Shape parameter
Inprobability theoryandstatistics,ashape parameter(also known asform parameter)[1]is a kind ofnumerical parameterof a parametric family ofprobability distributions[2] that is neither alocation parameternor ascale parameter(nor a function of these, such as arate parameter). Such a parameter must affect theshapeof 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]
Estimation[edit]
Manyestimatorsmeasure 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 likelihoodestimation can also be used.
Examples[edit]
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 donothave 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) theskewnessandkurtosisof 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
See also[edit]
References[edit]
- ^Ekawati, Dian; Warsono; Kurniasari, Dian (December 2014)."On the Moments, Cumulants, and Characteristic Function of the Log-Logistic Distribution"(PDF).The Journal for Technology and Science.25.
- ^Everitt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP.ISBN0-521-81099-X
- ^Birnbaum, Z. W. (1948)."On Random Variables with Comparable Peakedness".The Annals of Mathematical Statistics.19(1). Institute of Mathematical Statistics: 76–81.doi:10.1214/aoms/1177730293.ISSN0003-4851.
