A proposal of a bivariate Conditional Tail Expectation

Transcription

Risk measures en dimension 1: VaR and CTE
Generalizations of Conditional Tail Expectation
Plug-in estimation of level sets
References
A proposal of a bivariate Conditional Tail
Expectation
a
Elena Di Bernardino
b
joint works with Areski Cousin , Thomas Laloë
c , Véronique
d
e
Maume-Deschamps and Clémentine Prieur
Aussois, 4eme Rencontres des Jeunes Statisticiens, 5-9/09/11
a, b, d Université Lyon 1, ISFA, Laboratoire SAF
c
Université de Nice Sophia-Antipolis, Laboratoire J-A Dieudonné
e
Université Joseph Fourier, Grenoble
References
Given an univariate continuous and strictly monotonic loss distribution
function
FX ,
X ) = QX (α) = FX−1 (α), ∀ α ∈ (0, 1).
VaRα (
Shortcoming of VaR measure:
Var is not a coherent risk measure (in the "Artzner sense", see
Artzner, 1999)
Var does not give any information about thickness of the upper tail
of the loss
To overcome problems of VaR
→
Conditional Tail Expectation (CTE):
CTEα (X ) = E[ X | X ≥ VaRα (X ) ] = E[ X | X ≥ QX (α) ],
(e.g. for properties of the
CTEα (X ) see Denuit et al., 2005).
References
Our Bivariate Conditional Tail Expectation
In the literature
A new bivariate Conditional Tail Expectation
Denition (Generalization of the CTE in dimension 2)
Consider a random vector
(X , Y )
associate distribution function
α ∈ (0, 1),
F.
satisfying regularity properties, with
Let
L(α) = {F (x , y ) ≥ α}.
For
we dene
CTEα (X, Y) =
E[ X | (X , Y ) ∈ L(α) ]
E[ Y | (X , Y ) ∈ L(α) ]
!
.
Distributional approach to risk model:
X , Y ) does not use an aggregate variable (sum, min, max . . .)
X,Y)
Our CTEα (
in order to analyze the bivariate risk's issue. Conversely CTEα (
deals with the simultaneous joint damages considering the bivariate
dependence structure of data in a specic risk's area (α-level set :
L(α)).
References
In the literature
Several bivariate generalizations of the classical univariate CTE have been
proposed; mainly using as conditioning events the total risk or some
univariate extreme risk in the portfolio. We recall for instance:
CTEα (X , Y )sum = E[ (X , Y ) | X + Y
> QX +Y (α) ],
CTEα (X , Y )min = E[ (X , Y ) | min{X , Y } > Qmin{X ,Y } (α) ],
CTEα (X , Y )max = E[ (X , Y ) | max{X , Y } > Qmax{X ,Y } (α) ].
Remark: Conditioning events are the total risk or some univariate
extreme risk in the portfolio (aggregate variable).
References
In the literature
Two dierent developments of research :
1
L(c ) = {F (x ) ≥ c }, with
c ∈ (0, 1), of an unknown distribution function F on R2+ with a
plug-in approach; in order to provide a consistent estimator for
CTEα (X , Y ) [Article Plug-in estimation of level sets in a non-compact
Problem of estimating the level sets
setting with applications in multivariate risk theory , with Laloë L.,
Maume-Deschamps V. and Prieur C., submitted].
2
Analysis of the CTEα (
X , Y ) as risk measure:
study of the classical
properties (monotonicity, translation invariance, positive
homogeneity,
. . .),
behavior with respect to dierent risk scenarios
and stochastic ordering of marginals risks [Article Some proposals
about bivariate risk measures , with Cousin A.].
References
Literature and background
Our framework
Tools and general aspects
Notation and preliminaries
Consistency result in terms of the Hausdor distance
L1 consistency
Estimation of our bivariate Conditional Tail Expectation
Perspectives
Estimation of the level sets of a density function: Polonik (1995),
Tsybakov (1997), Baíllo
et al.
(2001), Biau
et al.
(2007).
Estimation of the level sets of a regression function in a compact setting:
Cavalier (1997), Laloë (2009).
Estimation of general compact level sets: Cuevas
et al.
(2006).
An alternative approach, based on the geometric properties of the
compact support sets: Cuevas and Fraiman (1997), Cuevas and
Rodríguez-Casal (2004).
Our framework
L1 consistency
Perspectives
References
Goal: We consider the problem of estimating the level sets of a bivariate
distribution function
estimator of
F.
More precisely our goal is to build a consistent
L(c ) := {F (x ) ≥ c },
Idea: We consider a
plug-in
for
approach that is
Ln (c ) := {Fn (x ) ≥ c },
where
Fn
c ∈ (0, 1).
is a consistent estimator of
for
L(c ) is estimated by
c ∈ (0, 1),
F.
References
From a parametric formulation of
∂ L(c )
Our framework
L1 consistency
Perspectives
(see Belzunce et al. 2007).
Case: Survival Clayton Copula with marginals (Burr(1), Burr(2))
References
Our framework
L1 consistency
Perspectives
We state consistency results with respect to two proximity criteria
between sets: the Hausdor distance and the volume of the symmetric
dierence (physical proximity between sets).
→
Problem of compactness property for the level sets we estimate.
Figure: (left) Hausdor distance between sets X and Y ; (right)
where
λ
stands for the Lebesgue measure on
R2
and
4
λ(X 4 Y ),
for the symmetric
dierence.
References
Our framework
L1 consistency
Perspectives
∗
R2+ := R2+ \ (0, 0), F the
f : R2+ → [0, 1] and F ∈ F .
Let
{Xi }ni=1 in R2+ with distribution function F ,
Fn (.) = Fn (X1 , X2 , . . . , Xn , .) an estimator of F .
Given an
denote
set of continuous distribution functions
i .i .d
Dene, for
estimator :
sample
c ∈ (0, 1), the upper c -level set of F ∈ F
and its
we
plug-in
L(c ) := {x ∈ R2+ : F (x ) ≥ c }, Ln (c ) := {x ∈ R2+ : Fn (x ) ≥ c },
{F = c } = {x ∈ R2+ : F (x ) = c }.
In addition, given
T > 0, we set
L(c )T = {x ∈ [0, T ]2 : F (x ) ≥ c },
Ln (c )T = {x ∈ [0, T ]2 : Fn (x ) ≥ c },
{F = c }T = {x ∈ [0, T ]2 : F (x ) = c }.
For any
A ⊂ R2+
we note
∂A
its boundary.
References
Our framework
L1 consistency
Perspectives
For
r > 0 and λ > 0, dene
E := B ({x ∈ R2+ : | F − c |≤ r }, λ),
mO := inf x ∈E k(∇F )x k,
MH := supx ∈E k(HF )x k,
B (x , ξ) is the closed ball centered on x and with positive radius ξ ,
(∇F )x is the gradient of F evaluated at x ,
k(HF )x k is the matrix norm induced by Euclidean distance of the
Hessian matrix in x .
Hausdor distance between A1 and A2 is dened by:
dH (A1 , A2 ) = inf {ε > 0 : A1 ⊂ B (A2 , ε), A2 ⊂ B (A1 , ε)},
where
B (S , ε) =
S
x ∈S B (x , ε);
A1
and
A2
are compact sets in
(R2+ , d ).
References
Our framework
L1 consistency
Perspectives
H: There exist
T >0
γ>0
such that
and A > 0 such that, if | t − c | ≤ γ
{F = c }T 6= ∅ and {F = t }T 6= ∅,
then
∀
dH ({F = c }T , {F = t }T ) ≤ A | t − c | .
Assumption H is satised under mild conditions.
Proposition
Let c ∈ (0, 1). Let F ∈ F be twice dierentiable on R2+∗ . Assume there
exist r > 0, ζ > 0 such that mO > 0 and MH < ∞. Then F satises
Assumption H, with A = m2O .
References
Our framework
L1 consistency
Perspectives
From now on we note, for n
kF − Fn k∞ =
sup
x ∈ R2+
∈ N∗ ,
and for T
> 0,
| F (x ) − Fn (x ) |, kF − Fn kT
∞ =
sup
x ∈ [0,T ]2
| F (x ) − Fn (x ) | .
Theorem (Consistency Hausdor distance)
∈ (0, 1). Let F ∈ F be twice dierentiable on R2+∗ . Assume that there
O
exist r > 0, ζ > 0 such that m > 0 and MH < ∞. Let T1 > 0 such that for
T
all t : | t − c | ≤ r , ∂ L(t ) 1 6= ∅. Let (Tn )n∈N∗ be an increasing sequence of
Let c
positive values. Assume that, for each n, Fn is continuous with probability one
and that
kF − Fn k∞ → 0,
a .s .
,
Then for n large enough
dH (∂ L(c )Tn , ∂ Ln (c )Tn ) ≤ 6 A kF − Fn kT∞n , a.s .,
where
A=
2
mO
.
References
Our framework
L1 consistency
Perspectives
dH (∂ L(c )
Tn , ∂ Ln (c )Tn )
converges to zero and the quality of our plug-in
estimator is obviously related to the quality of the estimator Fn .
Note that in the case c is close to one the constant A
= m2O
could be
large. In this case, we will need a large number of data to get a
reasonable estimation.
In order to overcome the problem of Fn is continuous with probability
one it can be considered a smooth version of estimator.
References
Our framework
L1 consistency
Perspectives
L1 consistency
Let us introduce the following assumption:
A1 There exist positive increasing sequences
vn
Z
[0,
Tn ]2
P
| F − Fn |p λ(dx ) →
n→∞
(vn )n∈N∗
0,
and
(Tn )n∈N∗
for some 1
such that
≤ p < ∞.
Theorem (Consistency volume)
Let c
exist
∈ (0, 1). Let F ∈ F be twice dierentiable on R2+∗ . Assume that there
O
r > 0, ζ > 0 such that m > 0 and MH < ∞. Let (vn )n∈N∗ and (Tn )n∈N∗
positive increasing sequences such that Assumption
all t
: |
t
− c | ≤ r , ∂ L(t )T1 6= ∅.
pn dλ (L(c )
A1
is satised and that for
Then it holds that
P
Tn , L (c )Tn ) →
0,
n
n→∞
with pn an increasing positive sequence such that pn
=o
1
p +1
vn
p /Tnp+1
.
References
Our framework
L1 consistency
Perspectives
L1 consistency
Theorem Consistency volume provides a convergence rate, which is closely
related to the choice of the sequence Tn . A sequence Tn whose
divergence rate is large implies a convergence rate pn quite slow.
Theorem Consistency volume does not require continuity assumption on
Fn .
Analysis in the case: Fn the bivariate empirical distribution function.
Problem: optimal criterion for the choice of Tn related with the p in
Assumption A1.
References
Our framework
L1 consistency
Perspectives
Estimation of CTEα (X , Y )
Denition
Consider a random vector
F ∈ F.
For
α ∈ (0, 1),
(X , Y )
with associate distribution function
we dene the estimated bivariate
α-Conditional
Tail Expectation
 Pn
i =1 Xi 1{(Xi ,Yi )∈Ln (α)}
 Pn 1
i =1 {(Xi ,Yi )∈Ln (α)}
[ α (X , Y ) = 
CTE
 Pn

Yi 1{(Xi ,Yi )∈Ln (α)}
Pi =n1



.

(1)
i =1 1{(Xi ,Yi )∈Ln (α)}
We introduce truncated versions of the theoretical and estimated CTEα :
T
T
[ α (X , Y ),
X , Y ) = E[(X , Y )|(X , Y ) ∈ L(α)T ], CTE
CTEα (
using
L(α)T
and
Ln (α)T
i.e. the truncated versions
L(α) and Ln (α).
References
Our framework
L1 consistency
Perspectives
Estimation of CTEα (X , Y )
[ α (X , Y ))
CTE
Under regularity properties of (X , Y ), Assumptions of Theorem
Consistency volume and with the same notation, it holds that
Theorem (Consistency of
P
T
Tn (X , Y ) − CTE
[ αn (X , Y ) →
βn CTEα
n→∞
0,
(2)
r
where βn = min{ pn2(1+r ) , an }, √
with r > 0 such that the density
f(X , Y ) ∈ L1+r (λ) and an = o n .
The convergence in (2) must be interpreted as a componentwise
f
convergence. In the case of a bounded density function (X , Y ) we obtain
√
βn = min{ pn , an }.
References
Our framework
L1 consistency
Perspectives
On the choice of Tn
2r
r
In the case of the bivariate empirical d.f.
Fn : βn = o n 6 (1+r ) /Tn3 (1+r )
f
In the case of a bounded density function (X ,Y ) ,
1
2
βn = o n 6 /Tn3
.
.
Obviously it could be interesting to consider the convergence:
T
CTEα (X , Y ) − CTE
[ α n (X , Y ) .
→
the speed of convergence will also depend on the convergence rate to
zero of
CTEα (X , Y ) − CTETαn (X , Y ) , then of
Y ≥ Tn ])−1 is increasing in Tn , whereas
Tn . This kind of compromise
provides an illustration on how to choose Tn , apart from satisfying the
We remark that
(P[X ≥ Tn
P[(X , Y ) ∈ L(α) \ L(α)Tn ].
or
the speed of convergence is decreasing in
assumptions of consistency results above.
Our framework
L1 consistency
Perspectives
References
Perspectives
Study of CTEα (
X , Y ) as risk measure (stochastic order for random
variables and random vectors)
X , Y ) and Kendall distribution function or bivariate
F (X , Y ).
CTEα (
probability integral transformation (BIPIT)
In this work we provide asymptotic results for a xed level
Problem of constant
A for c ∼ 1:
b
Ln (c ) := {(x , y ) ∈ R2+ : Fb ∗ (x , y ) ≥ c },
for
c ∼ 1,
F
(Bivariate extreme value theory; rst
with
Fb ∗
c ...
Plug in method with
a tail estimator of
part of my PhD thesis).
References
Artzner, P. and Delbaen, F. and Eber, J. M. and Heath, D., Coherent
measures of risk. Math. Finance 9(3), (1999), 203228.
A. Baíllo, J.A. Cuesta-Albertos, A. Cuevas, Convergence rates in
nonparametric estimation of level sets. Statistics and Probability Letters
53, (2001), 27-35.
F. Belzunce, A. Castaño, A. Olvera-Cervantes, A. Suárez-Llorens, Quantile
curves and dependence structure for bivariate distributions. Computational
Statistics & Data Analysis 51, (2007), 5112-5129.
G. Biau, B. Cadre, B. Pelletier, A graph-based estimator of the number of
clusters. ESAIM, Probability and Statistics 11, (2007), 272-280.
J. Cai, H. Li, Conditional tail expectations for multivariate phase-type
distributions. Journal of Applied Probability 42(3), (2005), 810-825.
L. Cavalier, Nonparametric estimation of regression level sets. Statistics
29, (1997), 131-160.
A. Cuevas, R. Fraiman, A plug-in approach to support estimation. Annals
of Statistics 25, (1997), 2300-2312.
References
A. Cuevas, W. González-Manteiga, A. Rodríguez-Casal, Plug-in estimation
of general level sets. Australian and New Zealand Journal of Statistics 48,
(2006), 7-19.
A. Cuevas, A. Rodríguez-Casal, On boundary estimation. Advances in
Applied Probability 36, (2004), 340-354.
M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas. Actuarial Theory for
Dependent Risks. Wiley, (2005).
W. Polonik, Measuring mass concentration and estimating density contour
clusters - an excess mass approach. Annals of Statistics 23, (1995),
855-881.
A. Rodríguez-Casal, Estimacíon de conjuntos y sus fronteras. Un enfoque
geometrico. Ph.D. Thesis, University of Santiago de Compostela, (2003).
A.B. Tsybakov, On nonparametric estimation of density level sets. Annals
of Statistics 25, (1997), 948-969.

A proposal of a bivariate Conditional Tail Expectation

Transcription

Documents pareils

gsr600 tail light cover

comitato provinciale di imperia asd bocciofila taggese

Procedura montaggio portatarga HONDA CBR 1000 RR 2008

Sequence Construction For Integral Estimation

Le Tariffe CONSULTORIO FAMILIARE SERVIZI SOCIOSANITARI

Leggendo e chiacchierando

1 Fondé à Paris en 2002 par Giorgia Fiorio, Reflexions Masterclass

OBJET DES RENCONTRES – OGGETTO DELLE INCONTRI

Projection du film: Marie Antoinette Sélection Prix italo