Definitions

# 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

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
• For $p=1$, the distribution exist on the non-negative integers
• For
• 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

• 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

Search another word or see Tweedie distributionson Dictionary | Thesaurus |Spanish