The multinormal distribution with mean
and covariance
has the density
![\begin{displaymath}f(x) = \vert 2 \pi \Sigma \vert^{-1/2} \exp \left \{ -\frac{1}{2}
(x- \mu)^{\top} \Sigma^{-1}(x- \mu) \right \}.
\end{displaymath}](mvahtmlimg1376.gif) |
(4.47) |
We write
.
How is this multinormal distribution with mean
and covariance
related to the multivariate standard normal
?
Through a linear transformation
using the results of Section 4.3, as shown in the next theorem.
THEOREM 4.5
Let
![$ X \sim N_p (\mu, \Sigma) $](mvahtmlimg1377.gif)
and
![$Y = \Sigma ^{-1/2}(X-\mu )$](mvahtmlimg1379.gif)
(Mahalanobis transformation
). Then
i.e., the elements
![$Y_j\in \mathbb{R}$](mvahtmlimg1381.gif)
are independent, one-dimensional
![$N(0,1)$](mvahtmlimg230.gif)
variables.
PROOF:
Note that
.
Application of (4.45) gives
, hence
![\begin{displaymath}f_Y(y) = (2 \pi)^{-p/2} \exp \left(-\frac{1}{2}y^{\top}y \right)
\end{displaymath}](mvahtmlimg1384.gif) |
(4.48) |
which is by (4.47) the pdf of a
.
Note that
the above Mahalanobis transformation yields
in fact a random variable
composed of independent
one-dimensional
since
Here each
is a standard normal density
.
From this it is clear that
and
.
How can we create
variables on the basis of
variables? We use the inverse linear transformation
![\begin{displaymath}X = \Sigma^{1/2}Y + \mu.
\end{displaymath}](mvahtmlimg1393.gif) |
(4.49) |
Using (4.11) and (4.23) we can also check that
and
.
The following theorem is useful because it presents the distribution
of a variable after it has been linearly transformed.
The proof is left as an exercise.
THEOREM 4.6
Let
![$ X \sim N_p (\mu, \Sigma) $](mvahtmlimg1377.gif)
and
![$\data{A}(p\times p),\; c \in \mathbb{R}^p$](mvahtmlimg1396.gif)
,
where
![$\data{A}$](mvahtmlimg319.gif)
is nonsingular.
Then
is again a
-variate Normal, i.e.,
![\begin{displaymath}
Y \sim N_p ( \data{A} \mu +c, \data{A} \Sigma \data{A}^{\top}).
\end{displaymath}](mvahtmlimg1398.gif) |
(4.50) |
From (4.47) we see that the density of the
distribution
is constant on ellipsoids of the form
![\begin{displaymath}
(x-\mu )^{\top} \Sigma^{-1}(x-\mu) = d^2.
\end{displaymath}](mvahtmlimg1399.gif) |
(4.51) |
EXAMPLE 4.14
Figure
4.3 shows the contour ellipses of a
two-dimensional normal distribution. Note that these contour ellipses
are the iso-distance curves (
2.34) from the mean of
this normal distribution corresponding to the metric
![$\Sigma^{-1}$](mvahtmlimg1400.gif)
.
Figure 4.3:
Scatterplot
of a normal sample and contour ellipses for
and
.
MVAcontnorm.xpl
|
According to Theorem 2.7 in Section 2.6 the
half-lengths of the axes in the contour
ellipsoid are
where
are
the eigenvalues of
and
are the eigenvalues of
.
The rectangle inscribing an ellipse has sides with length
and is thus
naturally proportional to the standard deviations of
.
The distribution of the quadratic form in (4.51) is given in
the next theorem.
THEOREM 4.8
The characteristic function (cf) of a multinormal
![$N_p(\mu,\Sigma)$](mvahtmlimg1282.gif)
is given by
![\begin{displaymath}\varphi_X(t) = \exp(\textrm{\bf i}\ t^{\top} \mu -\frac{1}{2}t^{\top}
\Sigma t). \end{displaymath}](mvahtmlimg1411.gif) |
(4.52) |
We can check Theorem 4.8 by transforming the cf back:
since
Note that if
(e.g., the Mahalanobis-transform), then
which is consistent with (4.33).
Singular Normal Distribution
Suppose that we have
, where
is
the dimension of
.
We define the (singular) density of
with the aid of the
-Inverse
of
,
![\begin{displaymath}
f(x) = \frac{(2\pi)^{-k/2}} {(\lambda_1 \cdots \lambda_k)^{1...
...ft \{ -\frac{1}{2} (x-\mu)^{\top} \Sigma^{-} (x-\mu) \right \}
\end{displaymath}](mvahtmlimg1418.gif) |
(4.53) |
where
- (1)
lies on the hyperplane
with
and
.
- (2)
is the
-Inverse of
, and
are the nonzero eigenvalues of
.
What is the connection to a multinormal with
-dimensions? If
![\begin{displaymath}
Y \sim N_k (0, \Lambda_1)\ \textrm{ and }\ \Lambda_1 = \mathop{\hbox{diag}}(\lambda_1,
\ldots, \lambda_k),
\end{displaymath}](mvahtmlimg1423.gif) |
(4.54) |
then there exists an orthogonal matrix
with
so that
where
has a singular pdf of the form (4.53).
The second important copula that we want to present is the
Gaussian or normal copula,
![\begin{displaymath}
C_{\rho}(u, v) =
\int_{- \infty}^{\Phi_1^{-1}(u)}
\int_{- \infty}^{\Phi_2^{-1}(v)} f_\rho(x_1,x_2) d x_2 d x_1\; ,
\end{displaymath}](mvahtmlimg1427.gif) |
(4.55) |
see Embrechts et al. (1999). In (4.55),
denotes the bivariate normal density
function with correlation
for
. The functions
and
in (4.55) refer to the corresponding
one-dimensional standard normal cdfs of the margins.
In the case of vanishing correlation,
, the Gaussian copula becomes
Summary
![$\ast$](mvahtmlimg108.gif)
- The pdf of a
-dimensional multinormal
is
The contour curves of a multinormal are ellipsoids with half-lengths
proportional to
, where
denotes the eigenvalues of
(
).
![$\ast$](mvahtmlimg108.gif)
-
The Mahalanobis transformation transforms
to
.
Going the other direction,
one can create a
from
via
.
![$\ast$](mvahtmlimg108.gif)
-
If the covariance matrix
is singular (i.e.,
),
then it defines a singular normal distribution.
![$\ast$](mvahtmlimg108.gif)
-
The density of a singular normal distribution is given by