In probability theory and statistics, a **shape parameter** (also known as **form parameter**)^{ [1] } is a kind of numerical parameter of a parametric family of probability distributions ^{ [2] } 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 * of a distribution rather than simply shifting it (as a location parameter does) or stretching/shrinking it (as a scale parameter does).

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), if the higher moments are defined and finite. 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.

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
- Mukherjee-Islam 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 **Cauchy distribution**, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the **Lorentz distribution**, **Cauchy–Lorentz distribution**, **Lorentz(ian) function**, or **Breit–Wigner distribution**. The Cauchy distribution is the distribution of the x-intercept of a ray issuing from with a uniformly distributed angle. It is also the distribution of the ratio of two independent normally distributed random variables with mean zero.

In probability theory and statistics, **kurtosis** is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtosis describes the shape of a probability distribution and there are different ways of quantifying it for a theoretical distribution and corresponding ways of estimating it from a sample from a population. Different measures of kurtosis may have different interpretations.

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.

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 Fréchet and first applied by Rosin & Rammler (1933) to describe a particle size distribution.

In probability theory and statistics, the **beta distribution** is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by *alpha* (*α*) and *beta* (*β*), that appear as exponents of the random variable and control the shape of the distribution. The generalization to multiple variables is called a Dirichlet 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 distribution. There are two different parameterizations in common use:

- With a shape parameter
*k*and a scale parameter*θ*. - With a shape parameter
*α*=*k*and an inverse scale parameter*β*= 1/*θ*, called a rate parameter.

In mathematics, the **moments** of a function are quantitative measures related to the shape of the function's graph. If the function represents mass, then the first moment is the center of the mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the first moment is the expected value, the second central moment is the variance, the third standardized moment is the skewness, and the fourth standardized moment is the kurtosis. The mathematical concept is closely related to the concept of moment in physics.

In probability theory and statistics, the **logistic distribution** is a continuous probability distribution. Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks. It resembles the normal distribution in shape but has heavier tails. The logistic distribution is a special case of the Tukey lambda distribution.

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.

In probability theory and statistics, the **hyperbolic secant distribution** is a continuous probability distribution whose probability density function and characteristic function are proportional to the hyperbolic secant function. The hyperbolic secant function is equivalent to the reciprocal hyperbolic cosine, and thus this distribution is also called the **inverse-cosh distribution**.

**Gauss Moutinho Cordeiro** is a Brazilian engineer, mathematician and statistician who has made significant contributions to the theory of statistical inference, mainly through asymptotic theory and applied probability.

The **normal-inverse Gaussian distribution (NIG)** is a continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution. The NIG distribution was noted by Blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by Ole Barndorff-Nielsen. In the next year Barndorff-Nielsen published the NIG in another paper. It was introduced in the mathematical finance literature in 1997.

In probability and statistics, the **log-logistic distribution** is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, as, for example, mortality rate from cancer following diagnosis or treatment. It has also been used in hydrology to model stream flow and precipitation, in economics as a simple model of the distribution of wealth or income, and in networking to model the transmission times of data considering both the network and the software.

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 distinguish the two families, they are referred to below as "version 1" and "version 2". However this is not a standard nomenclature.

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 quantities analogous to standard deviation, skewness and kurtosis, termed the L-scale, L-skewness and L-kurtosis respectively. Standardised L-moments are called **L-moment ratios** and are analogous to standardized moments. Just as for conventional moments, a theoretical distribution has a set of population L-moments. Sample L-moments can be defined for a sample from the population, and can be used as estimators of the population L-moments.

The **Lomax distribution**, conditionally also called the **Pareto Type II distribution**, is a heavy-tail probability distribution used in business, economics, actuarial science, queueing theory and Internet traffic modeling. It is named after K. S. Lomax. It is essentially a Pareto distribution that has been shifted so that its support begins at zero.

In probability theory, an **exponentially modified Gaussian distribution** describes the sum of independent normal and exponential random variables. An exGaussian random variable *Z* may be expressed as *Z* = *X* + *Y*, where *X* and *Y* are independent, *X* is Gaussian with mean *μ* and variance *σ*^{2}, and *Y* is exponential of rate *λ*. It has a characteristic positive skew from the exponential component.

In probability theory and statistics, the **harmonic distribution** is a continuous probability distribution. It was discovered by Étienne Halphen, who had become interested in the statistical modeling of natural events. His practical experience in data analysis motivated him to pioneer a new system of distributions that provided sufficient flexibility to fit a large variety of data sets. Halphen restricted his search to distributions whose parameters could be estimated using simple statistical approaches. Then, Halphen introduced for the first time what he called the harmonic distribution or harmonic law. The harmonic law is a special case of the generalized inverse Gaussian distribution family when .

- ↑ http://repository.lppm.unila.ac.id/120/1/23%20On%20the%20Moments,%20Cumulants,%20and%20Characteristic%20Function%20of%20the%20Log-Logistic%20Distribution.pdf
^{[ bare URL PDF ]} - ↑ Everitt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP. ISBN 0-521-81099-X

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