Normal variance-mean mixture
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In probability theory a normal variance-mean mixture with mixing probability density g is the continuous probability distribution of a random variable Y of the form
where α and β are real numbers and σ > 0. The random variables X and V are independent, X is normal distributed with mean zero and variance one, and V is a continuous probability distribution 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 observated 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.
The probability density function of a normal variance-mean mixture with mixing probability density g is
and its moment generating function is
where Mg is the moment generating function of the probability distribution with density function g, i.e.
[edit] Reference
O.E Barndorff-Nielsen, J. Kent and M. Sørensen (1982): Normal variance-mean mixtures and z-distributions.International Statistical Review, 50, pp. 145 - 159.
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