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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.## Applications

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

## Further reading

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:

- $p=0$ is the normal distribution
- $p=1$ with $phi=1$ is the Poisson distribution
- $p=2$ is the gamma distribution
- $p=3$ is the inverse Gaussian distribution.

Tweedie distributions exist for all real values of $p$ except for $01\; math>.\; Apart\; from\; the\; four\; special\; cases\; identified\; above,\; their\; probability\; density\; function\; have\; no\; closed\; form.\; However,\; software\; is\; available\; that\; enables\; the\; accurate\; computation\; of\; the\; Tweedie\; densities\; (and\; probability\; distribution\; functions).$

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 $01\; math>,\; no\; distributions\; exist$
- For $p=1$, the distribution exist on the non-negative integers
- For $12\; math>,\; the\; distribution\; is\; continuous\; on\; the\; positive\; reals,\; plus\; an\; added\; mass\; (exact\; zero)\; at$ Y=0$.\; For\; example,\; consider\; monthly\; rainfall.\; When\; no\; rain\; is\; recorded,\; an\; exact\; zero\; is\; recorded.\; If\; rain\; is\; recorded,\; a\; continuous\; amount\; results.\; These\; distributions\; are\; also\; called\; the\; Poisson-gamma\; distributions,\; since\; they\; can\; be\; represented\; as\; the\; Poisson\; sum\; ofgamma\; distributions.\; They\; are\; therefore\; a\; type\; ofcompound\; Poisson\; distribution.$
- 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.

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

- 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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Last updated on Saturday September 20, 2008 at 06:56:49 PDT (GMT -0700)

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