Normal variance-mean mixture
In probability theory and statistics, a normal variance-mean mixture with mixing probability density is the continuous probability distribution of a random variable
of the form
where ,
and
are real numbers, and random variables
and
are independent,
is normally distributed with mean zero and variance one, and
is continuously distributed on the positive half-axis with probability density function
. The conditional distribution of
given
is thus a normal distribution with mean
and variance
. A normal variance-mean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a Wiener process (Brownian motion) with drift
and infinitesimal variance
observed at a random time point independent of the Wiener process and with probability density function
. An important example of normal variance-mean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.
The probability density function of a normal variance-mean mixture with mixing probability density is
and its moment generating function is
where is the moment generating function of the probability distribution with density function
, i.e.
See also
References
O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", International Statistical Review, 50, 145–159.