Kind of numerical parameter of a parametric family of probability distributions
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).
For example, "peakedness" refers to how round the main peak is.[3]
Probability density functions for selected distributions with mean 0 and variance 1.
^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. ISBN 0-521-81099-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. ISSN 0003-4851.
probability theory and statistics, a shapeparameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability...
use: With a shapeparameter k and a scale parameter θ With a shapeparameter α = k {\displaystyle \alpha =k} and an inverse scale parameter β = 1 / θ {\displaystyle...
{\displaystyle \varphi (r)=\exp(-(\varepsilon r)^{2})} , the Gaussian, with a shapeparameter of ε = 3 {\displaystyle \varepsilon =3} , we can then solve the matrix...
roles, including the following: location parameter dispersion parameter or scale parametershapeparameter Where a probability distribution has a domain...
&x\geq 0,\\0,&x<0,\end{cases}}} where k > 0 is the shapeparameter and λ > 0 is the scale parameter of the distribution. Its complementary cumulative distribution...
parameters, denoted by alpha (α) and beta (β), that appear as exponents of the variable and its complement to 1, respectively, and control the shape of...
that describe the system are called parameters. For example, in mechanics, the masses, the dimensions and shapes (for solid bodies), the densities and...
including: The compactness measure of a shape In statistics: The shapeparameter, sometimes referred to as the shape factor, of some probability distributions...
continuous probability distributions on the real line. Both families add a shapeparameter to the normal distribution. To distinguish the two families, they are...
is specified by only scale and shape and sometimes only by its shapeparameter. Some references give the shapeparameter as κ = − ξ {\displaystyle \kappa...
a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions. The larger the scale parameter, the more...
shown that the Wishart distribution can be defined if and only if the shapeparameter n belongs to the set Λ p := { 0 , … , p − 1 } ∪ ( p − 1 , ∞ ) . {\displaystyle...
Weibull distribution with fixed shapeparameter k is an exponential family. Unlike in the previous examples, the shapeparameter does not affect the support;...
distribution, which is a special case of the gamma distribution with integral shapeparameter, developed to predict waiting times in queuing systems The inverse-gamma...
two shapeparameters (and a scale parameter). It is a generalization of the gamma distribution which has one shapeparameter (and a scale parameter). Since...
\lambda \geq 0,} The parameter k is called the shapeparameter, and the parameter λ {\displaystyle \lambda } is called the rate parameter. An alternative,...
{\displaystyle \mu >0} is the mean and λ > 0 {\displaystyle \lambda >0} is the shapeparameter. The inverse Gaussian distribution has several properties analogous...
of many free parameters, including peak shape, unit cell dimensions and coordinates of all atoms in the crystal structure. Other parameters can be guessed...
1 {\displaystyle \alpha <1} and β < 1 {\displaystyle \beta <1} are shapeparameters. Smooth transition function normalized to (-1,1): f ( x ) = { 2 1 +...
total intensity (I), (fractional) degree of polarization (p), and the shapeparameters of the polarization ellipse. The effect of an optical system on the...
X, and α is a positive parameter. The type I Pareto distribution is characterized by a scale parameter xm and a shapeparameter α, which is known as the...
1)} is the quantile function of a gamma distribution with shapeparameter n and scale parameter 1.: 176-178 This interval is 'exact' in the sense that...
known as a Pareto Type II distribution, with shapeparameter α {\displaystyle \alpha } and scale parameter λ {\displaystyle \lambda } , then X λ ∼ β ′...
as an extension of the Weibull family obtained by adding a second shapeparameter. The cumulative distribution function for the exponentiated Weibull...
depending on the author, either the parameter r or its reciprocal α is referred to as the "dispersion parameter", "shapeparameter" or "clustering coefficient"...
distribution if both shapeparameters are >= 1. The Weibull distribution if the shapeparameter is >= 1. Note that all of the parameter restrictions have...
Global Information