Normal variance-mean mixture explained

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 formY = \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.g isf(x) = \int ^{inf ty} _{0} (1)/(\sqrt{ 2 \pi \sigma ^{2} v) } \exp \left(\frac \right) g(v) \, dv

and its moment generating function is

M(s) = \exp(\alpha s) M _{g} \left(\beta s + (12)/(\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 ^{inf ty} _{0} \exp(s v) 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.