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 o ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. that is neither a

location parameter In geography, location or place are used to denote a region (point, line, or area) on Earth's surface or elsewhere. The term ''location'' generally implies a higher degree of certainty than ''place'', the latter often indicating an entity with an ...

nor a

scale parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...

(nor a function of these, such as a

rate parameter In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more spread out the distribution. Definition If a family o ...

). Such a parameter must affect the ''

shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ...

'' 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 In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined. For a unimodal ...

'' (3rd moment) or ''

kurtosis In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...

'' (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 (zer ...

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

L-moment In statistics, L-moments are a sequence of statistics used to summarize the shape of a probability distribution. They are linear combinations of order statistics ( L-statistics) analogous to conventional moments, and can be used to calculate qu ...

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 In probability theory, statistics and econometrics, the Burr Type XII distribution or simply the Burr distribution is a continuous probability distribution for a non-negative random variable. It is also known as the Singh–Maddala distribution a ...

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 ar ...

Erlang distribution The Erlang distribution is a two-parameter family of continuous probability distributions with support x \in independent exponential distribution">exponential variables with mean 1/\lambda each. Equivalently, it is the distribution of the tim ...

ExGaussian distribution In probability theory, an exponentially modified Gaussian distribution (EMG, also known as exGaussian distribution) describes the sum of independent normal and exponential random variables. An exGaussian random variable ''Z'' may be expressed as ...

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 dis ...

Fréchet distribution The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function :\Pr(X \le x)=e^ \text x>0. where ''α'' > 0 is a ...

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 d ...

Generalized extreme value distribution In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known a ...

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 eve ...

Log-t distribution In probability theory, a log-t distribution or log-Student t distribution is a probability distribution of a random variable whose logarithm is distributed in accordance with a Student's t-distribution. If ''X'' is a random variable with a Stude ...

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 In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,∞). Its probability density function is given by : f(x;\mu, ...

* Pareto distribution *

Pearson distribution The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. History The Pearson system ...

Skew normal distribution In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness. Definition Let \phi(x) denote the standard normal probability d ...

Lognormal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...

* Student's t-distribution *

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 In probability theory and statistics, the Weibull distribution is a continuous probability distribution. It is named after Swedish mathematician Waloddi Weibull, who described it in detail in 1951, although it was first identified by Maurice Re ...

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

and

of these distribution are constants, as skewness and kurtosis are independent of location and scale parameters. * Exponential distribution *

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 Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, ...

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 In probability theory and statistics, the raised cosine distribution is a continuous probability distribution supported on the interval mu-s,\mu+s/math>. The probability density function (PDF) is :f(x;\mu,s)=\frac \left +\cos\left(\frac\,\pi\rig ...

* Uniform distribution *

Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on minus;''R'', ''R''whose probability density function ''f'' is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): :f(x)=\sq ...

References

{{Statistics, inference Statistical parameters

Estimation

Examples

See also

References