Both families add a shape parameter to the normal distribution.
To distinguish the two families, they are referred to below as "symmetric" and "asymmetric"; however, this is not a standard nomenclature.
It includes all normal and Laplace distributions, and as limiting cases it includes all continuous uniform distributions on bounded intervals of the real line.
It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal (
), and a continuum of symmetric, leptokurtic densities spanning from the Laplace (
Parameter estimation via maximum likelihood and the method of moments has been studied.
[3] The estimates do not have a closed form and must be obtained numerically.
Estimators that do not require numerical calculation have also been proposed.
[4] The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C∞ of smooth functions) only if
It is possible to fit the generalized normal distribution adopting an approximate maximum likelihood method.
is estimated by using a Newton–Raphson iterative procedure, starting from an initial guess of
[7] The symmetric generalized normal distribution has been used in modeling when the concentration of values around the mean and the tail behavior are of particular interest.
[8][9] Other families of distributions can be used if the focus is on other deviations from normality.
If the symmetry of the distribution is the main interest, the skew normal family or asymmetric version of the generalized normal family discussed below can be used.
If the tail behavior is the main interest, the student t family can be used, which approximates the normal distribution as the degrees of freedom grows to infinity.
The t distribution, unlike this generalized normal distribution, obtains heavier than normal tails without acquiring a cusp at the origin.
It finds uses in plasma physics under the name of Langdon Distribution resulting from inverse bremsstrahlung.
For any non-negative integer k, the plain central moments are[2] From the viewpoint of the Stable count distribution,
[11] The multivariate generalized normal distribution, i.e. the product of
parameters, is the only probability density that can be written in the form
[13] The results for the special case of the Multivariate normal distribution is originally attributed to Maxwell.
[14] The asymmetric generalized normal distribution is a family of continuous probability distributions in which the shape parameter can be used to introduce asymmetry or skewness.
[15][16] When the shape parameter is zero, the normal distribution results.
Positive values of the shape parameter yield left-skewed distributions bounded to the right, and negative values of the shape parameter yield right-skewed distributions bounded to the left.
Since the sample space (the set of real numbers where the density is non-zero) depends on the true value of the parameter, some standard results about the performance of parameter estimates will not automatically apply when working with this family.
The skew normal distribution is another distribution that is useful for modeling deviations from normality due to skew.
Kullback-Leibler divergence (KLD) is a method using for compute the divergence or similarity between two probability density functions.
So there is no strong reason to prefer the "generalized" normal distribution of type 1, e.g. over a combination of Student-t and a normalized extended Irwin–Hall – this would include e.g. the triangular distribution (which cannot be modeled by the generalized Gaussian type 1).
A symmetric distribution which can model both tail (long and short) and center behavior (like flat, triangular or Gaussian) completely independently could be derived e.g. by using X = IH/chi.
The Tukey g- and h-distribution also allows for a deviation from normality, both through skewness and fat tails.