Tweedie distribution

Tweedie distribution is a family of probability distributions that includes a wide range of other distributions as special cases. It is named after Maurice Tweedie, a statistician who contributed to its development. The Tweedie distribution is particularly notable for its application in statistical modeling and actuarial science, where it is used to model data that can be characterized by a mixture of continuous and discrete probability distributions. This makes it highly relevant in fields such as insurance, where it can model claim sizes that include both exact zeros (no claim) and continuous positive values (claims of varying sizes).

Definition

A random variable \(Y\) is said to follow a Tweedie distribution if its probability density function (pdf) or probability mass function (pmf), depending on the context, can be expressed in a specific form that involves a dispersion parameter (\(\phi\)) and a power parameter (\(p\)). The Tweedie distribution is a member of the exponential dispersion model (EDM) family, and its general form is not expressed in a closed form except for special cases. The parameter \(p\) determines the type of distribution within the Tweedie family:

• For \(0 < p < 1\), the distribution of \(Y\) is a compound Poisson distribution.
• For \(p = 0\) or \(p = 1\), \(Y\) follows a normal or Poisson distribution, respectively.
• For \(1 < p < 2\), \(Y\) has a positive mass at zero and is continuous and positive for values greater than zero, resembling a gamma distribution for larger values.
• For \(p = 2\), the distribution is a gamma distribution.
• For \(2 < p < 3\), the distribution models data that are continuous, strictly positive, and have heavier tails than a gamma distribution.

Applications

The flexibility of the Tweedie distribution makes it useful for modeling various types of data. In insurance mathematics, it is used to model aggregate claim amounts, which can be zero (no claims) or positive. In ecology, it can model animal abundance and count data, where observations may include exact zeros (no sightings) or counts. The Tweedie distribution is also used in finance for modeling monetary data and in medicine for modeling healthcare costs, which can include both zero costs (no treatment) and continuous costs for treatment.

Estimation

Estimating the parameters of a Tweedie distribution, particularly the power parameter \(p\) and the dispersion parameter \(\phi\), can be challenging due to the lack of a closed-form expression for the distribution's likelihood. Methods such as the method of moments, maximum likelihood estimation (MLE), and quasi-likelihood estimation are commonly used. Software implementations for estimating Tweedie distribution parameters are available in statistical packages like R, which includes functions specifically designed for this purpose.

See also

• Exponential dispersion model
• Compound Poisson distribution
• Gamma distribution
• Poisson distribution