Inprobability theory andstatistics, ashape parameter (also known asform parameter)[1] is a kind ofnumerical parameter of a parametric family ofprobability distributions[2]that is neither alocation parameter nor ascale parameter (nor a function of these, such as arate parameter). Such a parameter must affect theshape of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does).For example, "peakedness" refers to how round the main peak is.[3]
Manyestimators measure location or scale; however, estimators for shape parameters also exist. Most simply, they can be estimated in terms of the highermoments, using themethod of moments, as in theskewness (3rd moment) orkurtosis (4th moment), if the higher moments are defined and finite. Estimators of shape often involvehigher-order statistics (non-linear functions of the data), as in the higher moments, but linear estimators also exist, such as theL-moments.Maximum likelihood estimation can also be used.
The following continuous probability distributions have a shape parameter:
By contrast, the following continuous distributions donot 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) theskewness andkurtosis of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters.
