p defines the type of distribution:

  • For p<0, the data y are supported on the whole real line (but, interestingly, mu>0). Applications for these distribution are unknown.
  • For p=0 (the normal distribution), the data y and the mean mu are supported on the whole real line.
  • For 0, no distributions exist
  • For p=1, the distribution exist on the non-negative integers
  • For 1, the distribution is continuous on the positive reals, plus an added mass (exact zero) at Y=0. For example, consider monthly rainfall. When no rain is recorded, an exact zero is recorded. If rain is recorded, a continuous amount results. These distributions are also called the Poisson-gamma distributions, since they can be represented as the Poisson sum of gamma distributions. They are therefore a type of compound Poisson distribution.
  • For p>2, the data y are supported on the non-negative reals, and mu>0. These distribution are like the gamma distribution (which corresponds to p=2), but are progressively more right-skewed as p gets larger.

Applications

Applications of Tweedie distributions (apart from the four special cases identified) include:

  • actuarial studies
  • assay analysis
  • survival analysis
  • ecology
  • analysis of alcohol consumption in British teenagers
  • medical applications
  • meteorology and climatology
  • fisheries

References

Further reading

  • Kaas, R. (2005). Compound Poisson distribution and GLM’s – Tweedie’s distribution. Handelingen van het contactforum 3rd Actuarial and Financial Mathematics Day (4 February 2005), 3-12. http://ucs.kuleuven.be/seminars_events/other/files/3afmd/Kaas.PDF
  • Ohlsson, E and Johansson, B. Exact Credibility and Tweedie Models, University of Stockholm, Research report , October 2003. http://www.math.su.se/matstat/reports/seriea/2003/rep15/report.pdf
  • Smith, CAB. (1997). Obituary: Maurice Charles Kenneth Tweedie, 1991-96 Journal of the Royal Statistical Society: Series A (Statistics in Society) 160 (1), 151–154. doi:10.1111/1467-985X.00052
  • Smyth, G. K., and Jørgensen, B. (2002). Fitting Tweedie's compound Poisson model to insurance claims data: dispersion modelling. ASTIN Bulletin 32, 143-157. 6/2002 http://www.statsci.org/smyth/pubs/insuranc.pdf
  • Tweedie, M. C. K. (1956) Some statistical properties of inverse Gaussian distributions. Virginia J. Sci. (N.S.) 7 (1956), 160--165.
  • Tweedie distributions. http://www.statsci.org/s/tweedie.html
  • Tweedie generalized linear model family. http://www.statsci.org/s/tweedief.html
  • Examples of use of the model. http://www.sci.usq.edu.au/staff/dunn/Datasets/tech-glms.html#Tweedie

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