probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...

and

statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...

, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon ...

sEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. 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 A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie ...

'' 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.

Estimation

Many

estimators In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data: thus the rule (the estimator), the quantity of interest (the estimand) and its result (the estimate) are distinguished. For example, the ...

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 In statistics, the term higher-order statistics (HOS) refers to functions which use the third or higher power of a sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zero ...

(non-linear functions of the data), as in the higher moments, but linear estimators also exist, such as the L-moments.

Maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed stat ...

estimation can also be used.

Examples

The following continuous probability distributions have a shape parameter: *

Beta distribution In probability theory and statistics, the beta distribution is a family of continuous probability distributions defined on the interval , 1in terms of two positive parameters, denoted by ''alpha'' (''α'') and ''beta'' (''β''), that appear as ...

* Burr distribution *

Dagum distribution The Dagum distribution (or Mielke Beta-Kappa distribution) is a continuous probability distribution defined over positive real numbers. It is named after Camilo Dagum, who proposed it in a series of papers in the 1970s. The Dagum distribution aro ...

Erlang distribution The Erlang distribution is a two-parameter family of continuous probability distributions with support x \in independent exponential_variables_with_mean_1/\lambda_each.__Equivalently,_it_is_the_distribution_of_the_time_until_the_''k''th_event_ ...

* ExGaussian distribution *

Exponential power distribution The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of parametric continuous probability distributions on the real line. Both families add a shape parameter to the normal distribution. To di ...

* Fréchet distribution *

Gamma distribution In probability theory and statistics, the gamma distribution is a two- parameter family of continuous probability distributions. The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma di ...

* Generalized extreme value distribution *

Log-logistic distribution In probability and statistics, the log-logistic distribution (known as the Fisk distribution in economics) is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for ...

* Log-t distribution *

Inverse-gamma distribution In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to ...

* Inverse Gaussian distribution *

Pareto distribution The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto ( ), is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, ac ...

* Pearson distribution * Skew normal distribution * Lognormal distribution *

Student's t-distribution In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arise when estimating the mean of a normally distributed population in situ ...

Tukey lambda distribution Formalized by John Tukey, the Tukey lambda distribution is a continuous, symmetric probability distribution defined in terms of its quantile function. It is typically used to identify an appropriate distribution (see the comments below) and not u ...

* 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 In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...

Cauchy distribution The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fun ...

* Logistic distribution *

Normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...

* Raised cosine distribution *

Uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence See also * * Homogeneous distribution In mathematics, a homogeneous distribution ...

* Wigner semicircle distribution

References

{{Statistics, inference Statistical parameters