Normal variance-mean mixture

In probability theory and statistics, a normal variance-mean mixture with mixing probability density g is the continuous probability distribution of a random variable Y of the form

$Y=\alpha + \beta V+\sigma \sqrt{V}X,$

where α and β are real numbers and σ > 0 and random variables X and V are independent, X is normally distributed with mean zero and variance one, and V is continuously distributed on the positive half-axis with probability density function g. The conditional distribution of Y given V is thus a normal distribution with mean α + βV and variance σ²V. 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 β and infinitesimal variance σ² observed at a random time point independent of the Wiener process and with probability density function g. 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 g is

$f(x) = \int_0^\infty \frac{1}{\sqrt{2 \pi \sigma^2 v}} \exp \left( \frac{-(x - \alpha - \beta v)^2}{2 \sigma^2 v} \right) g(v) \, dv$

and its moment generating function is

$M(s) = \exp(\alpha s) \, M_g \left(\beta s + \frac12 \sigma^2 s^2 \right),$

where M_g is the moment generating function of the probability distribution with density function g, i.e.

$M_g(s) = E\left(\exp( s V)\right) = \int_0^\infty \exp(s v) g(v) \, dv.$

See also

• Normal-inverse Gaussian distribution

References

O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): "Normal variance-mean mixtures and z-distributions", International Statistical Review, 50, 145–159.