<named-content content-type="header-section-number">4.10</named-content>
Copulas
In many applications we wish to model the joint behavior of several
random variables while allowing flexible and possibly different models
for their marginal distributions. Classical multivariate families (for
example, the bivariate normal in Section 3.3 of Casella & Berger
(2002)) tie the joint and marginal distributions together quite
rigidly. Copulas provide a way to separate these two aspects: the
margins and the dependence structure (Joe, 1997; Nelsen, 1999).
In the years leading up to the 2008 financial crisis, a particular
dependence model—the Gaussian copula—became widely used on Wall Street
to price and manage the risk of complex credit products such as
collateralized debt obligations (Salmon, 2009). By summarizing default
dependence between thousands of loans in a single, tractable
parameter, it allowed rating agencies and banks to turn heterogeneous
pools of mortgages into highly rated securities. The simplicity of the
formula helped mask the fact that it relied on a very specific and
fragile assumption about joint tail behavior: that extreme losses
across different loans were, in effect, nearly independent once the
linear correlation was set. When housing prices fell nationwide and
defaults became simultaneously likely across many regions, this
assumption broke down, and the models severely understated the
probability of joint defaults. Copulas themselves are not at fault,
but this episode illustrates how important it is to understand what a
dependence model captures—and, just as importantly, what it leaves
out.
In the following we consider solely the bivariate case; higher
dimensions are similar in spirit but technically more involved (Joe,
1997; Nelsen, 1999).
<named-content content-type="header-section-number">4.10.1</named-content>
Definition and Basic Properties
Recall from Section 4.5 of Casella & Berger (2002) that a
bivariate distribution function
FX,Y
must be 2-increasing and right-continuous and must have appropriate
limits at infinity, see Definition 4.1.1 and the discussion
preceding Example 4.5.1 in Casella & Berger (2002). Copulas are
special bivariate distribution functions on
[0,1]2
with uniform marginals (Nelsen, 1999).
Definition 4.10.1. Let
I=[0,1].
A bivariate copula (or 2-copula) is a
function
C:I2→I
such that:
(i)
(Margins) For every
u,v∈I,
C(u,0)=0,C(0,v)=0(i.e. C is grounded),C(u,1)=u,C(1,v)=v.
(ii)
(2-increasing) For every
u1≤u2
and
v1≤v2
in
I,
C(u2,v2)−C(u2,v1)−C(u1,v2)+C(u1,v1)≥0.
Thus a copula is itself a bivariate cdf on
[0,1]2
with uniform
[0,1]
marginals, in the sense of Definition 4.1.1 of Casella & Berger
(2002), see (Nelsen, 1999). Three special copulas will be
useful:
Definition 4.10.2. The following functions
from
I2
to
I
are called the product, lower
Fréchet, and upper Fréchet copulas,
respectively:
Π(u,v)=uv,W(u,v)=max{u+v−1,0},M(u,v)=min{u,v},
for
u,v∈I.
These correspond, respectively, to independence, perfect negative
monotone dependence, and perfect positive monotone dependence
(Nelsen, 1999). As in the inequalities for general bivariate
distributions (Section 4.5 of Casella & Berger (2002)), one can
show that for any copula
C,
W(u,v)≤C(u,v)≤M(u,v),u,v∈I.
Proof: Let
C
be a copula on
I2,
and fix
u,v∈I.
Since
C
is a bivariate distribution function, it is nondecreasing in each
coordinate. If
u≤v,
then by monotonicity in the second coordinate and the boundary
condition
C(u,1)=u,
C(u,v)≤C(u,1)=u=min{u,v}.
If
v≤u,
then by monotonicity in the first coordinate and
C(1,v)=v,
C(u,v)≤C(1,v)=v=min{u,v}.
Thus
C(u,v)≤M(u,v)=min{u,v}.
For the lower bound, use the 2-increasing property with
u1=u,
u2=1,
v1=v,
and
v2=1
to obtain
C(1,1)−C(1,v)−C(u,1)+C(u,v)≥0.
Using the boundary conditions
C(1,1)=1,
C(1,v)=v,
and
C(u,1)=u,
this becomes
1−v−u+C(u,v)≥0,
or
C(u,v)≥u+v−1.
Because any copula is a cdf,
C(u,v)≥0,
so
C(u,v)≥max{u+v−1,0}=W(u,v).
Combining the two inequalities yields
W(u,v)≤C(u,v)≤M(u,v)
for all
u,v∈I.
▫
The copula
Π
corresponds to the case of independent random variables (see
Definition 1.3.3 of Casella & Berger (2002));
M
and
W
correspond to the Fréchet bounds in the general dependence ordering
(Nelsen, 1999).
<named-content content-type="header-section-number">4.10.2</named-content>
Sklar’s Theorem
The central result connecting copulas with joint distributions is
due to Sklar (1959) . It shows how any joint cdf
FX,Y
studied in Sections 4.1 and 4.5 of Casella & Berger (2002) can
be decomposed into its marginals (Sections 4.1 and 4.2) and a
copula.
Theorem 4.10.3 (Sklar’s Theorem). Let
FX,Y
be the joint cdf of a pair of random variables
(X,Y)
with marginal cdfs
FX
and
FY.
(a)
There exists a copula
C
such that
FX,Y(x,y)=C(FX(x),FY(y)),x,y∈ℝ.
(b)
If
FX
and
FY
are continuous, then
C
is unique.
Conversely, if
C
is any copula and
FX,FY
are univariate cdfs, then
F(x,y)=C(FX(x),FY(y))
is a bivariate cdf with marginals
FX
and
FY.
Proof of the continuous margin case: For
x,y∈ℝ,
define
u=FX(x),v=FY(y),
and let
H(x,y)=FX,Y(x,y)
denote the joint cdf of
(X,Y).
For part (a), we must construct a function
C
on
[0,1]2
such that
H(x,y)=C(FX(x),FY(y)).
First suppose that
FX
and
FY
are continuous and strictly increasing. Then each cdf has a
(generalized) inverse
FX−1
and
FY−1,
and for any
u,v∈(0,1)
we may define
C(u,v)=H(FX−1(u),FY−1(v)).
If
u=FX(x)
and
v=FY(y),
then
FX−1(u)=x
and
FY−1(v)=y,
so
C(FX(x),FY(y))=H(x,y)=FX,Y(x,y),
which gives the desired representation. It is routine to check from
the properties of
H
and the marginals that
C
satisfies the copula axioms and has uniform margins on
(0,1).
For part (b), now suppose that
FX
and
FY
are continuous. If
C1
and
C2
are copulas such that
H(x,y)=C1(FX(x),FY(y))=C2(FX(x),FY(y))
for all
x,y,
then for any
u,v∈(0,1)
we can write
u=FX(x)
and
v=FY(y)
for some
x,y,
and hence
C1(u,v)=C2(u,v).
Thus
C1=C2
on
(0,1)2,
so the copula is unique when the marginals are continuous.
For the converse, let
C
be any copula and let
FX
and
FY
be univariate cdfs. Define
F(x,y)=C(FX(x),FY(y)).
Since
C
is a bivariate cdf on
[0,1]2
and
FX,FY
are cdfs on
ℝ,
the composition
F
is nondecreasing in each argument, right continuous, and has the
correct limits at
±∞,
so
F
is a bivariate cdf. Moreover,
limy→∞F(x,y)=limy→∞C(FX(x),FY(y))=C(FX(x),1)=FX(x),
and similarly
limx→∞F(x,y)=FY(y),
so the marginals of
F
are
FX
and
FY.
This completes the proof.
▫
If the marginal cdfs are continuous but not strictly increasing,
the same formula, interpreted with generalized inverses, yields the
copula uniquely on the relevant range and extends uniquely by
continuity to
[0,1]2.
If one or both marginal cdfs are discontinuous, Sklar’s
representation in part (a) remains valid, but constructing a copula
on the whole unit square requires an additional extension argument.
In that case the copula is uniquely determined only on
Ran(FX)×Ran(FY),
see, for example, (Nelsen, 1999). For the continuous models
considered below, the preceding construction gives the required
copula directly and uniquely.
Example 4.10.4 (Exponential margins with
dependence). Let
X
and
Y
be nonnegative random variables with exponential marginals
FX(x)=1−e−λx,FY(y)=1−e−μy,x,y≥0,
and suppose their joint cdf is
FX,Y(x,y)=C(FX(x),FY(y)),
for some copula
C.
(a)
If
C(u,v)=uv
(the product copula), then
FX,Y(x,y)=FX(x)FY(y)
for all
x,y≥0,
and
X
and
Y
are independent (Section 1.3).
(b)
If instead we take, for some
0]]>
θ>0,
the so-called Clayton copula
Cθ(u,v)=(u−θ+v−θ−1)−1/θ,
we obtain a model with the same exponential marginals but
positive lower tail dependence
(Section 4.10.4).
In this case, small simultaneous values of
X
and
Y
are more likely than under independence, even though the
marginal distributions are unchanged.
This example shows how the copula isolates the
dependence structure from the choice of marginal
distributions.
Thus any bivariate distribution with continuous marginals can be
decomposed into its margins and a copula that encodes the dependence
structure (Nelsen, 1999). Conversely, by choosing any margins (for
example, from the families in Chapter 3 of Casella & Berger
(2002)) and any copula, we obtain a valid bivariate
distribution.
<named-content content-type="header-section-number">4.10.3</named-content>
Probability Integral Transform and Invariance
Sklar’s theorem is closely tied to the probability integral
transform, which appeared implicitly in Section 2.1 of Casella &
Berger (2002) when we discussed distributions of functions of a
random variable (see, e.g., Nelsen, 1999).
If
X
has continuous cdf
FX,
then
U=FX(X)
is Uniform
(0,1);
similarly, if
Y
has cdf
FY,
then
V=FY(Y)
is Uniform
(0,1).
(Compare this with the construction of random variables from
uniforms in Section 5.5 of Casella & Berger (2002)) If
(X,Y)
has copula
C,
then
P(U≤u,V≤v)=C(u,v),0<u,v<1.
Hence copulas are exactly the joint distributions of the transformed
pair
(U,V)=(FX(X),FY(Y)).
This is a specific instance of the general transformation ideas in
Chapter 2 of Casella & Berger (2002).
Copulas are invariant under strictly increasing transformations
of the margins. This invariance is analogous to the invariance
principles discussed later in Chapter 6 of Casella & Berger
(2002), but here at the level of probability models rather than
estimators.
Theorem 4.10.5 (Invariance). Let
(X,Y)
be continuous with copula
C.
Let
g
and
h
be strictly increasing functions, and define
Z=g(X),
T=h(Y).
Then
(Z,T)
has the same copula
C
(Nelsen, 1999).
Proof: Let
FX
and
FY
be the marginal cdfs of
X
and
Y,
and let
FX,Y
be their joint cdf. Since
(X,Y)
has copula
C,
Sklar’s Theorem
(Theorem 4.10.3)
implies
FX,Y(x,y)=C(FX(x),FY(y)),x,y∈ℝ.
Because
g
and
h
are strictly increasing, the events
{g(X)≤z}
and
{h(Y)≤t}
are equivalent to
{X≤g−1(z)}
and
{Y≤h−1(t)},
respectively. Thus, for
z,t∈ℝ,
P(Z≤z,T≤t)=P(X≤g−1(z),Y≤h−1(t))=FX,Y(g−1(z),h−1(t)).
Let
FZ
and
FT
be the marginal cdfs of
Z
and
T.
Since
g
and
h
are strictly increasing,
FZ(z)=P(Z≤z)=P(X≤g−1(z))=FX(g−1(z)),
FT(t)=P(T≤t)=P(Y≤h−1(t))=FY(h−1(t)).
Hence, for
z,t∈ℝ,
FZ,T(z,t)=P(Z≤z,T≤t)=FX,Y(g−1(z),h−1(t))=C(FX(g−1(z)),FY(h−1(t)))=C(FZ(z),FT(t)).
This is exactly the Sklar representation of
(Z,T)
with copula
C,
so
(Z,T)
has the same copula as
(X,Y).
▫
Example 4.10.6 (Ranks and the empirical copula).
Suppose
(Xi,Yi),
i=1,…,n,
is a random sample from a continuous bivariate distribution with
copula
C.
Let
Ri
be the rank of
Xi
among
X1,…,Xn,
and let
Si
be the rank of
Yi
among
Y1,…,Yn.
Define the pseudo-observations
Ui=Rin+1,Vi=Sin+1,i=1,…,n.
Each
(Ui,Vi)
lies in
(0,1)2,
and these points can be viewed as a sample from an approximation
to
(U,V)=(FX(X),FY(Y)),
whose joint distribution is the copula
C.
The empirical copula is then
Cn(u,v)=1n∑i=1n𝟏{Ui≤u,Vi≤v},0≤u,v≤1.
As
n
increases,
Cn(u,v)
converges to
C(u,v)
for each fixed
(u,v),
in a manner analogous to the convergence of the empirical cdf to
the true cdf later in Section 5.4. Thus, the dependence structure
can be studied using only the ranks, without specifying the
marginal distributions.
Thus the copula depends only on the rank structure of the data
and not on the particular marginal scales. This is the basis for
rank-based inference for copula models (see Section 7.2 of Casella
& Berger (2002), where rank-based procedures are used for
correlation and nonparametric measures of association).
If
g
or
h
is decreasing, the copula transforms in a simple way (for example,
via reflections), but the dependence structure remains fully
described at the copula level (Nelsen, 1999).
<named-content content-type="header-section-number">4.10.4</named-content>
Dependence Concepts and Measures
Copulas formalize several notions of dependence and yield
measures that are invariant under monotone transformations (Joe,
1997; Nelsen, 1999). Earlier in the book, dependence was described
in terms of independence (Section 1.3 of Casella & Berger
(2002)), covariance, and correlation (Section 2.2 of Casella &
Berger (2002)). These classical measures depend on second moments
and are not invariant under nonlinear transformations. Copula-based
measures remedy this.
Definition 4.10.7. Random variables
X
and
Y
are said to be positively quadrant dependent (PQD) if
P(X≤x,Y≤y)≥P(X≤x)P(Y≤y)
for all
x,y.
Equivalently, in terms of the copula
C,
C(u,v)≥uv,0<u,v<1.
Negative quadrant dependence (NQD) is defined by reversing the
inequality. Because PQD can be expressed entirely in terms of the
copula, it is invariant under strictly increasing transformations of
the margins, in contrast with Pearson correlation (see the
discussion at the end of Section 2.2 of Casella & Berger
(2002)).
Definition 4.10.8. Kendall’s
τ
is a widely used measure of concordance that depends only on the
copula. For a continuous bivariate distribution with copula
C,
τ=4∫01∫01C(u,v)dC(u,v)−1.
0\bigr)
-P\bigl((X_1-X_2)(Y_1-Y_2)<0\bigr),]]>
Equivalently,τ=P((X1−X2)(Y1−Y2)>0)−P((X1−X2)(Y1−Y2)<0),
where
(X1,Y1)
and
(X2,Y2)
are independent copies.
Kendall’s
τ
satisfies:
−1≤τ≤1,
with
τ=1
(resp.
−1)
if and only if
C=M
(resp.
C=W),
that is, perfect increasing (resp. decreasing) functional
dependence.
τ=0
if
X
and
Y
are independent (i.e.,
C=Π);
compare this with
Corr(X,Y)=0
in Section 2.2 of Casella & Berger (2002).
τ
is invariant under strictly monotone transformations of
X
and/or
Y.
In practice,
τ
is estimated by the sample proportion of concordant minus discordant
pairs, based only on the ranks (see also rank-based methods in
Chapter 7 of Casella & Berger (2002)), cf. (Nelsen, 1999).
Definition 4.10.9. Spearman’s
ρ
is the ordinary correlation of the rank variables and can also be
expressed in terms of the copula. Let
U=FX(X),
V=FY(Y);
then
ρS=Corr(U,V)=12∫01∫01(C(u,v)−uv)dudv.
Example 4.10.10 (Pearson correlation versus
Spearman’s
ρ).
Let
X
be standard normal, and define
Y=X3.
Then
Y
is a strictly increasing function of
X,
so the copula of
(X,Y)
is the upper Fréchet bound
M(u,v)=min{u,v}.
In particular, both Kendall’s
τ
and Spearman’s
ρ
are equal to
1
for this pair.
However, Pearson’s correlation coefficient between
X
and
Y
is strictly less than
1,
because the relationship between
X
and
Y
is nonlinear. This illustrates how copula-based measures such as
τ
and
ρS
capture monotone dependence, while Pearson correlation measures
only linear association (Section 2.2 of Casella & Berger
(2002)).
Spearman’s
ρ
shares properties analogous to Kendall’s
τ:
it is bounded between
−1
and
1,
equals
±1
for
C=M
or
C=W,
is
0
under independence, and is invariant under strictly increasing
transformations of the margins (Nelsen, 1999). Unlike Pearson
correlation
ρ
(Section 2.2 of Casella & Berger (2002)),
ρS
is defined without second-moment assumptions.
These measures are especially useful for parametric copula
families, where
τ
or
ρS
can be expressed as simple functions of the dependence parameter and
inverted to give rank-based estimators (cf. the method of moments in
Section 7.2 of Casella & Berger (2002)).
Definition 4.10.11. Copulas also describe
extremal dependence well. The lower tail dependence
coefficient is
λL=limu↓0P(Y≤FY−1(u)∣X≤FX−1(u))=limu↓0C(u,u)u,
when the limit exists. The upper tail dependence
coefficient is
λU=limu↑11−2u+C(u,u)1−u.
These quantities depend only on the copula and measure the
strength of dependence in the joint lower and upper tails,
respectively. For example, the Gaussian copula (see next example)
with correlation
|ρ|<1
has
λL=λU=0,
so extreme events are asymptotically independent even when the
linear correlation is high (Joe, 1997).
Example 4.10.12 (Gaussian and Gumbel tail
behavior). Consider two dependence models with standard
normal marginals:
(a)
A bivariate normal model with correlation
ρ,
whose copula is the Gaussian copula
Cρ,
Cρ(u,v)=Φ2(Φ−1(u),Φ−1(v);ρ),
where
Φ2(⋅,⋅;ρ)
denotes the bivariate normal cdf with zero means, unit
variances, and correlation coefficient
ρ∈(−1,1)
as in Example 3.3.5 of Casella & Berger
(2002).
(b)
A Gumbel copula
Cθ
with parameter
θ≥1,
Cθ(u,v)=exp(−((−logu)θ+(−logv)θ)1/θ),
combined with standard normal marginals via Sklar’s
Theorem.
In both cases, Kendall’s
τ
and Spearman’s
ρ
can be chosen to be similar by an appropriate choice of
θ,
so the overall strength of dependence is comparable. However, the
Gaussian copula has
λL=λU=0,
so the probability of very large simultaneous exceedances decays
as if the tails were asymptotically independent. In contrast, the
Gumbel copula has
0]]>
λU=2−21/θ>0
(and
λL=0),
making simultaneous large positive extremes substantially more
likely.
Concretely, as for the Gaussian copula Kendall’s
τ
is
τGauss(ρ)=2πarcsin(ρ),
setting
ρ=0.8
yields
τGauss(0.8)=2π⋅0.9273≈1.85463.1416≈0.59.
For the Gumbel copula Kendall’s
τ
is
τGumbel(θ)=1−1θ.
So we set
τGumbel(θ)=τGauss(0.8)=1−1θ≈0.59.
Solving for
θ,
1θ≈1−0.59=0.41,θ≈10.41≈2.44,
and corresponding
λU=0.67.
This example underscores how copulas allow us to tune not only
the overall dependence but also the extremal (tail) dependence.
<named-content content-type="header-section-number">4.10.5</named-content>
Visualization of Copulas
Copulas may be visualized most naturally through their behavior
on the unit square. One approach is to plot the copula cdf
C(u,v)
as a surface over
(u,v)∈[0,1]2;
for example, the independence copula appears as the smooth surface
uv,
while strong positive dependence pulls the surface upward toward the
upper Fréchet bound
M(u,v)=min{u,v}.
For absolutely continuous copulas it is also useful to plot the
so-called copula density
c(u,v)=∂2∂u∂vC(u,v),
where regions of high density indicate values of
(U,V)
that occur more frequently. In applied work, empirical copulas are
often examined by scatterplots of the pseudo-observations
(Ui,Vi),
or by contour plots and heatmaps on the unit square, which display
the dependence structure independently of the marginal
distributions.
Scatterplots of pseudo-observations from the Clayton
copula (
θ=2.88),
Gaussian copula (
ρ=0.8),
and Gumbel copula (
θ=2.44).
![]()
In Figure 4.10.1 we
plot three such scatterplots generated by the Mathematica code given
in Example A.0.9 in the Appendix. These illustrate how different
copulas, even with the same margins and similar Kendall’s
τ,
can encode very different dependence structures, particularly in the
tails (Joe, 1997; Nelsen, 1999).
<named-content content-type="header-section-number">7.6</named-content>
Inference for Copula Models
Suppose that
(X1,Y1),…,(Xn,Yn)
is a random sample from a distribution with continuous marginals
FX,FY
and copula
Cθ
indexed by a parameter
θ.
The likelihood for
(X,Y)
decomposes into marginal and copula parts, paralleling the
decomposition of models into components that underlies the likelihood
principle in Chapter 6 of Casella & Berger (2002), see, for
example, Joe (1997).
Because the copula is invariant under strictly increasing
transformations of the margins, it is natural to base inference about
θ
on the ranks (or, equivalently, on empirical versions of
Ui=FX(Xi)
and
Vi=FY(Yi)).
Let
Ri
and
Si
be the ranks of
Xi
and
Yi
among
X1,…,Xn
and
Y1,…,Yn,
respectively. The empirical copula is defined on
[0,1]2
by
Cn(u,v)=1n∑i=1n𝟏(Rin+1≤u,Sin+1≤v).
This is a rank-based estimator of
C;
under mild conditions,
Cn(u,v)→C(u,v)
for each fixed
(u,v)
(Genest & Favre, 2007). The use of ranks parallels the rank-based
procedures for nonparametric inference introduced in Chapter 8 of
Casella & Berger (2002).
Although full maximum likelihood estimation is certainly possible,
it could be numerically unstable. Several other rank-based estimation
strategies are used in practice:
Method of moments via Kendall’s tau or Spearman’s
rho. If
τ=τ(θ)
(or
ρS=ρS(θ))
is known in closed form, one can estimate
θ
by solving
τ(θ̂)=τ̂
(or
ρS(θ̂)=ρ̂S),
where
τ̂
or
ρ̂S
is computed from the sample ranks (Section 7.2 of Casella &
Berger (2002)), cf. (Genest & Favre, 2007).
Maximum pseudolikelihood. If the copula has
density
cθ(u,v),
one may maximize
ℓ(θ)=∑i=1nlogcθ(Rin+1,Sin+1)
with respect to
θ.
Under suitable conditions, the maximizer
θ̂
is consistent and asymptotically normal, and its large-sample
behavior can be analyzed using the methods of Chapter 10 of
Casella & Berger (2002), cf. (Joe, 1997).
Two-step methods. One may first estimate
marginal parameters by univariate methods (Chapter 7 of Casella
& Berger (2002)) and then estimate
θ
by maximizing a “copula likelihood” built from the transformed
data, but this approach is sensitive to misspecification of the
margins (Joe, 1997).
Rank-based methods have the advantage of depending only on the
copula and remaining valid under arbitrary strictly increasing
marginal transformations, in line with the invariance considerations
emphasized in Chapter 6 of Casella & Berger (2002), cf. (Genest
& Favre, 2007).
Example 7.6.1 (Estimating a Clayton copula
parameter). Suppose
(Xi,Yi),
i=1,…,n,
is a sample from a model with continuous marginals and Clayton
copula
0.]]>
Cθ(u,v)=(u−θ+v−θ−1)−1/θ,θ>0.
For this family, Kendall’s tau is a simple function of
θ:
τ(θ)=θθ+2.
Let
τ̂
be the sample Kendall’s tau computed from the ranks of the data. A
method-of-moments estimator of
θ
is obtained by solving
τ(θ̂)=τ̂,
which gives
θ̂=2τ̂1−τ̂.
This estimator depends only on the copula (through the
ranks) and not on the specific marginal distributions, in contrast
with likelihood-based estimators that require a full specification
of the marginals (Chapter 7 of Casella & Berger
(2002)).
Exercises for Section 4.10
4.66
(Copulas and monotone transformations.) Let
(X,Y)
be continuous with copula
C,
and let
Z=g(X),
T=h(Y)
with
g,h
strictly increasing.
(a)
Using the change-of-variables ideas from
Section 2.1, express the cdf of
(Z,T)
in terms of
FX,Y.
(b)
Show that
(Z,T)
has the same copula
C.
(c)
Let
(X,Y)
be the bivariate normal pair of Example 4.5.3 of
Casella & Berger (2002) with correlation
ρ.
Take
g(x)=ex
and
h(y)=e3y.
Simulate a moderate sample from this model (see
Section 5.5 of Casella & Berger (2002)) and
compare the scatter plots of
(X,Y)
and
(Z,T).
Explain why the copula, and thus Kendall’s tau, is the
same in both cases.
4.67
(PQD and covariance sign.) Suppose
(X,Y)
is continuous and positively quadrant dependent:
P(X≤x,Y≤y)≥P(X≤x)P(Y≤y)for all x,y.
(a)
Express this condition in terms of the copula
C.
(b)
Show that if
X
and
Y
have finite second moments, then
𝖢𝗈𝗏(X,Y)≥0
(hint: proceed as in the proof of Theorem 2.2.6 of
Casella & Berger (2002), but use the PQD
condition).
(c)
Consider the model of Example 4.5.2 of Casella
& Berger (2002), where
X
and
Y
are independent exponentials with parameter
λ
(hence
𝖢𝗈𝗏(X,Y)=0).
Modify this model by introducing a Clayton copula with
parameter
0]]>
θ>0
while keeping the same exponential marginals. Argue
that the resulting pair is PQD, and discuss how the
covariance changes.
4.68
(Gaussian copula and tail independence.) Let
(Z1,Z2)
be bivariate normal with correlation
ρ
(Section 3.3 of Casella & Berger (2002)), and define
U=Φ(Z1),
V=Φ(Z2).
(a)
Argue that the copula of
(U,V)
is the Gaussian copula
Cρ.
(b)
Using the tail dependence formulas in
Section 4.10.4,
show heuristically that for
|ρ|<1,
λL=λU=0.
(c)
Explain in words why this means that even strongly
correlated Gaussian models may understate the
probability of joint extreme events.
Exercises for Section 7.6
7.67
(Kendall’s
τ
and Archimedean copulas.) An Archimedean copula has the form
C(u,v)=φ−1(φ(u)+φ(v)),
where
φ
is a convex, decreasing generator.
(a)
For the Clayton copula with parameter
0]]>
θ>0,
verify that the generator is
φ(t)=t−θ−1.
(b)
Using the following formula for Kendall’s
τ
in terms of the generator
τ=1+4∫01φ(t)φ′(t)dt,
derive
τ(θ)
and base an estimator for
θ
on it.
7.68
(A complete copula analysis of rainfall and
runoff)
The following display contains
n=30
invented paired observations of rainfall
X,
measured in millimetres, and same-day runoff volume
Y,
measured in
103m3,
from substantial rainfall events in a small catchment. For the
purposes of this exercise, regard the pairs as an i.i.d.
sample from a continuous bivariate distribution. There are no
ties.
Invented rainfall–runoff data for Exercise
7.68.
|
i |
xi |
yi |
i |
xi |
yi |
i |
xi |
yi |
|
|
|
| 1 |
34.7 |
64.1 |
11 |
114.2 |
105.0 |
21 |
35.6 |
58.1 |
|
|
|
| 2 |
26.8 |
19.1 |
12 |
15.5 |
17.5 |
22 |
50.8 |
20.3 |
|
|
|
| 3 |
42.1 |
71.1 |
13 |
14.5 |
21.5 |
23 |
97.4 |
87.7 |
|
|
|
| 4 |
12.9 |
27.3 |
14 |
33.3 |
42.0 |
24 |
57.2 |
117.8 |
|
|
|
| 5 |
20.1 |
24.8 |
15 |
21.1 |
13.1 |
25 |
34.0 |
17.2 |
|
|
|
| 6 |
73.5 |
88.3 |
16 |
38.1 |
45.6 |
26 |
16.9 |
33.9 |
|
|
|
| 7 |
111.9 |
119.3 |
17 |
20.3 |
41.3 |
27 |
13.1 |
11.5 |
|
|
|
| 8 |
15.7 |
26.9 |
18 |
19.1 |
22.8 |
28 |
27.6 |
17.4 |
|
|
|
| 9 |
14.0 |
10.6 |
19 |
45.6 |
39.2 |
29 |
40.3 |
110.0 |
|
|
|
| 10 |
37.9 |
58.3 |
20 |
23.8 |
22.9 |
30 |
33.6 |
42.5 |
|
|
|
(a)
Draw a scatterplot of
(Xi,Yi).
Describe the marginal distributions and the apparent
association. Explain why fitting a bivariate normal
distribution on the original measurement scale may be
questionable.
(b)
Compute Pearson’s sample correlation
r,
Kendall’s sample tau
τ̂,
and Spearman’s sample rho
ρ̂S.
Explain why
τ̂
and
ρ̂S
are particularly natural summaries in a copula
analysis.
(c)
Let
Ri
and
Si
denote the ranks of
Xi
and
Yi,
respectively. Construct the pseudo-observations
Ui=Rin+1,Vi=Sin+1,i=1,…,n.
Plot the points
(Ui,Vi)
in the unit square. Explain why this plot, and hence
the empirical copula, is unchanged if
X
and
Y
are replaced by
logX
and
logY,
respectively.
(d)
Define the empirical copula by
Cn(u,v)=1n∑i=1n𝟏{Ui≤u,Vi≤v},0≤u,v≤1.
Compute
Cn(0.5,0.5),
Cn(0.8,0.8),
and
Cn(0.9,0.9).
Compare these values with those of the independence
copula
Π(u,v)=uv.
(e)
Fit a Gaussian copula and a Gumbel copula by
inversion of Kendall’s tau. Use
τGauss(ρ)=2πarcsin(ρ),−1<ρ<1,
and
τGumbel(θ)=1−1θ,θ≥1.
Obtain the estimates
ρ̂τ
and
θ̂τ.
(f)
Estimate the same two copulas by maximum
pseudolikelihood. For a copula density
cη,
maximize
ℓn(η)=∑i=1nlogcη(Rin+1,Sin+1).
For the Gaussian copula, you may use
cρ(u,v)=ϕ2(Φ−1(u),Φ−1(v);ρ)ϕ(Φ−1(u))ϕ(Φ−1(v)),
where
ϕ
is the standard normal density and
ϕ2(⋅,⋅;ρ)
is the bivariate standard normal density with
correlation
ρ.
For the Gumbel copula
Cθ(u,v)=exp{−[(−logu)θ+(−logv)θ]1/θ},θ≥1,
you may evaluate the mixed derivative
cθ(u,v)=∂2Cθ(u,v)/∂u∂v
numerically, or use software that implements this
density like Mathematica. Report the
maximum-pseudolikelihood estimates
ρ̂MPL
and
θ̂MPL,
and compare them with the estimates from part (e).
(g)
State the lower- and upper-tail dependence
coefficients for each fitted copula. In particular,
show that the Gaussian copula has
λL=λU=0
whenever
|ρ|<1,
whereas the Gumbel copula has
λL=0,λU=2−21/θ.
Explain why copulas with similar fitted values of
Kendall’s tau can nevertheless make substantially
different assertions about simultaneous extreme
rainfall and runoff events.
(h)
Using the pseudo-observation plot together with the
maximized pseudo-log-likelihoods and the
tail-dependence coefficients obtained above, discuss
which model is more appropriate for describing the
data. Distinguish carefully between overall positive
association and upper-tail dependence.
(i)
Let
qX(0.90)
and
qY(0.90)
be the empirical
0.90
quantiles of rainfall and runoff. For each fitted
copula, estimate
q_X(0.90),\,Y>q_Y(0.90)\}
=1-0.90-0.90+C(0.90,0.90).]]>
Pr{X>qX(0.90),Y>qY(0.90)}=1−0.90−0.90+C(0.90,0.90).
Compare the resulting estimates with the independence
benchmark
0.102=0.01.
Give a practical interpretation in terms of the
occurrence of a high-rainfall event together with high
runoff.
Further Reading
For a systematic treatment of copulas, their properties, and many
families and examples, see Nelsen (1999). A more advanced monograph
with emphasis on multivariate dependence and extreme-value
structures is Joe (1997). For an accessible introduction to
inference for copula models based on ranks, with a detailed worked
example and another hydrological application, see Genest & Favre
(2007).
Acknowledgement
The author thanks Perplexity AI for assistance with
English-language editing and with the preparation and refinement of
selected Mathematica code snippets. This assistance was limited to
matters of presentation, wording, and coding style. Responsibility
for the scientific content, mathematical correctness, numerical
calculations, and final form of the manuscript rests entirely with
the author.
Solutions
Solution to 4.66.
Let
(X,Y)
be a continuous bivariate random vector with joint cdf
FX,Y
and copula
C,
and let
g,h:ℝ→ℝ
be strictly increasing functions. Define
Z=g(X),T=h(Y).
(a) Cdf of
(Z,T)
in terms of
FX,Y.
For any
z,t∈ℝ,
FZ,T(z,t)=ℙ(Z≤z,T≤t)=ℙ(g(X)≤z,h(Y)≤t).
Since
g
and
h
are strictly increasing, the inequalities
g(X)≤z
and
h(Y)≤t
are equivalent to
X≤g−1(z),Y≤h−1(t),
where
g−1
and
h−1
denote the (strictly increasing) inverses of
g
and
h.
Hence
FZ,T(z,t)=ℙ(X≤g−1(z),Y≤h−1(t))=FX,Y(g−1(z),h−1(t)).
(b) Copula of
(Z,T).
Let
FX
and
FY
be the marginal cdfs of
X
and
Y,
and let
FZ
and
FT
be the marginal cdfs of
Z
and
T.
From part (a),
FZ(z)=ℙ(Z≤z)=ℙ(g(X)≤z)=ℙ(X≤g−1(z))=FX(g−1(z)),FT(t)=ℙ(T≤t)=ℙ(h(Y)≤t)=ℙ(Y≤h−1(t))=FY(h−1(t)).
Let
u,v∈(0,1)
and define
z=FZ−1(u),t=FT−1(v).
Using
FZ(z)=FX(g−1(z))
and the strict monotonicity of
g,
FX(g−1(z))=u⇒g−1(z)=FX−1(u),
so that
z=g(FX−1(u)).
Similarly,
t=h(FY−1(v)).
The copula
CZ,T
of
(Z,T)
is defined by
CZ,T(u,v)=FZ,T(FZ−1(u),FT−1(v)).
Using part (a) and the relations above,
CZ,T(u,v)=FZ,T(z,t)=FX,Y(g−1(z),h−1(t))=FX,Y(FX−1(u),FY−1(v)).
By definition of the copula
C
of
(X,Y),
C(u,v)=FX,Y(FX−1(u),FY−1(v)),
hence
CZ,T(u,v)=C(u,v),(u,v)∈[0,1]2.
Therefore
(Z,T)
has the same copula
C
as
(X,Y).
(c) Bivariate normal example and invariance of the
copula.
Let
(X,Y)
be bivariate normal with correlation
ρ,
as in Example 4.5.3 of Casella & Berger (2002), and define
Z=g(X)=eX,T=h(Y)=e3Y.
To simulate a sample of size
n
from this model, one can:
Generate
n
i.i.d. observations
(Xi,Yi)
from the bivariate normal distribution with correlation
ρ.
For each
i,
set
Zi=eXi,Ti=e3Yi.
The scatter plot of
(Zi,Ti)
is obtained from that of
(Xi,Yi)
by applying strictly increasing transformations in each coordinate.
These change the marginal distributions (normal to lognormal and
scaled lognormal) and the scale of the axes, but they preserve the
ordering of the points in each coordinate and hence the dependence
structure encoded by the copula.
Part (b) shows rigorously that strictly increasing transformations
in each margin leave the copula unchanged. Kendall’s tau is a measure
of concordance that depends only on the copula (it is invariant under
strictly increasing transformations of each coordinate). Therefore
(X,Y)
and
(Z,T)
have the same copula and the same value of Kendall’s tau; empirically,
the estimated tau from the two scatter plots will (up to sampling
error) be equal.
Solution to 4.67.
Assume
(X,Y)
is continuous with joint cdf
FX,Y,
marginals
FX,FY,
and copula
C.
(a) PQD in terms of the copula.
Positive quadrant dependence (PQD) means
P(X≤x,Y≤y)≥P(X≤x)P(Y≤y)for all x,y∈ℝ.
In terms of the cdfs this is
FX,Y(x,y)≥FX(x)FY(y)∀x,y.
By Sklar’s theorem,
FX,Y(x,y)=C(FX(x),FY(y)).
Let
u=FX(x)
and
v=FY(y);
then
u,v∈[0,1]
and the PQD condition becomes
C(u,v)≥uvfor all (u,v)∈[0,1]2.
Thus
(X,Y)
is PQD if and only if its copula
C
satisfies
C(u,v)≥uv
on
[0,1]2.
(b) PQD implies nonnegative covariance.
Assume
X
and
Y
have finite second moments. We show
𝖢𝗈𝗏(X,Y)≥0.
Recall that for square–integrable
(X,Y)
one can write
x,Y>y)-P(X>x)\,P(Y>y)\bigr\}\,dx\,dy,]]>
𝖢𝗈𝗏(X,Y)=𝔼(XY)−𝔼X𝔼Y=∬ℝ2{P(X>x,Y>y)−P(X>x)P(Y>y)}dxdy,
which is obtained by expanding
XY
into indicator integrals and interchanging expectation and integration
(as in the proof of Theorem 2.2.6).
Note that
x,Y>y)
= 1-F_X(x)-F_Y(y)+F_{X,Y}(x,y),]]>
P(X>x,Y>y)=1−FX(x)−FY(y)+FX,Y(x,y),
and
x)\,P(Y>y)
= (1-F_X(x))(1-F_Y(y))
= 1-F_X(x)-F_Y(y)+F_X(x)F_Y(y).]]>
P(X>x)P(Y>y)=(1−FX(x))(1−FY(y))=1−FX(x)−FY(y)+FX(x)FY(y).
Hence
x,Y>y)-P(X>x)\,P(Y>y)
= F_{X,Y}(x,y)-F_X(x)F_Y(y).]]>
P(X>x,Y>y)−P(X>x)P(Y>y)=FX,Y(x,y)−FX(x)FY(y).
Therefore
𝖢𝗈𝗏(X,Y)=∬ℝ2(FX,Y(x,y)−FX(x)FY(y))dxdy.
If
(X,Y)
is PQD, then
FX,Y(x,y)≥FX(x)FY(y)
for all
x,y,
so the integrand is everywhere nonnegative. The integral of a
nonnegative function is nonnegative, hence
𝖢𝗈𝗏(X,Y)≥0.
(c) Clayton copula with exponential marginals.
In Example 4.5.2,
X
and
Y
are independent exponentials with parameter
λ:
FX(x)=FY(x)=1−e−λx,x≥0,
and
FX,Y(x,y)=FX(x)FY(y)⇒𝖢𝗈𝗏(X,Y)=0.
Now keep these marginals but introduce dependence via a Clayton
copula with parameter
0]]>
θ>0:
0.]]>
Cθ(u,v)=(u−θ+v−θ−1)−1/θ,u,v∈(0,1],θ>0.
The joint cdf of the new pair
(Xθ,Yθ)
is
FXθ,Yθ(x,y)=Cθ(FX(x),FY(y)),x,y≥0.
PQD: For Clayton copulas with
0]]>
θ>0
one has
Cθ(u,v)≥uv,(u,v)∈[0,1]2.
Thus the copula
Cθ
lies above the independence copula
uv,
so by part (a) the resulting pair
(Xθ,Yθ)
is positively quadrant dependent.
Covariance: By part (b), PQD and finite second
moments imply
𝖢𝗈𝗏(Xθ,Yθ)≥0.
When
θ=0
(interpreted as a limit), the copula reduces to the product copula
uv,
i.e. independence, so
Cov(X0,Y0)=0.
For
0]]>
θ>0,
the resulting pair is positively quadrant dependent and hence has
nonnegative covariance. Increasing
θ
corresponds to stronger positive dependence within the Clayton family.
Thus introducing a Clayton copula with parameter
0]]>
θ>0,
while keeping the exponential marginals, turns the originally
independent pair into a positively quadrant dependent pair with
nonnegative covariance.
Solution to 4.68.
Let
(Z1,Z2)
be bivariate normal with correlation
ρ,
and let
U=Φ(Z1),V=Φ(Z2),
where
Φ
is the standard normal cdf.
(a) Copula of
(U,V).
By construction,
U=FZ1(Z1),V=FZ2(Z2),
where
FZi=Φ
are the marginal cdfs of
Zi.
Thus
U
and
V
are standard uniform random variables (probability integral
transform). Their joint cdf is
P(U≤u,V≤v)=P(Φ(Z1)≤u,Φ(Z2)≤v)=P(Z1≤Φ−1(u),Z2≤Φ−1(v))=F(Z1,Z2)(Φ−1(u),Φ−1(v)),
where
F(Z1,Z2)
is the bivariate normal cdf with correlation
ρ.
By definition, the copula
Cρ
associated with this bivariate normal distribution is
Cρ(u,v):=F(Z1,Z2)(Φ−1(u),Φ−1(v)).
Comparing the two displays, we see that
(U,V)
has copula
Cρ
(the Gaussian copula with parameter
ρ).
(b) Tail dependence for
|ρ|<1.
Recall the upper and lower tail dependence coefficients for a
copula
C:
λU=limu↑11−2u+C(u,u)1−u,λL=limu↓0C(u,u)u.
For the Gaussian copula
Cρ,
we have
Cρ(u,u)=P(U≤u,V≤u)=P(Z1≤z,Z2≤z),where z=Φ−1(u).
Equivalently,
z, Z_2>z)
= 1 + C_\rho(u,u) - 2u.]]>
P(Z1>z,Z2>z)=1+Cρ(u,u)−2u.
Upper tail. Write
u, V>u)
= P(Z_1>z, Z_2>z),
\quad u\to 1,\;z\to\infty.]]>
P(U>u,V>u)=P(Z1>z,Z2>z),u→1,z→∞.
For a bivariate normal with
|ρ|<1,
it is known (and can be derived via asymptotic analysis of the joint
normal tail) that
z, Z_2>z)
\sim k(\rho)\,\frac{1}{z}\exp\!\left(-\frac{z^2}{1+\rho}\right),
\qquad z\to\infty,]]>
P(Z1>z,Z2>z)∼k(ρ)1zexp(−z21+ρ),z→∞,
for some positive constant
k(ρ)
depending on
ρ.
On the other hand,
z)
= 1-u
\sim \frac{1}{z}\exp\!\left(-\frac{z^2}{2}\right),
\qquad z\to\infty.]]>
P(Z1>z)=1−u∼1zexp(−z22),z→∞.
Therefore
z, Z_2>z)}{P(Z_1>z)}
\sim k(\rho)\exp\!\left(-z^2\left[\frac{1}{1+\rho} - \frac{1}{2}\right]\right).]]>
P(Z1>z,Z2>z)P(Z1>z)∼k(ρ)exp(−z2[11+ρ−12]).
If
|ρ|<1,
then
\frac{1}{2}]]>
11+ρ>12,
so the exponent is negative and the ratio decays to
0
as
z→∞.
Translating back to the copula formulation, this means
u\mid V>u)
= 0,\qquad |\rho|<1.]]>
λU=limu↑1P(U>u∣V>u)=0,|ρ|<1.
Lower tail. By symmetry of the bivariate normal
about
(0,0),
(Z1,Z2)
and
(−Z1,−Z2)
have the same joint distribution (with the same correlation
ρ).
Thus lower–tail events
(Z1≤−z,Z2≤−z)
correspond to upper–tail events for
(−Z1,−Z2);
the same asymptotic analysis applies and yields
λL=0,|ρ|<1.
Heuristically, for any fixed
ρ
strictly less than
1
in magnitude, the joint tail of the bivariate normal decays faster
than the marginal tail, so the conditional probability of a joint
extreme given a marginal extreme goes to zero in both tails.
(c) Interpretation for joint extremes.
The fact that
λL=λU=0
for
|ρ|<1
means that, in a Gaussian copula model, the probability of observing
one variable in an extreme tail given that the other is in that tail
tends to
0
as we move further into the tail. In other words, even if the linear
correlation
ρ
is large (say
ρ=0.9),
the Gaussian dependence structure does not create asymptotic
clustering of extremes: very large values of
X
and
Y
do not tend to occur together with positive limiting
probability.
In practical terms, this implies that strongly correlated Gaussian
models can substantially understate the probability of joint extreme
events. They may fit well in the center of the distribution (capturing
linear correlation), but they impose tail independence: extreme events
in one margin are, asymptotically, almost never accompanied by
simultaneous extremes in the other margin. For applications where
joint tail behavior is critical (e.g. financial or environmental
risk), the Gaussian copula may therefore give overly optimistic (too
small) estimates of joint extreme risks.
Solution to 7.67
An Archimedean copula has the form
C(u,v)=φ−1(φ(u)+φ(v)),
where
φ
is a convex, decreasing generator.
(a)
Clayton generator.
The (bivariate) Clayton copula with parameter
0]]>
θ>0
is
Cθ(u,v)=(u−θ+v−θ−1)−1/θ,u,v∈(0,1].
We claim that this can be written in Archimedean form with
generator
φ(t)=t−θ−1,t∈(0,1].
First compute its inverse:
s=φ(t)=t−θ−1⇒t−θ=s+1⇒t=(1+s)−1/θ.
Hence
φ−1(s)=(1+s)−1/θ,s≥0.
Now form the Archimedean copula associated with this
generator:
C(u,v)=φ−1(φ(u)+φ(v))=(1+(u−θ−1)+(v−θ−1))−1/θ=(u−θ+v−θ−1)−1/θ.
This is exactly the (alternative) Archimedean representation
of the Clayton copula, so
φ(t)=t−θ−1
is indeed a valid generator for the Clayton family.
(b)
Kendall’s
τ
and an estimator for
θ.
For an Archimedean copula with generator
φ,
Kendall’s
τ
is given by
τ=1+4∫01φ(t)φ′(t)dt.
For
φ(t)=t−θ−1
we have
φ′(t)=−θt−θ−1.
Therefore
φ(t)φ′(t)=t−θ−1−θt−θ−1=−1θ(t−tθ+1).
Plugging into the formula for
τ,
τ(θ)=1+4∫01φ(t)φ′(t)dt=1+4∫01(−1θ(t−tθ+1))dt=1−4θ∫01(t−tθ+1)dt.
Compute the integral:
∫01tdt=12,∫01tθ+1dt=1θ+2.
Hence
∫01(t−tθ+1)dt=12−1θ+2=θ2(θ+2).
Substitute back:
τ(θ)=1−4θ⋅θ2(θ+2)=1−2θ+2=θθ+2.
Thus Kendall’s
τ
for this Clayton copula (with generator
φ(t)=t−θ−1)
is
τ(θ)=θθ+2.
To base an estimator for
θ
on Kendall’s
τ,
let
τ̂
denote the sample (empirical) Kendall’s
τ.
We invert the above relationship:
τ̂=θθ+2⇒τ̂(θ+2)=θ⇒θ(1−τ̂)=2τ̂⇒θ̂=2τ̂1−τ̂.
This
θ̂
is the method-of-moments–type estimator of the Clayton parameter
based on Kendall’s
τ.
Solution to 7.68
Write
n=30.
The data are listed in random observation order; that order has no
statistical significance. There are no ties. In the copula
calculations below, we use the rank pseudo-observations
Ui=Rin+1=Ri31,Vi=Sin+1=Si31,i=1,…,30.
(a)
The scatterplot shows a clear positive association, but it is
far from a deterministic functional relationship. There is
substantial variation in runoff at comparable rainfall amounts;
for example, the observations with rainfall near
34–
40
mm have runoff values ranging from about
17
to
110
in the stated units. This is plausible because runoff also
depends on antecedent wetness, rainfall intensity and duration,
catchment storage, routing, and measurement variation.
Both variables are positive and right-skewed. Consequently, a
bivariate normal model on the original scale is unattractive:
normal margins are symmetric and place positive probability on
negative rainfall and runoff. Logarithmic transformations may
improve marginal plots, but they do not change the copula,
because they are strictly increasing on the positive
half-line.
(b)
The numerical summaries are
r=0.7948,τ̂=0.5540,ρ̂S=0.7339.
Thus the sample displays moderately strong positive association.
Pearson’s
r
measures linear association on the original measurement scales
and can change under nonlinear increasing transformations. In
contrast, Kendall’s tau and Spearman’s rho depend only on the
joint ranks and hence are invariant under strictly increasing
transformations of either margin. This makes them especially
natural for a copula analysis.
(c)
The rank pseudo-observations are obtained from the following
ranks:
i12345678910Ri1813231927296320Si237241412263013122
i11121314151617181920Ri30541511211082412Si27691832017101611
i21222324252627282930Ri192528261772142216Si2182529415252819
The plot of
(Ui,Vi)
in the unit square shows positive rank association with
appreciable scatter. If rainfall and runoff are replaced by
logX
and
logY,
respectively, their ranks do not change because the logarithm is
strictly increasing. Therefore neither the pseudo-observations
nor the empirical copula changes.
(d)
Direct counting yields
Cn(0.5,0.5)=1330=0.4333,Cn(0.8,0.8)=2330=0.7667,Cn(0.9,0.9)=2530=0.8333.
Under independence, the corresponding values are
Π(0.5,0.5)=0.25,Π(0.8,0.8)=0.64,Π(0.9,0.9)=0.81.
The empirical copula exceeds the independence copula at all
three selected points, which is consistent with positive
quadrant dependence in the sample. The comparison at
(0.9,0.9)
must be interpreted cautiously: with only 30 observations, very
few data points concern the most extreme part of the unit
square.
(e)
Inversion of the stated Kendall-tau relationships gives
ρ̂τ=sin(πτ̂2)=0.7645,
and
θ̂τ=11−τ̂=2.2423.
Thus both copulas are fitted to the same global rank-dependence
summary.
(f)
Maximizing the rank-based pseudo-log-likelihood gives,
numerically,
ρ̂MPL=0.7707,θ̂MPL=2.2446.
At their respective maxima, the pseudo-log-likelihood values are
approximately
ℓGauss=11.4359,ℓGumbel=12.1508.
The two estimation methods give broadly similar dependence
levels. For this sample, the fitted Gumbel copula has the larger
maximized pseudo-log-likelihood. This comparison is descriptive;
with
n=30,
it should not be interpreted as decisive evidence for an
upper-tail-dependent model.
(g)
For every Gaussian copula with
|ρ|<1,
λL=λU=0.
For a Gumbel copula,
λL=0,λU=2−21/θ.
Using the maximum-pseudolikelihood estimate yields
λ̂U=2−21/θ̂MPL=2−21/2.2446≈0.6382.
Using the inversion-of-tau estimate instead gives
2−21/2.2423≈0.6378.
Hence the Gaussian and Gumbel copulas can describe broadly
comparable ordinary rank association while making different
assertions about extreme events. The Gaussian copula is
asymptotically tail independent: as the threshold tends to one,
joint exceedances become negligible relative to a marginal
exceedance. The Gumbel copula retains positive upper-tail
dependence.
(h)
The pseudo-observations show a broadly increasing cloud
rather than a concentration exactly on the diagonal. Both fitted
families capture this central positive association reasonably
well. The Gumbel copula is substantively attractive when joint
high rainfall and high runoff are the principal scientific
concern because it permits upper-tail dependence. Moreover, for
this sample its maximized pseudo-log-likelihood,
ℓGumbel=12.1508,
is slightly larger than that of the Gaussian copula,
ℓGauss=11.4359.
Thus the Gumbel copula is the preferred model among these two
fitted families on this descriptive criterion. Nevertheless,
with only
n=30
observations, the data provide limited information in the
extreme upper-right corner of the unit square. The preference
for Gumbel should therefore be interpreted cautiously and not as
conclusive evidence of asymptotic upper-tail dependence.
(i)
For
q=0.90,
the fitted-copula estimates are
p̂Gauss=1−2q+Cρ̂MPL(q,q)≈0.0532,
and
p̂Gumbel=1−2q+Cθ̂MPL(q,q)≈0.0663.
Both are substantially greater than the independence benchmark
0.102=0.01.
The Gumbel copula gives the larger estimated probability of
coincident high rainfall and high runoff. Its distinction from
the Gaussian copula becomes still more consequential when
extrapolating to more extreme thresholds, because it has
positive upper-tail dependence whereas the Gaussian copula is
asymptotically tail independent.
Mathematica code snippets. The code below
reproduces the calculations. It is written using elementary built-in
functions. Numerical optimization is restricted to the interior of the
parameter spaces because the copula densities are singular or
problematic at the boundaries.
(* --------------------------------------------------------------- *)
(* 1. Data, scatterplots, ranks, and pseudo-observations *)
(* --------------------------------------------------------------- *)
data = {
{34.7, 64.1}, {26.8, 19.1}, {42.1, 71.1}, {12.9, 27.3},
{20.1, 24.8}, {73.5, 88.3}, {111.9, 119.3}, {15.7, 26.9},
{14.0, 10.6}, {37.9, 58.3}, {114.2, 105.0}, {15.5, 17.5},
{14.5, 21.5}, {33.3, 42.0}, {21.1, 13.1}, {38.1, 45.6},
{20.3, 41.3}, {19.1, 22.8}, {45.6, 39.2}, {23.8, 22.9},
{35.6, 58.1}, {50.8, 20.3}, {97.4, 87.7}, {57.2, 117.8},
{34.0, 17.2}, {16.9, 33.9}, {13.1, 11.5}, {27.6, 17.4},
{40.3, 110.0}, {33.6, 42.5}
};
{x, y} = Transpose[data];
n = Length[data];
ListPlot[data, Frame -> True, PlotRange -> All,
FrameLabel -> {"rainfall (mm)", "runoff (10^3 m^3)"},
PlotStyle -> Directive[Blue, PointSize[0.016]]]
(* Because there are no ties, Ordering[Ordering[...]] gives ranks. *)
rX = Ordering[Ordering[x]];
rY = Ordering[Ordering[y]];
u = N[rX/(n + 1)];
v = N[rY/(n + 1)];
uv = Transpose[{u, v}];
ListPlot[uv, Frame -> True, AspectRatio -> 1,
FrameLabel -> {"u", "v"}, PlotRange -> {{0, 1}, {0, 1}},
PlotStyle -> Directive[Red, PointSize[0.016]]]
(* --------------------------------------------------------------- *)
(* 2. Pearson correlation, Spearman rho, and Kendall tau *)
(* --------------------------------------------------------------- *)
rPearson = Correlation[x, y];
(* Spearman rho, valid because there are no ties. *)
rhoSHat = 1 - 6 Total[(rX - rY)^2]/(n (n^2 - 1));
(* Kendall tau: concordant minus discordant pairs, divided by n choose 2. *)
pairs = Subsets[Range[n], {2}];
tauHat = Total[Sign[(x[[#[[1]]]] - x[[#[[2]]]])*
(y[[#[[1]]]] - y[[#[[2]]]])] & /@ pairs]/Binomial[n, 2];
N[{rPearson, tauHat, rhoSHat}]
(* {0.794774, 0.554023, 0.733927} *)
(* --------------------------------------------------------------- *)
(* 3. Empirical copula and inversion of Kendall's tau *)
(* --------------------------------------------------------------- *)
empCopula[a_?NumericQ, b_?NumericQ] :=
Total[
Boole[#1 <= a && #2 <= b] & @@@ Transpose[{u, v}]
]/n;
N[empCopula @@@ {{.5, .5}, {.8, .8}, {.9, .9}}]
(* {0.433333, 0.766667, 0.833333} *)
rhoTau = Sin[Pi tauHat/2];
thetaTau = 1/(1 - tauHat);
N[{rhoTau, thetaTau}]
(* {0.764495, 2.24227} *)
(* --------------------------------------------------------------- *)
(* 4. Copula densities and maximum pseudolikelihood *)
(* --------------------------------------------------------------- *)
gaussianDensity[uu_?NumericQ, vv_?NumericQ, rho_?NumericQ] :=
Module[{a, b},
a = InverseCDF[NormalDistribution[], uu];
b = InverseCDF[NormalDistribution[], vv];
Exp[(2 rho a b - rho^2 (a^2 + b^2))/(2 (1 - rho^2))]/
Sqrt[1 - rho^2]
];
gumbelDensity[uu_?NumericQ, vv_?NumericQ, theta_?NumericQ] :=
Module[{a, b, s, t},
a = -Log[uu];
b = -Log[vv];
s = a^theta + b^theta;
t = s^(1/theta);
Exp[-t] (a b)^(theta - 1) s^(1/theta - 2) (t + theta - 1)/(uu vv)
];
logLikGaussian[rho_?NumericQ] :=
Total[Log[gaussianDensity[#[[1]], #[[2]], rho]] & /@ uv];
logLikGumbel[theta_?NumericQ] :=
Total[Log[gumbelDensity[#[[1]], #[[2]], theta]] & /@ uv];
fitGaussian = FindMaximum[
{logLikGaussian[rho], -.999 < rho < .999}, {rho, rhoTau}];
fitGumbel = FindMaximum[
{logLikGumbel[theta], 1.000001 < theta < 20}, {theta, thetaTau}];
N[{fitGaussian, fitGumbel}]
(* {{11.4359, {rho -> 0.770723}},
{12.1508, {theta -> 2.24462}}} *)
(* --------------------------------------------------------------- *)
(* 5. Tail dependence and fitted joint exceedance probability *)
(* --------------------------------------------------------------- *)
rhoMPL = rho /. Last[fitGaussian];
thetaMPL = theta /. Last[fitGumbel];
lambdaLGaussian = 0;
lambdaUGaussian = 0;
lambdaLGumbel = 0;
lambdaUGumbel = 2 - 2^(1/thetaMPL);
N[lambdaUGumbel]
(* 0.638205 *)
gaussianCopula[q_?NumericQ, rho_?NumericQ] := CDF[
MultinormalDistribution[{0, 0}, {{1, rho}, {rho, 1}}],
{InverseCDF[NormalDistribution[], q],
InverseCDF[NormalDistribution[], q]}];
gumbelCopula[q_?NumericQ, theta_?NumericQ] :=
Exp[-(2 (-Log[q])^theta)^(1/theta)];
q = .90;
pGauss = 1 - 2 q + gaussianCopula[q, rhoMPL];
pGumbel = 1 - 2 q + gumbelCopula[q, thetaMPL];
N[{pGauss, pGumbel}]
(* {0.053241, 0.0663387} *)
Note that the population data-generating mechanism used to create
this teaching sample was a Gumbel copula with
θ0=2.5,
for which
τ0=0.60
and
λU=2−21/2.5≈0.68,
combined with lognormal margins. The realized sample need not
reproduce the generating parameter exactly; in fact, its moderate size
is useful for illustrating sampling variation and the limits of
tail-dependence inference from a small dataset.