Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.

Definition

A shape parameter is any parameter of a probability distribution that is neither a location parameter nor a scale parameter (nor a function of either or both of these only, such as a rate parameter). Such a parameter must affect the shape of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does).

Estimation

Many estimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the higher moments, using the method of moments, as in the skewness (3rd moment) or kurtosis (4th moment). Estimators of shape often involve higher-order statistics (non-linear functions of the data), as in the higher moments, but linear estimators also exist, such as the L-moments. Maximum likelihood estimation can also be used.

Examples

The following continuous probability distributions have a shape parameter:

• Beta distribution
• Burr distribution
• Erlang distribution
• ExGaussian distribution
• Exponential power distribution
• Gamma distribution
• Generalized extreme value distribution
• Log-logistic distribution
• Inverse-gamma distribution
• Pareto distribution
• Pearson distribution
• Skew normal distribution
• Lognormal distribution
• Student's t-distribution
• Tukey lambda distribution
• Weibull distribution

By contrast, the following continuous distributions do not have a shape parameter, so their shape is fixed and only their location or their scale or both can change. It follows that (where they exist) the skewness and kurtosis of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters.

• Exponential distribution
• Cauchy distribution
• Logistic distribution
• Normal distribution
• Raised cosine distribution
• Uniform distribution
• Wigner semicircle distribution

See also

• Skewness
• Kurtosis
• Location parameter