ⓘ Normal variance-mean mixture. In probability theory and statistics, a normal variance-mean mixture with mixing probability density g {\displaystyle g} is the co ..


ⓘ Normal variance-mean mixture

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

Y = α + β V + σ V X, {\displaystyle Y=\alpha +\beta V+\sigma {\sqrt {V}}X,}

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

f x = ∫ 0 ∞ 1 2 π σ 2 v exp ⁡ − x − α − β v 2 σ 2 v) g v d v {\displaystyle fx=\int _{0}^{\infty }{\frac {1}{\sqrt {2\pi \sigma ^{2}v}}}\exp \left{\frac {-x-\alpha -\beta v^{2}}{2\sigma ^{2}v}}\right)gv\,dv}

and its moment generating function is

M s = exp ⁡ α s M g β s + 1 2 σ 2 s 2, {\displaystyle Ms=\exp\alpha s\,M_{g}\left\beta s+{\frac {1}{2}}\sigma ^{2}s^{2}\right,}

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

M g s = E exp ⁡ s V) = ∫ 0 ∞ exp ⁡ s v g v d v. {\displaystyle M_{g}s=E\left\expsV\right)=\int _{0}^{\infty }\expsvgv\,dv.}
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