Normal variance-mean mixture

In probability theory and statistics, a normal variance-mean mixture with mixing probability density [math]\displaystyle{ g }[/math] is the continuous probability distribution of a random variable [math]\displaystyle{ Y }[/math] of the form

[math]\displaystyle{ Y=\alpha + \beta V+\sigma \sqrt{V}X, }[/math]

where [math]\displaystyle{ \alpha }[/math], [math]\displaystyle{ \beta }[/math] and [math]\displaystyle{ \sigma \gt 0 }[/math] are real numbers, and random variables [math]\displaystyle{ X }[/math] and [math]\displaystyle{ V }[/math] are independent, [math]\displaystyle{ X }[/math] is normally distributed with mean zero and variance one, and [math]\displaystyle{ V }[/math] is continuously distributed on the positive half-axis with probability density function [math]\displaystyle{ g }[/math]. The conditional distribution of [math]\displaystyle{ Y }[/math] given [math]\displaystyle{ V }[/math] is thus a normal distribution with mean [math]\displaystyle{ \alpha + \beta V }[/math] and variance [math]\displaystyle{ \sigma^2 V }[/math]. 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 [math]\displaystyle{ \beta }[/math] and infinitesimal variance [math]\displaystyle{ \sigma^2 }[/math] observed at a random time point independent of the Wiener process and with probability density function [math]\displaystyle{ g }[/math]. 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 [math]\displaystyle{ g }[/math] is

[math]\displaystyle{ 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 }[/math]

and its moment generating function is

[math]\displaystyle{ M(s) = \exp(\alpha s) \, M_g \left(\beta s + \frac12 \sigma^2 s^2 \right), }[/math]

where [math]\displaystyle{ M_g }[/math] is the moment generating function of the probability distribution with density function [math]\displaystyle{ g }[/math], i.e.

[math]\displaystyle{ M_g(s) = E\left(\exp( s V)\right) = \int_0^\infty \exp(s v) g(v) \, dv. }[/math]

See also

• Normal-inverse Gaussian distribution
• Variance-gamma distribution
• Generalised hyperbolic 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.