Elliptically Contoured Distributions
Elliptically Contoured Distributions
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Recall: if X ∼ Np (µ, Σ), then
1
1
0 −1
exp − (x − µ) Σ (x − µ)
fX (x) = √
2
det 2πΣ
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So fX (x) depends on x only through (x − µ)0 Σ−1 (x − µ), and is
therefore constant on the ellipsoidal surfaces
(x − µ)0 Σ−1 (x − µ) = k.
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Each ellipsoid is centered at µ and its principal axes are in the
directions
of the eigenvectors of Σ, with lengths proportional to
√
eigenvalues of Σ.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The general elliptically contoured distribution has pdf
√
1
g (x − ν)0 Λ−1 (x − ν)
det Λ
where:
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Λ is positive definite;
g (·) ≥ 0, with
Z ∞Z ∞
Z
∞
g (y0 y) dy1 dy2 . . . dyp = 1.
...
−∞
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−∞
−∞
If X has this distribution and Y = L−1 (X − ν), where LL0 = Λ, then
Y has pdf g (y0 y), spherically contoured.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Polar Coordinates
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Every y can be written in polar coordinates as
y1 = r sin θ1 ,
y2 = r cos θ1 sin θ2 ,
y3 = r cos θ1 cos θ2 sin θ3 ,
..
.
yp−1 = r cos θ1 cos θ2 . . . cos θp−2 sin θp−1 ,
yp = r cos θ1 cos θ2 . . . cos θp−2 cos θp−1
where:
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r ≥ 0;
− 21 π < θi ≤ 21 π, 1 ≤ i ≤ p − 1, and −π < θp−1 ≤ π.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The Jacobian
J(r , θ) =
r p−1 [cos (θ1 )]p−2 [cos (θ2 )]p−3 . . . [cos (θp−3 )]2 cos (θp−2 )
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If the pdf of Y is g (y0 y), then the pdf of R, Θ is
J(r , θ)g r 2 =
r p−1 g r 2 [cos (θ1 )]p−2 [cos (θ2 )]p−3 . . . [cos (θp−3 )]2 cos (θp−2 )
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R and Θ are independent, and the marginal pdf of R is
1
2π 2 p
× r p−1 g r 2 = C (p) × r p−1 g r 2 .
1
Γ 2p
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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Here C (p) is the ”surface area” of the p-dimensional unit sphere.
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The vector U = Y/R is uniformly distributed on that surface, and,
because it is a function of Θ, is independent of R.
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So Y can be written as RU, where R and U are independent, R has
density
C (p) × r p−1 g r 2 ,
and U is uniformly distributed on the surface of the p-dimensional
unit sphere.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Moments
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If O is an orthogonal matrix (O0 O = Ip ), and V = OU then
V0 V = U0 U = 1, so V also takes values on the surface of the unit
sphere.
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Also the Jacobian is | det O| = 1, so V is uniformly distributed on the
surface of the p-dimensional unit sphere, the same as U.
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So E(U) = E(OU) = OE(U) for every orthogonal O
⇒ E(U) = 0 (take O = −Ip , for instance).
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
Similarly, if
E (UU0 ) = Σ, then
Σ = E VV0 = OΣO0
for every orthogonal O ⇒ Σ = kIp for some k (not so trivial. . . )
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. . . and U0 U = 1 ⇒
kp = trace(Σ) = E trace UU0 = E U0 U = 1
⇒ Σ = E (UU0 ) = p −1 Ip .
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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Finally:
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if E(R) < ∞, then
E(Y) = E(RU) = E(R)E(U) = 0;
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if
E R
2
< ∞, then
1
p
E (YY0 ) = E R 2 UU0 = E R 2 E (UU0 ) = E R 2 Ip
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and since X = LY + ν, under the same conditions
E(X) = ν
and
1
p
C(X) = LC(Y)L0 = E
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1
R 2 LL0 = E R 2 Λ
p
Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Marginal Distributions
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If Y is spherically contoured with density g (y0 y), and Y is partitioned
as
(1) Y
Y=
Y(2)
where Y(1) contains Y1 , Y2 , . . . , Yq , and Y(2) contains
Yq+1 , Yq+2 , . . . , Yp , the marginal density of Y(2) is
Z ∞
Z ∞
g(q+1):p (yq+1 , yq+2 , . . . , yp ) =
...
g u0 u + y20 y2 du
−∞
Z ∞−∞
g r12 + y20 y2 r1q−1 dr1
= C (q)
0
= g2 y20 y2 , say.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
So the marginal distribution of Y(2) is spherically contoured.
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Now suppose that X has the elliptically contoured density
√
1
g (x − ν)0 Λ−1 (x − ν)
det Λ
and is similarly partitioned, and as for the multivariate normal
distribution we let
(2)
Z(1) = X(1) − Σ12 Σ−1
22 X
(2)
= X(1) − Λ12 Λ−1
22 X
Z(2) = X(2)
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
Then the density of Z is
√
1
det Λ11·2 det Λ22
g
z(1) − τ (1)
+ z
0
(2)
(1)
(1)
Λ−1
z
−
τ
11·2
−ν
(2)
0
Λ−1
22
z
(2)
−ν
(2)
where
(2)
τ (1) = ν (1) − Λ12 Λ−1
22 ν
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Note that Z(1) and Z(2) are uncorrelated, but independent only for
the multivariate normal distribution.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The marginal density of Z(2) = X(2) is
0
1
(2)
(2)
g2 x(2) − ν (2) Λ−1
x
−
ν
22
det Λ22
Z ∞ 0
−1
2
(2)
(2)
(2)
(2)
g r1 + x − ν
= C (q)
Λ22 x − ν
r1q−1 dr1
√
0
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So the marginal density of X(2) is also elliptically contoured.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Uniform Distribution on a Sphere
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√
Consider the special case Y = pU, uniformly distributed on the
√
sphere of radius p.
√
√
Then Y1 = pU1 = p sin (Θ1 ), and the pdf of Θ1 is proportional to
[cos (θ1 )]p−1 .
So the pdf of Y1 (and hence of every Yi ) is proportional to
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y2
1−
p
13 / 28
p−3
2
Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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This is the Beta density with parameters α = β = (p − 1)/2, scaled
√ √
to the interval [− p, p].
Special cases:
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p = 1: the “density” isn’t integrable, so the argument fails. Clearly Y1
takes the values ±1 with probability 21 .
√
p = 2: the density is U-shaped, unbounded
at ± 2.
√
p = 3: the density is uniform on ± 3.
p → ∞: the density converges to N(0, 1).
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
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More generally, for any fixed q, the marginal density of Y(1)
(consisting of Y1 , Y2 , . . . , Yq ) converges to Nq (0, Iq ) as p → ∞.
√
√
If the radius is a random R p instead of the fixed p, the density of
Y(1) is the corresponding mixture of scaled distributions, and as
p → ∞ converges to the corresponding mixture of normal
distributions.
So if Y is spherically contoured because it is part of a larger random
vector that is also spherically contoured, of arbitrarily large dimension,
its distribution must be a mixture of spherical normal distributions.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Conditional Distributions
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The conditional density of Y(1) given Y(2) = y2 is
g y10 y1 + r22
g (y10 y1 + y20 y2 )
=
g2 (y20 y2 )
g2 r22
where r22 = y20 y2 .
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This is a spherically contoured density, so Y(1) has conditional mean
zero and conditional covariance matrix
1 0
C Y(1) Y(2) = y(2) = E Y(1) Y(1) R22 = r22 Iq
q
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
After simplification,
Z
E
0
Y(1) Y(1) R22 = r22
∞
= Z0 ∞
0
r1q+1 g r12 + r22 dr1
r1q−1 g r12 + r22 dr1
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In general, this depends on the conditional value r22 .
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It does not depend on r22 for the multivariate normal, because in that
case
g r12 g r22
2
2
g r1 + r2 =
g (0)
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The conditional density of X(1) given X(2) = x(2) is
g2
1
√
g
2
r2
det Λ11·2
h
i0
x(1) − ν (1) − B x(2) − ν (2)
Λ−1
11·2
o
h
i
× x(1) − ν (1) − B x(2) − ν (2) + r22
where
0
(2)
(2)
r22 = x(2) − ν (2) Λ−1
x
−
ν
,
22
B = Λ12 Λ−1
22
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
This conditional density is also elliptically contoured, with
E X(1) X(2) = x(2) = ν (1) + B x(2) − ν (2)
and
C
1
X(1) X(2) = x(2) = E R12 R22 = r22 Λ11·2
q
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The linearity of the conditional mean is shared with the multivariate
normal distribution.
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The conditional covariance matrix is in general not constant, the
exception being the multivariate normal distribution.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
Example: Multivariate t
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Suppose that Z ∼ Np (0, Ip ), and ms 2 ∼ χ2m , independent of Z. Then
Y = (1/s)Z has the multivariate t density with m degrees of freedom,
Γ
m
2
Γ
and
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m+p 2
(mπ)p/2
y0 y
1+
m
− m+p
2
χ2p /p
R2
Y0 Y
=
∼ Fp,m = 2
p
p
χm /m
Every normalized linear combination c0 Y with c0 c = 1 has the
univariate t-distribution with m degrees of freedom.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
Note: if m > 2,
E
R 2 /p = m/(m − 2), so
C(Y) =
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The standardized multivariate t-distribution is defined by
r
r
m−2
m−2
Z
Ys =
Y=
m
ms 2
so that
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m
Ip .
m−2
E (Ys0 Ys /p) = 1 and C(Ys ) = Ip .
The standardized multivariate t-distribution is convenient for
simulating long-tailed data with a given mean vector and covariance
matrix, but it is not the familiar version.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
For the multivariate t-distribution, the conditional density of Y(1)
given Y(2) = y2 is also multivariate t, with m∗ = m + (p − q) degrees
of freedom, but with a scale factor that depends on r22 = y20 y2 .
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Conditionallly on Y(2) = y2 ,
(1)
Y∗ = q
1
1+
r22 −(p−q)
m∗
Y(1)
has the conventional multivariate t-distribution in q dimensions with
m∗ degrees of freedom.
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(1)
Since this distribution does not depend on y2 , Y∗
Y(2) .
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is independent of
Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
So if m∗ > 2,
C
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Y
r22 − (p − q)
(1) (2)
(2)
C Y ∗ Y = y2
Y = y2 = 1 +
m∗
2
r2 − (p − q)
m∗
= 1+
Iq
m∗
m∗ − 2
m∗ + r22 − (p − q)
=
Iq
m∗ − 2
m + r22
Iq .
=
m∗ − 2
(1) That is, large values of Y(2) (in magnitude) increase the conditional
covariance matrix of Y(1) , and small values decrease it.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The reason is that the magnitude of Y(2) provides information about
s 2 : conditionally on Y(2) = y2 ,
m + r22 s 2 ∼ χ2m∗ .
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Heuristically, a large value of Y(2) suggests a small value of s 2 , which
in turn implies a large value of Y(1) .
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Similarly, a small value of Y(2) suggests a large value of s 2 , which in
turn implies a small value of Y(1) .
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
For the general elliptically contoured multivariate t-distribution with
m degrees of freedom, the density function is
m+p 2 √
m
p/2 det Λ
(mπ)
2
Γ
Γ
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− m+p
2
(x − µ)0 Λ−1 (x − µ)
1+
m
The conditional density of X(1) given X(2) = x2 is correspondingly
multivariate t-distribution with m∗ degrees of freedom.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The conditional distribution is centered at
µ(1) + B x(2) − µ(2)
where
B = Λ12 Λ−1
22 .
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If m∗ > 1 (as it must be), this is the conditional mean.
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The matrix of the quadratic form in the conditional density is
m + r22
Λ11·2
m∗
where now
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0
(2)
(2)
r22 = x(2) − µ(2) Λ−1
x
−
µ
22
If m∗ > 2, the conditional covariance matrix is
m + r22
Λ11·2
m∗ − 2
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Statistics 784
Multivariate Analysis
Elliptically Contoured Distributions
I
The conditional structure of the multivariate t-distribution differs
from that of the multivariate normal distribution in a surprisingly
simple way:
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The conditional mean is the same linear function of x(2) ;
The conditional covariance matrix is multiplied by a (scalar) quadratic
function of x(2) , instead of being constant.
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Statistics 784
Multivariate Analysis