Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions
that is neither a location parameter nor a scale parameter. Such a parameter must affect the shape of a distribution rather than simply shifting it or stretching/shrinking it.
For example, "peakedness" refers to how round the main peak is.
Image:Standard symmetric pdfs.svg|300px|thumb|Probability density functions for selected distributions with mean 0 and variance 1.

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 moments (statistics)|method of moments], as in the skewness or kurtosis, if the higher moments are defined and finite. Estimators of shape often involve higher-order statistics, 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:

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 the skewness and kurtosis of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters.