In Nelson (1991), which develops the Exponential GARCH model, he refers (p. 352) to the «Generalized Error Distribution (GED)» and provides this density function:
$$f(z; nu) = frac{nucdotexpleft[ -frac{1}{2}|z/lambda|^{nu} right]}{lambda 2^{(1+1/nu)}Gamma(1/nu)},$$ where
begin{equation}
lambdaequiv sqrt{frac{2^{(-2/nu)}Gamma(1/nu)}{Gamma(3/nu)}},
end{equation}
and $Gamma(cdot)$ is the gamma function.
When $nu=2$ this is equivalent to the standard Normal distribution, since $lambda=1$ and
$$f(z; 2) = frac{1}{sqrt{2pi}} expleft(-frac{z^2}{2}right).$$
This distribution also appears in Leemis and McQueston (2008), which lists many distributions, where they call it the «Error (exponential power, general error)» distribution and write it as
$$f(z) = frac{expleft[-frac{1}{2}(|z-a|/b)^{2/c}right]}{b(2^{c/2+1})Gamma(1+c/2)}.$$
Letting $a=0$, $b=lambda$, and $c=2/nu$ we can see that these are the same distributions. Clearly $a$ is the mean parameter, which was assumed to be 0 in Nelson’s paper.
I’m trying to understand the relation between this distribution and what Wikipedia calls the «generalized error distribution», with pdf
$$g(z) = frac{beta}{2alphaGamma(1/beta)} ; e^{-(|z-mu|/alpha)^beta}.$$
Trying to write the Leemis and McQuestion version in the same form as Wikipedia, let $beta=2/c$, $a=mu$, and $alpha=b$. We have
$$f(z) = frac{expleft[-frac{1}{2}(|z-mu|/alpha)^{beta}right]}{2alpha(2^{1/beta})Gamma(1+1/beta)} =
frac{beta expleft[-frac{1}{2}(|z-mu|/alpha)^{beta}right]}{2alphaGamma(1/beta)(2^{1/beta})},$$
where I’ve used the fact that $Gamma(1+x)=xGamma(x)$.
These do not look like the same distributions. There’s an extra $2^{1/beta}$ term in the denominator, and the $-frac{1}{2}$ term in the exponential.
The question is, are these actually the same distributions? Have I made a mistake somewhere?
References
Nelson, Daniel B. «Conditional heteroskedasticity in asset returns: A new approach.» Econometrica (1991): 347-370.
Leemis, Lawrence M., and Jacquelyn T. McQueston. «Univariate distribution relationships.» The American Statistician 62, no. 1 (2008): 45-53.
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 «symmetric» and «asymmetric»; however, this is not a standard nomenclature.
Symmetric versionEdit
|
Probability density function
|
|
|
Cumulative distribution function
|
|
| Parameters |
location (real) scale (positive, real) shape (positive, real) |
|---|---|
| Support | |
|
denotes the gamma function |
|
| CDF |
where is a shape parameter, is a scale parameter and is the unnormalized incomplete lower gamma function. |
| Quantile |
where is the quantile function of Gamma distribution[1] |
| Mean | |
| Median | |
| Mode | |
| Variance | |
| Skewness | 0 |
| Ex. kurtosis | |
| Entropy | [2] |
The symmetric generalized normal distribution, also known as the exponential power distribution or the generalized error distribution, 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 (with mean and variance ) and it includes the Laplace distribution when . As , the density converges pointwise to a uniform density on .
This family allows for tails that are either heavier than normal (when ) or lighter than normal (when ). It is a useful way to parametrize a continuum of symmetric, platykurtic densities spanning from the normal ( ) to the uniform density ( ), and a continuum of symmetric, leptokurtic densities spanning from the Laplace ( ) to the normal density ( ).
The shape parameter also controls the peakedness in addition to the tails.
Parameter estimationEdit
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 is a positive, even integer. Otherwise, the function has continuous derivatives. As a result, the standard results for consistency and asymptotic normality of maximum likelihood estimates of only apply when .
Maximum likelihood estimatorEdit
It is possible to fit the generalized normal distribution adopting an approximate maximum likelihood method.[5][6] With initially set to the sample first moment ,
is estimated by using a Newton–Raphson iterative procedure, starting from an initial guess of ,
where
is the first statistical moment of the absolute values and is the second statistical moment. The iteration is
where
and
and where and are the digamma function and trigamma function.
Given a value for , it is possible to estimate by finding the minimum of:
Finally is evaluated as
For , median is a more appropriate estimator of . Once is estimated, and can be estimated as described above. [7]
ApplicationsEdit
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.
PropertiesEdit
MomentsEdit
Let be zero mean generalized Gaussian distribution of shape and scaling parameter . The moments of exist and are finite for any k greater than −1. For any non-negative integer k, the plain central moments are[2]
Connection to Stable Count DistributionEdit
From the viewpoint of the Stable count distribution, can be regarded as Lévy’s stability parameter. This distribution can be decomposed to an integral of kernel density where the kernel is either a Laplace distribution or a Gaussian distribution:
where is the Stable count distribution and
is the Stable vol distribution.
Connection to Positive-Definite FunctionsEdit
The probability density function of the symmetric generalized normal distribution is a positive-definite function for .[10][11]
Infinite divisibilityEdit
The symmetric generalized Gaussian distribution is an infinitely divisible distribution if and only if .[12]
GeneralizationsEdit
The multivariate generalized normal distribution, i.e. the product of exponential power distributions with the same and parameters, is the only probability density that can be written in the form and has independent marginals.[13] The results for the special case of the Multivariate normal distribution is originally attributed to Maxwell.[14]
Asymmetric versionEdit
|
Probability density function
|
|
|
Cumulative distribution function
|
|
| Parameters |
location (real) scale (positive, real) shape (real) |
|---|---|
| Support |
|
|
, where is the standard normal pdf |
|
| CDF |
, where is the standard normal CDF |
| Mean | |
| Median | |
| Variance | |
| Skewness | |
| Ex. kurtosis |
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. 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 estimationEdit
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.
ApplicationsEdit
The asymmetric generalized normal distribution 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.
Edit
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 log-normal, 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.
Actually all distributions with finite variance are in the limit highly related to the normal distribution. The Student-t distribution, the Irwin–Hall distribution and the Bates distribution also extend the normal distribution, and include in the limit the normal distribution. 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.
See alsoEdit
- Complex normal distribution
- Skew normal distribution
ReferencesEdit
- ^ a b Griffin, Maryclare. «Working with the Exponential Power Distribution Using gnorm». Github, gnorm package. Retrieved 26 June 2020.
- ^ a b Nadarajah, Saralees (September 2005). «A generalized normal distribution». Journal of Applied Statistics. 32 (7): 685–694. doi:10.1080/02664760500079464. S2CID 121914682.
- ^ Varanasi, M.K.; Aazhang, B. (October 1989). «Parametric generalized Gaussian density estimation». Journal of the Acoustical Society of America. 86 (4): 1404–1415. Bibcode:1989ASAJ…86.1404V. doi:10.1121/1.398700.
- ^
Domínguez-Molina, J. Armando; González-Farías, Graciela; Rodríguez-Dagnino, Ramón M. «A practical procedure to estimate the shape parameter in the generalized Gaussian distribution» (PDF). Retrieved 2009-03-03. - ^ Varanasi, M.K.; Aazhang B. (1989). «Parametric generalized Gaussian density estimation». J. Acoust. Soc. Am. 86 (4): 1404–1415. Bibcode:1989ASAJ…86.1404V. doi:10.1121/1.398700.
- ^ Do, M.N.; Vetterli, M. (February 2002). «Wavelet-based Texture Retrieval Using Generalised Gaussian Density and Kullback-Leibler Distance». Transaction on Image Processing. 11 (2): 146–158. Bibcode:2002ITIP…11..146D. doi:10.1109/83.982822. PMID 18244620.
- ^ Varanasi, Mahesh K.; Aazhang, Behnaam (1989-10-01). «Parametric generalized Gaussian density estimation». The Journal of the Acoustical Society of America. 86 (4): 1404–1415. Bibcode:1989ASAJ…86.1404V. doi:10.1121/1.398700. ISSN 0001-4966.
- ^
Liang, Faming; Liu, Chuanhai; Wang, Naisyin (April 2007). «A robust sequential Bayesian method for identification of differentially expressed genes». Statistica Sinica. 17 (2): 571–597. Archived from the original on 2007-10-09. Retrieved 2009-03-03. - ^
Box, George E. P.; Tiao, George C. (1992). Bayesian Inference in Statistical Analysis. New York: Wiley. ISBN 978-0-471-57428-6. - ^
Dytso, Alex; Bustin, Ronit; Poor, H. Vincent; Shamai, Shlomo (2018). «Analytical properties of generalized Gaussian distributions». Journal of Statistical Distributions and Applications. 5 (1): 6. doi:10.1186/s40488-018-0088-5. - ^
Bochner, Salomon (1937). «Stable laws of probability and completely monotone functions». Duke Mathematical Journal. 3 (4): 726–728. doi:10.1215/s0012-7094-37-00360-0. - ^
Dytso, Alex; Bustin, Ronit; Poor, H. Vincent; Shamai, Shlomo (2018). «Analytical properties of generalized Gaussian distributions». Journal of Statistical Distributions and Applications. 5 (1): 6. doi:10.1186/s40488-018-0088-5. - ^ Sinz, Fabian; Gerwinn, Sebastian; Bethge, Matthias (May 2009). «Characterization of the p-Generalized Normal Distribution». Journal of Multivariate Analysis. 100 (5): 817–820. doi:10.1016/j.jmva.2008.07.006.
- ^ Kac, M. (1939). «On a characterization of the normal distribution». American Journal of Mathematics. 61 (3): 726–728. doi:10.2307/2371328. JSTOR 2371328.
- ^ Hosking, J.R.M., Wallis, J.R. (1997) Regional frequency analysis: an approach based on L-moments, Cambridge University Press. ISBN 0-521-43045-3. Section A.8
- ^ Documentation for the lmomco R package
Format: Error(m, s, n)
The Error distribution goes by variety of names:
Exponential Power Distribution
Generalized Error Distribution
(GED)
Generalized Gaussian
distribution (GGD)
Subbotin distribution
To add to the confusion, you will also see a wide range of parameterizations.
In ModelRisk, we have chosen to use the mean m,
standard deviation s and power index
n
to parameterize the distribution, because it make comparisons with the
Normal, Laplace,
Hyperbolic-Secant and
other symmetric distributions easier.
This three parameter distribution offers a variety of symmetric shapes,
as shown in the figures below. The first pane shows the effect on the
distribution’s shape of varying parameter n
. Note n
= 2
is a Normal distribution, n
=1 is a Laplace distribution and
the distribution approaches a Uniform as n
approaches infinity. The second pane shows the change in the distribution’s
spread by varying parameter s, its standard
deviation. Parameter m
is simply the location of the distribution’s peak, and the distribution’s
mean.
Uses
The Error distribution finds quite a lot of use as a prior
distribution in Bayesian
inference because it has greater flexibility than a Normal prior,
in that the Error distribution is flatter than a Normal (platykurtic)
when n
> 2, and more peaked than a Normal distribution (leptokurtic) when
n
< 2. Thus, using the GED allows one to maintain the same mean and variance,
but vary the distribution’s shape (via the parameter n)
as required.
The Error distribution has also used to model variations in historic
UK property market returns.
The ‘Error Function’ distribution,
distinct from the distribution described here, is another format for the
Normal distribution with a zero mean, i.e. Erf(h) = Normal(0, 1/(h*SQRT(2)))
ModelRisk
functions added to Microsoft Excel for the Error distribution
VoseError
generates random values from this distribution for Monte
Carlo simulation, or calculates a percentile if used with a
U
parameter.
VoseErrorObject
constructs a distribution object for this distribution.
VoseErrorProb
returns the probability density or cumulative distribution function for
this distribution.
VoseErrorProb10
returns the log10 of the probability density or cumulative distribution
function.
VoseErrorFit
generates values from this distribution fitted to data, or calculates
a percentile from the fitted distribution.
VoseErrorFitObject
constructs a distribution object of this distribution fitted to data.
VoseErrorFitP
returns the parameters of this distribution fitted to data.
Error distribution equations
