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

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

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

f(x) = \int_0^\infty \frac{1}{\sqrt{2 \pi \sigma^2 v}} \exp \left( -(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 Mg 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.

[edit] References

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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