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 $\alpha$, $\beta$ and $\sigma >0$ are real numbers, 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 $\alpha +\beta V$ and variance $\sigma ^{2}V$. 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 $\beta$ and infinitesimal variance $\sigma ^{2}$ 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

$M(s)=\exp(\alpha s)\,M_{g}\left(\beta s+{\frac {1}{2}}\sigma ^{2}s^{2}\right),$

where $M_{g}$ is the moment generating function of the probability distribution with density function $g$, i.e.

$M_{g}(s)=E\left(\exp(sV)\right)=\int _{0}^{\infty }\exp(sv)g(v)\,dv.$

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.