Normal-inverse Gaussian distribution
Encyclopedia
Nig: The normal-inverse Gaussian distribution (NIG) is continuous probability distribution that is defined as the normal variance-mean mixture where the mixing density is the inverse Gaussian distribution
. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The normal-inverse Gaussian distributions form a subclass of the generalised hyperbolic distributions.
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If
and 
are independent
random variable
s that are NIG-distributed with the same values of the parameters
and 
, but possibly different values of the location and scale parameters, 
, 
and 
, respectively, then 
is NIG-distributed with parameters 
and 
The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (Wiener process
),
, we can define the inverse Gaussian process 
Then given a second independent drifting Brownian motion, 
, the normal-inverse Gaussian process is the time-changed process 
. The process 
at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.
The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.
Inverse Gaussian distribution
| cdf = \Phi\left +\exp\left \Phi\left...
. The tails of the distribution decrease more slowly than the normal distribution. It is therefore suitable to model phenomena where numerically large values are more probable than is the case for the normal distribution. Examples are returns from financial assets and turbulent wind speeds. The normal-inverse Gaussian distributions form a subclass of the generalised hyperbolic distributions.
The fact that there is a simple expression for the moment generating function implies that simple expressions for all moments are available. The class of normal-inverse Gaussian distributions is closed under convolution in the following sense. If
and are independentStatistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s that are NIG-distributed with the same values of the parameters
and , but possibly different values of the location and scale parameters, , and , respectively, then is NIG-distributed with parameters and The normal-inverse Gaussian distribution can also be seen as the marginal distribution of the normal-inverse Gaussian process which provides an alternative way of explicitly constructing it. Starting with a drifting Brownian motion (Wiener process
Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
),
, we can define the inverse Gaussian process Then given a second independent drifting Brownian motion, , the normal-inverse Gaussian process is the time-changed process . The process at time 1 has the normal-inverse Gaussian distribution described above. The NIG process is a particular instance of the more general class of Lévy processes.The parameters of the normal-inverse Gaussian distribution are often used to construct a heaviness and skewness plot called the NIG-triangle.