Shape parameter

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Shape parameter

In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions. [Everitt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. ISBN 0-521-81099-x]

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 of both or 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).

Examples

The following continuous probability distributions have a shape parameter:
* Beta distribution
* Burr distribution
* Erlang distribution
* Exponential power distribution
* Gamma distribution
* Generalized extreme value distribution
* Log-logistic distribution
* Inverse-gamma distribution
* Pareto distribution
* Pearson distribution
* Weibull distributionBy 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

* skewness
* kurtosis

References

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