Template:Probability 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 distinguish the two families, they are referred to below as "version 1" and "version 2." However this is not a standard nomenclature.
Version 1
Known also as the exponential power distribution, or the generalized error distribution, this is a parametric family of symmetric distributions. It includes all normal and Laplace distributions, and as limiting cases it includes all continuous uniform distributions on bounded intervals of the real line.
This family includes the normal distribution when $\backslash textstyle\backslash beta=2$ (with mean $\backslash textstyle\backslash mu$ and variance $\backslash textstyle\; \backslash frac\{\backslash alpha^2\}\{2\}$) and it includes the Laplace distribution when $\backslash textstyle\backslash beta=1$. As $\backslash textstyle\backslash beta\backslash rightarrow\backslash infty$, the density converges pointwise to a uniform density on $\backslash textstyle\; (\backslash mu-\backslash alpha,\backslash mu+\backslash alpha)$.
This family allows for tails that are either heavier than normal (when $\backslash beta<2$) or lighter than normal (when $\backslash beta>2$). It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal ($\backslash textstyle\backslash beta=2$) to the uniform density ($\backslash textstyle\backslash beta=\backslash infty$), and a continuum of symmetric, leptokurtic densities spanning from the Laplace ($\backslash textstyle\backslash beta=1$) to the normal density ($\backslash textstyle\backslash beta=2$).
Parameter estimation
Parameter estimation via maximum likelihood and the method of moments has been studied.^{[1]} The estimates do not have a closed form and must be obtained numerically. Estimators that do not require numerical calculation have also been proposed.^{[2]}
The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C^{∞} of smooth functions) only if $\backslash textstyle\backslash beta$ is a positive, even integer. Otherwise, the function has $\backslash textstyle\backslash lfloor\; \backslash beta\; \backslash rfloor$ continuous derivatives. As a result, the standard results for consistency and asymptotic normality of maximum likelihood estimates of $\backslash beta$ only apply when $\backslash textstyle\backslash beta\backslash ge\; 2$.
Maximum likelihood estimator
It is possible to fit the generalized normal distribution adopting an approximate maximum likelihood method.^{[3]}^{[4]} With $\backslash mu$ initially set to the sample first moment $m\_1$,
$\backslash textstyle\backslash beta$ is estimated by using a Newton-Raphson iterative procedure, starting from an initial guess of $\backslash textstyle\backslash beta=\backslash textstyle\backslash beta\_0$,
- $\backslash beta\; \_0\; =\; \backslash frac\{m\_1\}\{\backslash sqrt\{m\_2\}\},$
where
- $m\_1=\{1\; \backslash over\; N\}\; \backslash sum\_\{i=1\}^N\; |x\_i|,$
is the first statistical moment of the absolute values and $m\_2$ is the second statistical moment. The iteration is
- $\backslash beta\; \_\{i+1\}\; =\; \backslash beta\; \_\{i\}\; -\; \backslash frac\{g(\backslash beta\; \_\{i\})\}\{g\text{'}(\backslash beta\; \_\{i\})\}\; ,$
where
- $g(\backslash beta)=\; 1\; +\; \backslash frac\{\backslash psi(1/\backslash beta)\}\{\backslash beta\}\; -\; \backslash frac\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\; \backslash log|x\_i-\backslash mu|\; \}\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\}\; +\; \backslash frac\{\backslash log(\; \backslash frac\{\backslash beta\}\{N\}\; \backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\})\}\{\backslash beta\}\; ,$
and
- $g\text{'}(\backslash beta)=\; -\backslash frac\{\backslash psi(1/\backslash beta)\}\{\backslash beta^2\}\; -\; \backslash frac\{\backslash psi\text{'}(1/\backslash beta)\}\{\backslash beta^3\}\; +\; \backslash frac\{1\}\{\backslash beta^2\}\; -\; \backslash frac\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\; (\backslash log|x\_i-\backslash mu|)^2\}\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\}\; +\; \backslash frac\{(\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\; \backslash log|x\_i-\backslash mu|)^2\}\{(\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\})^2\}\; +\; \backslash frac\{\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\; \backslash log|x\_i-\backslash mu|\}\{\backslash beta\; \backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\}\; -\; \backslash frac\{\backslash log(\backslash frac\{\backslash beta\}\{N\}\; \backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}\; )\}\{\backslash beta^2\}\; ,$
and where $\backslash psi()$ and $\backslash psi\text{'}()$ are the digamma function and trigamma function.
Given a value for $\backslash textstyle\backslash beta$, it is possible to estimate $\backslash mu$ by finding the minimum of:
- $min\_\{\backslash mu\}=\backslash sum\_\{i=1\}^\{N\}\; |x\_i-\backslash mu|^\{\backslash beta\}$
Finally $\backslash textstyle\backslash alpha$ is evaluated as
- $\backslash alpha\; =\; (\; \backslash frac\{\backslash beta\}\{N\}\; \backslash sum\_\{i=1\}^\{N\}|x\_i-\backslash mu|^\{\backslash beta\})^\{\backslash frac\{1\}\{\; \backslash beta\}\}\; .$
Applications
This version of the generalized normal distribution has been used in modeling when the concentration of values around the mean and the tail behavior are of particular interest.^{[5]}^{[6]} 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 version 2 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.
Properties
The multivariate generalized normal distribution, i.e. the product of $n$ exponential power distributions with the same $\backslash beta$ and $\backslash alpha$ parameters, is the only probability density that can be written in the form $p(\backslash mathbf\; x)=g(\backslash |\backslash mathbf\; x\backslash |\_\backslash beta)$ and has independent marginals.^{[7]} The results for the special case of the Multivariate normal distribution is originally attributed to Maxwell.^{[8]}
Template:Probability distribution \text{ sign}(\kappa) |
kurtosis =$e^\{4\; \backslash kappa^2\}\; +\; 2\; e^\{3\; \backslash kappa^2\}\; +\; 3\; e^\{2\; \backslash kappa^2\}\; -\; 6$|
entropy =|
mgf =|
char =|
}}
Version 2
This is a family of continuous probability distributions in which the shape parameter can be used to introduce skew.^{[9]}^{[10]} 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. Only when the shape parameter is zero is the density function for this distribution positive over the whole real line: in this case the distribution is a normal distribution, otherwise the distributions are shifted and possibly reversed log-normal distributions.
Parameter estimation
Parameters can be estimated via maximum likelihood estimation or the method of moments. The parameter estimates do not have a closed form, so numerical calculations must be used to compute the estimates. 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.
Applications
This family of distributions can be used to model values that may be normally distributed, or that may be either right-skewed or left-skewed relative to the normal distribution. The skew normal distribution is another distribution that is useful for modeling deviations from normality due to skew. Other distributions used to model skewed data include the gamma, lognormal, and Weibull distributions, but these do not include the normal distributions as special cases.
Other distributions related to the normal
The two generalized normal families described here, like the skew normal family, are parametric families that extends the normal distribution by adding a shape parameter. Due to the central role of the normal distribution in probability and statistics, many distributions can be characterized in terms of their relationship to the normal distribution. For example, the lognormal, folded normal, and inverse normal distributions are defined as transformations of a normally-distributed value, but unlike the generalized normal and skew-normal families, these do not include the normal distributions as special cases.
See also
References
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