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Concept# Shape parameter

Summary

In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributions
that is neither a location parameter nor a scale parameter (nor a function of these, 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).
For example, "peakedness" refers to how round the main peak is.
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), if the higher moments are defined and finite. 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.
The following continuous probability distributions have a shape parameter:
Beta distribution
Burr distribution
Dagum distribution
Erlang distribution
ExGaussian distribution
Exponential power distribution
Fréchet distribution
Gamma distribution
Generalized extreme value distribution
Log-logistic distribution
Log-t distribution
Inverse-gamma distribution
Inverse Gaussian 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
Uni

Official source

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