- Source: 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
2
σ
2
v
)
g
(
v
)
d
v
{\displaystyle 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
(
α
s
)
M
g
(
β
s
+
1
2
σ
2
s
2
)
,
{\displaystyle M(s)=\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(\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.