slides CIRM copulas, extremes and actuarial science

Arthur CHARPENTIER - Probit transformation for nonparametric kernel estimation of the copula
Probit transformation for nonparametric kernel estimation of the copula
A. Charpentier (Université de Rennes 1 & UQAM),
joint work with G. Geenens (UNSW) & D. Paindaveine (ULB)
CIRM Workshop “Extremes - Copulas - Actuarial science”, February 2016
http://arxiv.org/abs/1404.4414, to appear in Bernoulli
url: http://freakonometrics.hypotheses.org
email : charpentier.arthur@uqam.ca
: @freakonometrics
1

Motivation
Consider some n-i.i.d. sample {(Xi, Yi)} with cu-
mulative distribution function FXY and joint den-
sity fXY . Let FX and FY denote the marginal
distributions, and C the copula,
FXY (x, y) = C(FX(x), FY (y))
so that
fXY (x, y) = fX(x)fY (y)c(FX(x), FY (y))
We want a nonparametric estimate of c on [0, 1]2
. 1e+01 1e+03 1e+05
1e+011e+021e+031e+041e+05
2

Notations
Define uniformized n-i.i.d. sample {(Ui, Vi)}
Ui = FX(Xi) and Vi = FY (Yi)
or uniformized n-i.i.d. pseudo-sample {( Ûi, ˆVi)}
Ûi =
n
n + 1
ˆFXn(Xi) and ˆVi =
n
n + 1
ˆFY n(Yi)
where ˆFXn and ˆFY n denote empirical c.d.f.
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
3

Standard Kernel Estimate
The standard kernel estimator for c, say ˆc∗
, at (u, v) ∈ I would be (see Wand &
Jones (1995))
ˆc∗
(u, v) =
1
n|HUV |1/2
n
i=1
K H
−1/2
UV
u − Ui
v − Vi
, (1)
where K : R2
→ R is a kernel function and HUV is a bandwidth matrix.
4

Standard Kernel Estimate
However, this estimator is not consistent along
boundaries of [0, 1]2
E(ˆc∗
(u, v)) =
1
4
c(u, v) + O(h) at corners
E(ˆc∗
(u, v)) =
1
2
c(u, v) + O(h) on the borders
if K is symmetric and HUV symmetric.
Corrections have been proposed, e.g. mirror reﬂec-
tion Gijbels (1990) or the usage of boundary kernels
Chen (2007), but with mixed results.
Remark: the graph on the bottom is ˆc∗
on the
(ﬁrst) diagonal.
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01234567
5

Mirror Kernel Estimate
Use an enlarged sample: instead of only {( Ûi, ˆVi)},
add {(− Ûi, ˆVi)}, {( Ûi, − ˆVi)}, {(− Ûi, − ˆVi)},
{( Ûi, 2 − ˆVi)}, {(2 − Ûi, ˆVi)},{(− Ûi, 2 − ˆVi)},
{(2 − Ûi, − ˆVi)} and {(2 − Ûi, 2 − ˆVi)}.
See Gijbels & Mielniczuk (1990).
That estimator will be used as a benchmark in the
simulation study.
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6

Using Beta Kernels
Use a Kernel which is a product of beta kernels
Kxi
(u)) ∝ u
x1,i
b
1 [1 − u1]
x1,i
b · u
x2,i
b
2 [1 − u2]
x2,i
b
See Chen (1999).
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7

Probit Transformation
See Devroye & Gyöfi (1985) and Marron & Ruppert
(1994).
Define normalized n-i.i.d. sample {(Si, Ti)}
Si = Φ−1
(Ui) and Ti = Φ−1
(Vi)
or normalized n-i.i.d. pseudo-sample {( ˆSi, ˆTi)}
Ûi = Φ−1
( Ûi) and ˆVi = Φ−1
( ˆVi)
where Φ−1
is the quantile function of N(0, 1)
(probit transformation). −3 −2 −1 0 1 2 3
−3−2−10123
8

Probit Transformation
FST (x, y) = C(Φ(x), Φ(y))
so that
fST (x, y) = φ(x)φ(y)c(Φ(x), Φ(y))
Thus
c(u, v) =
fST (Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
.
So use
ˆc(τ)
(u, v) =
ˆfST (Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
−3 −2 −1 0 1 2 3
−3−2−10123
9

The naive estimator
Since we cannot use
ˆf∗
ST (s, t) =
1
n|HST |1/2
n
i=1
K H
−1/2
ST
s − Si
t − Ti
,
where K is a kernel function and HST is a band-
width matrix, use
ˆfST (s, t) =
1
n|HST |1/2
n
i=1
K H
−1/2
ST
s − ˆSi
t − ˆTi
.
and the copula density is
ˆc(τ)
(u, v) =
ˆfST (Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
0.0
0.2
0.4
0.6
0.8
1.0 0.0
0.2
0.4
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0.8
1.0
0
2
4
6
0.0 0.2 0.4 0.6 0.8 1.0
01234567
10

The naive estimator
ˆc(τ)
(u, v) =
1
n|HST |1/2φ(Φ−1(u))φ(Φ−1(v))
n
i=1
K H
−1/2
ST
Φ−1
(u) − Φ−1
( ˆUi)
Φ−1(v) − Φ−1( ˆVi)
as suggested in C., Fermanian & Scaillet (2007) and Lopez-Paz . et al. (2013).
Note that Omelka . et al. (2009) obtained theoretical properties on the
convergence of ˆC(τ)
(u, v) (not c).
11

Improved probit-transformation copula density estimators
When estimating a density from pseudo-sample, Loader (1996) and Hjort &
Jones (1996) deﬁne a local likelihood estimator
Around (s, t) ∈ R2
, use a polynomial approximation of order p for log fST
log fST (ˇs, ˇt) a1,0(s, t) + a1,1(s, t)(ˇs − s) + a1,2(s, t)(ˇt − t)
.
= Pa1
(ˇs − s, ˇt − t)
log fST (ˇs, ˇt) a2,0(s, t) + a2,1(s, t)(ˇs − s) + a2,2(s, t)(ˇt − t)
+ a2,3(s, t)(ˇs − s)2
+ a2,4(s, t)(ˇt − t)2
+ a2,5(s, t)(ˇs − s)(ˇt − t)
.
= Pa2 (ˇs − s, ˇt − t).
12

Improved probit-transformation copula density estimators
Remark Vectors a1(s, t) = (a1,0(s, t), a1,1(s, t), a1,2(s, t)) and
a2(s, t)
.
= (a2,0(s, t), . . . , a2,5(s, t)) are then estimated by solving a weighted
maximum likelihood problem.
˜ap(s, t) = arg max
ap
n
i=1
K H
−1/2
ST
s − ˆSi
t − ˆTi
Pap
( ˆSi − s, ˆTi − t)
−n
R2
K H
−1/2
ST
s − ˇs
t − ˇt
exp Pap (ˇs − s, ˇt − t) dˇs dˇt ,
The estimate of fST at (s, t) is then ˜f
(p)
ST (s, t) = exp(˜ap,0(s, t)), for p = 1, 2.
The Improved probit-transformation kernel copula density estimators are
˜c(τ,p)
(u, v) =
˜f
(p)
ST (Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
13

Improved probit-transformation
copula density estimators
For the local log-linear (p = 1) approximation
˜c(τ,1)
(u, v) =
exp(˜a1,0(Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
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0.2
0.4
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0.8
1.0 0.0
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0.4
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0.8
1.0
0
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4
6
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01234567
14

Improved probit-transformation
copula density estimators
For the local log-quadratic (p = 2) approximation
˜c(τ,2)
(u, v) =
exp(˜a2,0(Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
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0.2
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0.8
1.0 0.0
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15

Asymptotic properties
A1. The sample {(Xi, Yi)} is a n- i.i.d. sample from the joint distribution FXY ,
an absolutely continuous distribution with marginals FX and FY strictly
increasing on their support;
(uniqueness of the copula)
16

A2. The copula C of FXY is such that (∂C/∂u)(u, v) and (∂2
C/∂u2
)(u, v) exist
and are continuous on {(u, v) : u ∈ (0, 1), v ∈ [0, 1]}, and (∂C/∂v)(u, v) and
(∂2
C/∂v2
)(u, v) exist and are continuous on {(u, v) : u ∈ [0, 1], v ∈ (0, 1)}. In
addition, there are constants K1 and K2 such that



∂2
C
∂u2
(u, v) ≤
K1
u(1 − u)
for (u, v) ∈ (0, 1) × [0, 1];
∂2
C
∂v2
(u, v) ≤
K2
v(1 − v)
for (u, v) ∈ [0, 1] × (0, 1);
A3. The density c of C exists, is positive and admits continuous second-order
partial derivatives on the interior of the unit square I. In addition, there is a
constant K00 such that
c(u, v) ≤ K00 min
1
u(1 − u)
,
1
v(1 − v)
∀(u, v) ∈ (0, 1)2
.
see Segers (2012).
17

Assume that K(z1, z2) = φ(z1)φ(z2) and HST = h2
I with h ∼ n−a
for some
a ∈ (0, 1/4). Under Assumptions A1-A3, the ‘naive’ probit transformation kernel
copula density estimator at any (u, v) ∈ (0, 1)2
is such that
√
nh2 ˆc(τ)
(u, v) − c(u, v) − h2 b(u, v)
φ(Φ−1(u))φ(Φ−1(v))
L
−→ N 0, σ2
(u, v) ,
where b(u, v) =
1
2
∂2
c
∂u2
(u, v)φ2
(Φ−1
(u)) +
∂2
c
∂v2
(u, v)φ2
(Φ−1
(v))
− 3
∂c
∂u
(u, v)Φ−1
(u)φ(Φ−1
(u)) +
∂c
∂v
(u, v)Φ−1
(v)φ(Φ−1
(v))
+ c(u, v) {Φ−1
(u)}2
+ {Φ−1
(v)}2
− 2 (2)
and σ2
(u, v) =
c(u, v)
4πφ(Φ−1(u))φ(Φ−1(v))
.
18

The Amended version
The last unbounded term in b be easily adjusted.
ˆc(τam)
(u, v) =
ˆfST (Φ−1
(u), Φ−1
(v))
φ(Φ−1(u))φ(Φ−1(v))
×
1
1 + 1
2 h2 ({Φ−1(u)}2 + {Φ−1(v)}2 − 2)
.
The asymptotic bias becomes proportional to
b(am)
(u, v) =
1
2
∂2
c
∂u2
(u, v)φ2
(Φ−1
(u)) +
∂2
c
∂v2
(u, v)φ2
(Φ−1
(v))
−3
∂c
∂u
(u, v)Φ−1
(u)φ(Φ−1
(u)) +
∂c
∂v
(u, v)Φ−1
(v)φ(Φ−1
(v)) .
19

A local log-linear probit-transformation kernel estimator
˜c∗(τ,1)
(u, v) = ˜f
∗(1)
ST (Φ−1
(u), Φ−1
(v))/ φ(Φ−1
(u))φ(Φ−1
(v))
Then
√
nh2 ˜c∗(τ,1)
(u, v) − c(u, v) − h2 b(1)
(u, v)
φ(Φ−1(u))φ(Φ−1(v))
L
−→ N 0, σ(1) 2
(u, v) ,
where b(1)
(u, v) =
1
2
∂2
c
∂u2
(u, v)φ2
(Φ−1
(u)) +
∂2
c
∂v2
(u, v)φ2
(Φ−1
(v))
−
1
c(u, v)
∂c
∂u
(u, v)
2
φ2
(Φ−1
(u)) +
∂c
∂v
(u, v)
2
φ2
(Φ−1
(v))
−
∂c
∂u
(u, v)Φ−1
(u)φ(Φ−1
(u)) +
∂c
∂v
(u, v)Φ−1
(v)φ(Φ−1
(v)) − 2c(u, v)
20

Using a higher order polynomial approximation
Locally ﬁtting a polynomial of a higher degree is known to reduce the asymptotic
bias of the estimator, here from order O(h2
) to order O(h4
), see Loader (1996) or
Hjort (1996), under suﬃcient smoothness conditions.
If fST admits continuous fourth-order partial derivatives and is positive at (s, t),
then
√
nh2 ˜f
∗(2)
ST (s, t) − fST (s, t) − h4
b
(2)
ST (s, t)
L
−→ N 0, σ
(2)
ST
2
(s, t) ,
where σ
(2)
ST
2
(s, t) =
5
2
fST (s, t)
4π
and
b
(2)
ST (s, t) = −
1
8
fST (s, t)
×
∂4
g
∂s4
+
∂4
g
∂t4
+ 4
∂3
g
∂s3
∂g
∂s
+
∂3
g
∂t3
∂g
∂t
+
∂3
g
∂s2∂t
∂g
∂t
+
∂3
g
∂s∂t2
∂g
∂s
+ 2
∂4
g
∂s2∂t2
(s, t),
with g(s, t) = log fST (s, t).
21

Using a higher order polynomial approximation
A4. The copula density c(u, v) = (∂2
C/∂u∂v)(u, v) admits continuous
fourth-order partial derivatives on the interior of the unit square [0, 1]2
.
Then
√
nh2 ˜c∗(τ,2)
(u, v) − c(u, v) − h4 b(2)
(u, v)
φ(Φ−1(u))φ(Φ−1(v))
L
−→ N 0, σ(2) 2
(u, v)
where σ(2) 2
(u, v) =
5
2
c(u, v)
4πφ(Φ−1(u))φ(Φ−1(v))
22

Improving Bandwidth choice
Consider the principal components decomposition of the (n × 2) matrix
[ˆS, ˆT ] = M.
Let W1 = (W11, W12)T
and W2 = (W21, W22)T
be the eigenvectors of MT
M. Set
Q
R
=


W11 W12
W21 W22

 S
T
= W
S
T
which is only a linear reparametrization of R2
, so
an estimate of fST can be readily obtained from an
estimate of the density of (Q, R)
Since { ˆQi} and { ˆRi} are empirically uncorrelated,
consider a diagonal bandwidth matrix HQR =
diag(h2
Q, h2
R).
−4 −2 0 2 4
−3−2−1012
23

Improving Bandwidth choice
Use univariate procedures to select hQ and hR independently
Denote ˜f
(p)
Q and ˜f
(p)
R (p = 1, 2), the local log-polynomial estimators for the
densities
hQ can be selected via cross-validation (see Section 5.3.3 in Loader (1999))
hQ = arg min
h>0
∞
−∞
˜f
(p)
Q (q)
2
dq −
2
n
n
i=1
˜f
(p)
Q(−i)( ˆQi) ,
where ˜f
(p)
Q(−i) is the ‘leave-one-out’ version of ˜f
(p)
Q .
24

Graphical Comparison (loss ALAE dataset)
c~(τ2)
Loss (X)
ALAE(Y)
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
1.5
1.5
2
2
4
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0.00.20.40.60.81.0
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0.25
0.5
0.5
0.75
0.75
1
1
1
1.25
1.25
1.5
1.5
2
2
4
c^
β
Loss (X)
ALAE(Y)
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
1.5
1.5
2
2
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.25
0.25
0.25 0.25
0.5
0.5
0.75
0.75
0.75
1
1
1
1
1
1.25
1.25
1.25
1.25
1.25
1.5
1.5
2
2
2
4
c^
b
Loss (X)
ALAE(Y)
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
1.5
1.5
2
2
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
1.5
1.5
2
2
c^
p
Loss (X)
ALAE(Y)
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1.25
1.25
1.5
1.5
2
2
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
0.25
0.25
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1
1.25
1.25
1.25
1.25
1.5
1.5
1.5
2
2
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
c~(τ2)
Loss (X)
ALAE(Y)
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1
1.25
1.25
1.5
1.5
2
2
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
c^
β
Loss (X)
ALAE(Y)
0.25
0.25
0.25 0.25
0.5
0.5
0.75
0.75
0.75
1
1
1
1
1
1.25
1.25
1.25
1.25
1.25
1.5
1.5
1.5
2
2
2
4
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
c^
b
Loss (X)
ALAE(Y)
0.25
0.25
0.5
0.5
0.75
0.751
1
1.25
1.25
1.5
1.5
2
2
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
c^
p
Loss (X)
ALAE(Y)
0.25
0.25
0.25
0.25
0.5
0.5
0.75
0.75
1
1
1
1.25
1.25
1.25
1.25
1.5
1.5
1.5
2
2
4
25

Simulation Study
M = 1, 000 independent random samples {(Ui, Vi)}n
i=1 of sizes n = 200, n = 500
and n = 1000 were generated from each of the following copulas:
· the independence copula (i.e., Ui’s and Vi’s drawn independently);
· the Gaussian copula, with parameters ρ = 0.31, ρ = 0.59 and ρ = 0.81;
· the Student t-copula with 4 degrees of freedom, with parameters ρ = 0.31,
ρ = 0.59 and ρ = 0.81;
· the Frank copula, with parameter θ = 1.86, θ = 4.16 and θ = 7.93;
· the Gumbel copula, with parameter θ = 1.25, θ = 1.67 and θ = 2.5;
· the Clayton copula, with parameter θ = 0.5, θ = 1.67 and θ = 2.5.
(approximated) MISE relative to the MISE of the mirror-reflection estimator
(last column), n = 1000. Bold values show the minimum MISE for the
corresponding copula (non-significantly different values are highlighted as well).
26

n = 1000 ˆc(τ) ˆc(τam) ˜c(τ,1) ˜c(τ,2) ˆc
(β)
1 ˆc
(β)
2 ˆc
(B)
1 ˆc
(B)
2 ˆc
(p)
1 ˆc
(p)
2 ˆc
(p)
3
Indep 3.57 2.80 2.89 1.40 7.96 11.65 1.69 3.43 1.62 0.50 0.14
Gauss2 2.03 1.52 1.60 0.76 4.63 6.06 1.10 1.82 0.98 0.66 0.89
Gauss4 0.63 0.49 0.44 0.21 1.72 1.60 0.75 0.58 0.62 0.99 2.93
Gauss6 0.21 0.20 0.11 0.05 0.74 0.33 0.77 0.37 0.72 1.21 2.83
Std(4)2 0.61 0.56 0.50 0.40 1.57 1.80 0.78 0.67 0.75 1.01 1.88
Std(4)4 0.21 0.27 0.17 0.15 0.88 0.51 0.75 0.42 0.75 1.12 2.07
Std(4)6 0.09 0.17 0.08 0.09 0.70 0.19 0.82 0.47 0.90 1.17 1.90
Frank2 3.31 2.42 2.57 1.35 7.16 9.63 1.70 2.95 1.31 0.45 0.49
Frank4 2.35 1.45 1.51 0.99 4.42 4.89 1.49 1.65 0.60 0.72 6.14
Frank6 0.96 0.52 0.45 0.44 1.51 1.19 1.35 0.76 0.65 1.58 7.25
Gumbel2 0.65 0.62 0.56 0.43 1.77 1.97 0.82 0.75 0.83 1.03 1.52
Gumbel4 0.18 0.28 0.16 0.19 0.89 0.41 0.78 0.47 0.81 1.10 1.78
Gumbel6 0.09 0.21 0.10 0.15 0.78 0.29 0.85 0.58 0.94 1.12 1.63
Clayton2 0.63 0.60 0.51 0.34 1.78 1.99 0.78 0.70 0.79 1.04 1.79
Clayton4 0.11 0.26 0.10 0.15 0.79 0.27 0.83 0.56 0.90 1.10 1.50
Clayton6 0.11 0.28 0.08 0.15 0.82 0.35 0.88 0.67 0.96 1.09 1.36
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slides CIRM copulas, extremes and actuarial science

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