# Shape Parameter

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A Shape Parameter is a function parameter of a probability distribution family.

**Context:**- It can range from being a Location Shape Parameter, to being a Scale Shape Parameter, to being ...

**Example(s):**- a Beta distribution shape parameter.
- a Burr distribution shape parameter.
- a Erlang distribution shape parameter.
- a ExGaussian distribution shape parameter.
- a Exponential power distribution shape parameter.
- a Gamma distribution shape parameter.
- a Generalized extreme value distribution shape parameter.
- a Log-logistic distribution shape parameter.
- a Inverse-gamma distribution shape parameter.
- a Pareto distribution shape parameter.
- a Pearson distribution shape parameter.
- a Skew normal distribution shape parameter.
- a Lognormal distribution shape parameter.
- a Student's t-distribution shape parameter.
- a Tukey lambda distribution shape parameter.
- a Weibull distribution shape parameter.

**See:**Maximum Likelihood Estimation Task, Numerical Parameter, Probability Distribution Parameter Estimation.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/shape_parameter Retrieved:2015-6-24.
- In probability theory and statistics, a
**shape parameter**is a kind of numerical parameter of a parametric family of probability distributions.^{[1]}

- In probability theory and statistics, a

- ↑ Everitt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. ISBN 0-521-81099-X

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/shape_parameter#Definition Retrieved:2015-6-24.
- 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).

- 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
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/shape_parameter#Estimation Retrieved:2015-6-24.
- 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.

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/shape_parameter#Examples Retrieved:2015-6-24.
- 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.

- The following continuous probability distributions have a shape parameter:

### 2000

- http://www.itl.nist.gov/div898/handbook/eda/section3/eda363.htm
- QUOTE: Many probability distributions are not a single distribution, but are in fact a family of distributions. This is due to the distribution having one or more shape parameters. Shape parameters allow a distribution to take on a variety of shapes, depending on the value of the shape parameter. These distributions are particularly useful in modeling applications since they are flexible enough to model a variety of data sets.
The Weibull distribution is an example of a distribution that has a shape parameter.

- QUOTE: Many probability distributions are not a single distribution, but are in fact a family of distributions. This is due to the distribution having one or more shape parameters. Shape parameters allow a distribution to take on a variety of shapes, depending on the value of the shape parameter. These distributions are particularly useful in modeling applications since they are flexible enough to model a variety of data sets.