The generalized normal distribution or generalized Gaussian distribution (GGD) is either of two families of
parametric continuous probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
s on the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
line. Both families add a
shape parameter
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP.
...
to the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
. To distinguish the two families, they are referred to below as "symmetric" and "asymmetric"; however, this is not a standard nomenclature.
Symmetric version
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 Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
and
Laplace
Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
distributions, and as limiting cases it includes all
continuous uniform distribution
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies betw ...
s on bounded intervals of the real line.
This family includes the
normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
when
(with mean
and variance
) and it includes the
Laplace distribution
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
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
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real number, real-valued random variable. Like skew ...
densities spanning from the normal (
) to the uniform density (
), and a continuum of symmetric,
leptokurtic
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurt ...
densities spanning from the Laplace (
) to the normal density (
).
The shape parameter
also controls the
peakedness
In probability theory and statistics, a shape parameter (also known as form parameter) is a kind of numerical parameter of a parametric family of probability distributionsEveritt B.S. (2002) Cambridge Dictionary of Statistics. 2nd Edition. CUP.
t ...
in addition to the tails.
Parameter estimation
Parameter estimation via
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
and the
method of moments has been studied. The estimates do not have a closed form and must be obtained numerically. Estimators that do not require numerical calculation have also been proposed.
The generalized normal log-likelihood function has infinitely many continuous derivates (i.e. it belongs to the class C
∞ of
smooth function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
s) 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
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
estimates of
only apply when
.
Maximum likelihood estimator
It is possible to fit the generalized normal distribution adopting an approximate
maximum likelihood
In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...
method. With
initially set to the sample first moment
,
is estimated by using a
Newton–Raphson
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valu ...
iterative procedure, starting from an initial guess of
,
:
where
:
is the first statistical
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
of the absolute values and
is the second statistical
moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...
. The iteration is
:
where
:
and
:
and where
and
are the
digamma function
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function:
:\psi(x)=\frac\ln\big(\Gamma(x)\big)=\frac\sim\ln-\frac.
It is the first of the polygamma functions. It is strictly increasing and strictly ...
and
trigamma function
In mathematics, the trigamma function, denoted or , is the second of the polygamma functions, and is defined by
: \psi_1(z) = \frac \ln\Gamma(z).
It follows from this definition that
: \psi_1(z) = \frac \psi(z)
where is the digamma functio ...
.
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.
Applications
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. Other families of distributions can be used if the focus is on other deviations from normality. If the
symmetry
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definit ...
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
A cusp is the most pointed end of a curve. It often refers to cusp (anatomy), a pointed structure on a tooth.
Cusp or CUSP may also refer to:
Mathematics
* Cusp (singularity), a singular point of a curve
* Cusp catastrophe, a branch of bifurca ...
at the origin.
Properties
Moments
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
:
Connection to Stable Count Distribution
From the viewpoint of the
Stable count distribution
In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn (Chinese: 藺鴻圖) in his 2017 study of daily distributions of the S&P 500 and the ...
,
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
In probability theory and statistics, the Laplace distribution is a continuous probability distribution named after Pierre-Simon Laplace. It is also sometimes called the double exponential distribution, because it can be thought of as two exponen ...
or a
Gaussian distribution:
:
where
is the
Stable count distribution
In probability theory, the stable count distribution is the conjugate prior of a one-sided stable distribution. This distribution was discovered by Stephen Lihn (Chinese: 藺鴻圖) in his 2017 study of daily distributions of the S&P 500 and the ...
and
is the
Stable vol distribution.
Connection to Positive-Definite Functions
The probability density function of the symmetric generalized normal distribution is a
positive-definite function
In mathematics, a positive-definite function is, depending on the context, either of two types of function.
Most common usage
A ''positive-definite function'' of a real variable ''x'' is a complex-valued function f: \mathbb \to \mathbb such th ...
for
.
Infinite divisibility
The symmetric generalized Gaussian distribution is an
infinitely divisible distribution In probability theory, a probability distribution is infinitely divisible if it can be expressed as the probability distribution of the sum of an arbitrary number of independent and identically distributed (i.i.d.) random variables. The characteri ...
if and only if
.
Generalizations
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. The results for the special case of the
Multivariate normal distribution
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One d ...
is originally attributed to
Maxwell.
Asymmetric version
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.
Documentation for the lmomco R package
/ref> 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
In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is
:
f(x) = \frac e^
The parameter \mu ...
, otherwise the distributions are shifted and possibly reversed log-normal distribution
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
s.
Parameter estimation
Parameters can be estimated via maximum likelihood estimation
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed statis ...
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
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
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous proba ...
is another distribution that is useful for modeling deviations from normality due to skew. Other distributions used to model skewed data include the gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter re ...
, lognormal
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...
, and Weibull
Weibull is a Swedish locational surname. The Weibull family share the same roots as the Danish / Norwegian noble family of Falsenbr>They originated from and were named after the village of Weiböl in Widstedts parish, Jutland, but settled in Sk ...
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 log-normal
In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a 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
In probability and statistics, the Irwin–Hall distribution, named after Joseph Oscar Irwin and Philip Hall, is a probability distribution for a random variable defined as the sum of a number of independent random variables, each having a unifo ...
and the Bates distribution
In probability and business statistics, the Bates distribution, named after Grace Bates, is a probability distribution of the mean of a number of statistically independent uniformly distributed random variables on the unit interval. This distr ...
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 also
* Complex normal distribution
In probability theory, the family of complex normal distributions, denoted \mathcal or \mathcal_, characterizes complex random variables whose real and imaginary parts are jointly normal. The complex normal family has three parameters: ''location' ...
* Skew normal distribution
In probability theory and statistics, the skew normal distribution is a continuous probability distribution that generalises the normal distribution
In statistics, a normal distribution or Gaussian distribution is a type of continuous proba ...
References
{{DEFAULTSORT:Generalized Normal Distribution
Continuous distributions