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IntroductionThe Quantile-Copula estimator
Comparison with competitors
A quantile-copula approach to conditional densityestimation.
Olivier P. Faugeras
Universite Paris VI - Laboratoire de Statistique Theorique et Appliquee
Universite Parix XIII - Laboratoire Lim&Bio
25/05/2008
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Outline
1 IntroductionWhy estimating the conditional density?Two classical approaches for estimation
2 The Quantile-Copula estimatorQuantile transform and copula functionConstruction of the estimatorAsymptotic results
3 Comparison with competitorsTheoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Setup and Motivation
Objective
observe a sample ((Xi, Yi); i = 1, . . . , n) i.i.d. of (X, Y ).predict the output Y for an input X at location x
with minimal assumptions on the law of (X, Y ) (Nonparametric setup).
Notation
(X, Y )→ joint c.d.f FX,Y , joint density fX,Y ;
X → c.d.f. F , density f ;
Y → c.d.f. G, density g.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Setup and Motivation
Objective
observe a sample ((Xi, Yi); i = 1, . . . , n) i.i.d. of (X, Y ).predict the output Y for an input X at location x
with minimal assumptions on the law of (X, Y ) (Nonparametric setup).
Notation
(X, Y )→ joint c.d.f FX,Y , joint density fX,Y ;
X → c.d.f. F , density f ;
Y → c.d.f. G, density g.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Setup and Motivation
Objective
observe a sample ((Xi, Yi); i = 1, . . . , n) i.i.d. of (X, Y ).predict the output Y for an input X at location x
with minimal assumptions on the law of (X, Y ) (Nonparametric setup).
Notation
(X, Y )→ joint c.d.f FX,Y , joint density fX,Y ;
X → c.d.f. F , density f ;
Y → c.d.f. G, density g.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Why estimating the conditional density ?
What is a good prediction ?
1 the regression function r(x) = E(Y |X = x),2 or the full conditional density f(y|x)?
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Why estimating the conditional density ?
What is a good prediction ?
1 the regression function r(x) = E(Y |X = x),2 or the full conditional density f(y|x)?
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Why estimating the conditional density ?
What is a good prediction ?
1 the regression function r(x) = E(Y |X = x),2 or the full conditional density f(y|x)?
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Why estimating the conditional density ?
What is a good prediction ?
1 the regression function r(x) = E(Y |X = x),2 or the full conditional density f(y|x)?
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Why estimating the conditional density ?
What is a good prediction ?
1 the regression function r(x) = E(Y |X = x),2 or the full conditional density f(y|x)?
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 1
A first density -based approach
f(y|x) =fX,Y (x, y)
f(x)← fX,Y (x, y)
f(x)
fX,Y , f : Parzen-Rosenblatt kernel estimators with kernels K, K ′,bandwidths h and h′.
The double kernel estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ ratio shaped
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 1
A first density -based approach
f(y|x) =fX,Y (x, y)
f(x)← fX,Y (x, y)
f(x)
fX,Y , f : Parzen-Rosenblatt kernel estimators with kernels K, K ′,bandwidths h and h′.
The double kernel estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ ratio shaped
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 1
A first density -based approach
f(y|x) =fX,Y (x, y)
f(x)← fX,Y (x, y)
f(x)
fX,Y , f : Parzen-Rosenblatt kernel estimators with kernels K, K ′,bandwidths h and h′.
The double kernel estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ ratio shaped
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 1
A first density -based approach
f(y|x) =fX,Y (x, y)
f(x)← fX,Y (x, y)
f(x)
fX,Y , f : Parzen-Rosenblatt kernel estimators with kernels K, K ′,bandwidths h and h′.
The double kernel estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ ratio shaped
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 2
A regression strategy
Fact: E(1|Y −y|≤h|X = x
)= F (y + h|x)− F (y − h|x) ≈ 2h.f(y|x)
Conditional density estimation problem → regression framework
Transform the data:Yi → Y ′
i := (2h)−11|Yi−y|≤h
Yi → Y ′i := Kh(Yi − y) smoothed version
Nonparametric regression of Y ′i on Xis by local averaging methods
(Nadaraya-Watson, local polynomial, orthogonal series,...)
Nadaraya-Watson estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ (same) ratio shape.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 2
A regression strategy
Fact: E(1|Y −y|≤h|X = x
)= F (y + h|x)− F (y − h|x) ≈ 2h.f(y|x)
Conditional density estimation problem → regression framework
Transform the data:Yi → Y ′
i := (2h)−11|Yi−y|≤h
Yi → Y ′i := Kh(Yi − y) smoothed version
Nonparametric regression of Y ′i on Xis by local averaging methods
(Nadaraya-Watson, local polynomial, orthogonal series,...)
Nadaraya-Watson estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ (same) ratio shape.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Estimating the conditional density - 2
A regression strategy
Fact: E(1|Y −y|≤h|X = x
)= F (y + h|x)− F (y − h|x) ≈ 2h.f(y|x)
Conditional density estimation problem → regression framework
Transform the data:Yi → Y ′
i := (2h)−11|Yi−y|≤h
Yi → Y ′i := Kh(Yi − y) smoothed version
Nonparametric regression of Y ′i on Xis by local averaging methods
(Nadaraya-Watson, local polynomial, orthogonal series,...)
Nadaraya-Watson estimator
f(y|x) =
n∑i=1
K ′h′(Xi − x)Kh(Yi − y)
n∑i=1
K ′h′(Xi − x)
→ (same) ratio shape.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Ratio shaped estimators
Drawbacks
quotient shape of estimator is tricky to study;
explosive behavior when the denominator is small → numericalimplementation delicate (trimming);
minoration hypothesis on the marginal density f(x) ≥ c > 0.
How to remedy these problems?→ build on the idea of using synthetic data:find a representation of the data more adapted to the problem.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Why estimating the conditional density?Two classical approaches for estimation
Ratio shaped estimators
Drawbacks
quotient shape of estimator is tricky to study;
explosive behavior when the denominator is small → numericalimplementation delicate (trimming);
minoration hypothesis on the marginal density f(x) ≥ c > 0.
How to remedy these problems?→ build on the idea of using synthetic data:find a representation of the data more adapted to the problem.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
Outline
1 IntroductionWhy estimating the conditional density?Two classical approaches for estimation
2 The Quantile-Copula estimatorQuantile transform and copula functionConstruction of the estimatorAsymptotic results
3 Comparison with competitorsTheoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
The quantile transform and copula function
What is the “best” transformation of the data in that context ?
The quantile transform
(X1, . . . , Xn) (Y1, . . . , Yn)↓ ↓
(U1 = F (X1), . . . , Un = F (Xn)) (V1 = G(Y1), . . . , Vn = G(Yn))
then Ui, Vi = pseudo-sample with a prescribed uniform marginaldistribution.
→ leads naturally to the copula function:
Sklar’s theorem [1959]
FX,Y (x, y) = C(F (x), G(y)) , −∞ ≤ x, y ≤ +∞.
with C the c.d.f. of (U, V ).
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
The quantile transform and copula function
What is the “best” transformation of the data in that context ?
The quantile transform
(X1, . . . , Xn) (Y1, . . . , Yn)↓ ↓
(U1 = F (X1), . . . , Un = F (Xn)) (V1 = G(Y1), . . . , Vn = G(Yn))
then Ui, Vi = pseudo-sample with a prescribed uniform marginaldistribution.
→ leads naturally to the copula function:
Sklar’s theorem [1959]
FX,Y (x, y) = C(F (x), G(y)) , −∞ ≤ x, y ≤ +∞.
with C the c.d.f. of (U, V ).
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
The quantile transform and copula function
What is the “best” transformation of the data in that context ?
The quantile transform
(X1, . . . , Xn) (Y1, . . . , Yn)↓ ↓
(U1 = F (X1), . . . , Un = F (Xn)) (V1 = G(Y1), . . . , Vn = G(Yn))
then Ui, Vi = pseudo-sample with a prescribed uniform marginaldistribution.
→ leads naturally to the copula function:
Sklar’s theorem [1959]
FX,Y (x, y) = C(F (x), G(y)) , −∞ ≤ x, y ≤ +∞.
with C the c.d.f. of (U, V ).
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
A product shaped estimator
A product form of the conditional density
fY |X(y|x) =fXY (x, y)
f(x)= g(y)c(F (x), G(y))
with c(u, v) = ∂2C(u,v)∂u∂v the copula density.
→ a product estimator
The quantile copula estimator
fn(y|x) := gn(y)cn(Fn(x), Gn(y))
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
A product shaped estimator
A product form of the conditional density
fY |X(y|x) =fXY (x, y)
f(x)= g(y)c(F (x), G(y))
with c(u, v) = ∂2C(u,v)∂u∂v the copula density.
→ a product estimator
The quantile copula estimator
fn(y|x) := gn(y)cn(Fn(x), Gn(y))
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
Construction of the estimator
with
density of Y : g ← kernel estimator gn(y) := 1nhn
n∑i=1
K0
(y−Yi
hn
)c.d.f. F,G← Fn, Gn empirical c.d.f.pseudo sample (Ui, Vi)← (Fn(Xi), Gn(Yi)) empirical approximatecopula density c(u, v)← cn(u, v) kernel density (pseudo) estimatorbased on the pseudo sample (Ui, Vi)cn(u, v)← cn(u, v) kernel density estimator based on the empiricalapproximate (Fn(Xi), Gn(Yi))
The quantile-copula estimator
fn(y|x) :=
[1
nhn
n∑i=1
K0
(y − Yi
hn
)].
[1
na2n
n∑i=1
K1
(Fn(x)− Fn(Xi)
an
)K2
(Fn(y)−Gn(Yi)
an
)]Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
Construction of the estimator
with
density of Y : g ← kernel estimator gn(y) := 1nhn
n∑i=1
K0
(y−Yi
hn
)c.d.f. F,G← Fn, Gn empirical c.d.f.pseudo sample (Ui, Vi)← (Fn(Xi), Gn(Yi)) empirical approximatecopula density c(u, v)← cn(u, v) kernel density (pseudo) estimatorbased on the pseudo sample (Ui, Vi)cn(u, v)← cn(u, v) kernel density estimator based on the empiricalapproximate (Fn(Xi), Gn(Yi))
The quantile-copula estimator
fn(y|x) :=
[1
nhn
n∑i=1
K0
(y − Yi
hn
)].
[1
na2n
n∑i=1
K1
(Fn(x)− Fn(Xi)
an
)K2
(Fn(y)−Gn(Yi)
an
)]Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
Construction of the estimator
with
density of Y : g ← kernel estimator gn(y) := 1nhn
n∑i=1
K0
(y−Yi
hn
)c.d.f. F,G← Fn, Gn empirical c.d.f.pseudo sample (Ui, Vi)← (Fn(Xi), Gn(Yi)) empirical approximatecopula density c(u, v)← cn(u, v) kernel density (pseudo) estimatorbased on the pseudo sample (Ui, Vi)cn(u, v)← cn(u, v) kernel density estimator based on the empiricalapproximate (Fn(Xi), Gn(Yi))
The quantile-copula estimator
fn(y|x) :=
[1
nhn
n∑i=1
K0
(y − Yi
hn
)].
[1
na2n
n∑i=1
K1
(Fn(x)− Fn(Xi)
an
)K2
(Fn(y)−Gn(Yi)
an
)]Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
Construction of the estimator
with
density of Y : g ← kernel estimator gn(y) := 1nhn
n∑i=1
K0
(y−Yi
hn
)c.d.f. F,G← Fn, Gn empirical c.d.f.pseudo sample (Ui, Vi)← (Fn(Xi), Gn(Yi)) empirical approximatecopula density c(u, v)← cn(u, v) kernel density (pseudo) estimatorbased on the pseudo sample (Ui, Vi)cn(u, v)← cn(u, v) kernel density estimator based on the empiricalapproximate (Fn(Xi), Gn(Yi))
The quantile-copula estimator
fn(y|x) :=
[1
nhn
n∑i=1
K0
(y − Yi
hn
)].
[1
na2n
n∑i=1
K1
(Fn(x)− Fn(Xi)
an
)K2
(Fn(y)−Gn(Yi)
an
)]Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Quantile transform and copula functionConstruction of the estimatorAsymptotic results
Asymptotic results
Under classical regularity assumptions on the densities and the kernels,with hn → 0, an → 0,
Consistency
weak consistency nhn →∞, na2n →∞ entail
fn(y|x) = f(y|x) + OP
(n−1/3
).
strong consistency nhn/(ln lnn)→∞ and na2n/(ln lnn)→∞
fn(y|x) = f(y|x) + Oa.s.
((ln lnn
n
)1/3)
.
asymptotic normality nhn →∞ and na2n →∞ entail√
na2n
(fn(y|x)− f(y|x)
)d N
(0, g(y)f(y|x)||K||22
).
hn ' n−1/5 and an ' n−1/6, we get
f(y|x) = f(y|x) + OP (n−1/3).
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Outline
1 IntroductionWhy estimating the conditional density?Two classical approaches for estimation
2 The Quantile-Copula estimatorQuantile transform and copula functionConstruction of the estimatorAsymptotic results
3 Comparison with competitorsTheoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Theoretical asymptotic comparison
all estimators have the same order of convergence ≈ n−1/3
Asymptotic Bias
c of compact support : the “classical” kernel method to estimate thecopula density induces bias on the boundaries of [0, 1]2
→ use boundary-correction techniques (boundary kernels, beta kernels,reflection and transformation methods,.. )
Main terms in the asymptotic variance
Ratio shaped estimators: f(y|x)f(x) → explosive variance for small value
of the density f(x), e.g. in the tail of the distribution of X.
Quantile-copula estimator: g(y)f(y|x) → does not suffer from theunstable nature of competitors.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Theoretical asymptotic comparison
all estimators have the same order of convergence ≈ n−1/3
Asymptotic Bias
c of compact support : the “classical” kernel method to estimate thecopula density induces bias on the boundaries of [0, 1]2
→ use boundary-correction techniques (boundary kernels, beta kernels,reflection and transformation methods,.. )
Main terms in the asymptotic variance
Ratio shaped estimators: f(y|x)f(x) → explosive variance for small value
of the density f(x), e.g. in the tail of the distribution of X.
Quantile-copula estimator: g(y)f(y|x) → does not suffer from theunstable nature of competitors.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Theoretical asymptotic comparison
all estimators have the same order of convergence ≈ n−1/3
Asymptotic Bias
c of compact support : the “classical” kernel method to estimate thecopula density induces bias on the boundaries of [0, 1]2
→ use boundary-correction techniques (boundary kernels, beta kernels,reflection and transformation methods,.. )
Main terms in the asymptotic variance
Ratio shaped estimators: f(y|x)f(x) → explosive variance for small value
of the density f(x), e.g. in the tail of the distribution of X.
Quantile-copula estimator: g(y)f(y|x) → does not suffer from theunstable nature of competitors.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Finite sample simulation
Model
Sample of n = 100 i.i.d. variables (Xi, Yi), from the following model:
X, Y is marginally distributed as N (0, 1)X, Y is linked via Frank Copula .
C(u, v, θ) =ln[(θ + θu+v − θu − θv)/(θ − 1)]
ln θ
with parameter θ = 100.
Practical implementation:
Beta kernels for copula estimator, Epanechnikov for other.
simple Rule-of-thumb method for the bandwidths.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Conclusions
Summary
ratio type into the product → consistency and limit results whereobtained by simple combination of the previous known ones on(unconditional) density estimation,
nonexplosive behavior in the tails of the marginal density,
no need for trimming or clipping.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Conclusions
Perspectives and work-in-progress
Adaptive bandwidth choices to the regularity of the model
Efficient kernel estimation of the copula density byBoundary-corrected kernels
Designing an extreme-specific estimation method. Applications toinsurance, Risk theory, Environmental Sciences.
Application to prediction: functional of the density∫
φ(y)f(y|x)dy,mode, predictive sets.
Extension to time series by coupling arguments for Markovianmodels.
Alternative nonparametric methods of estimation: projections,wavelets,..
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
Bibliography
Reference
O. P. Faugeras. A quantile-copula approach to conditional densityestimation. Submitted, in revision, 2008. Available onhttp://hal.archives-ouvertes.fr/hal-00172589/fr/.
Related work:J. Fan and Q. Yao. Nonlinear time series. Springer Series inStatistics. Springer-Verlag, New York, second edition, 2005.Nonparametric and parametric methods.J. Fan, Q. Yao, and H. Tong. Estimation of conditional densitiesand sensitivity measures in nonlinear dynamical systems. Biometrika,83(1):189–206, 1996.L. Gyorfi and M. Kohler. Nonparametric estimation of conditionaldistributions. IEEE Trans. Inform. Theory, 53(5):1872–1879, 2007.R. J. Hyndman, D. M. Bashtannyk, and G. K. Grunwald. Estimatingand visualizing conditional densities. J. Comput. Graph. Statist.,5(4):315–336, 1996.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.
IntroductionThe Quantile-Copula estimator
Comparison with competitors
Theoretical comparisonFinite sample simulation
References
R. J. Hyndman and Q. Yao. Nonparametric estimation andsymmetry tests for conditional density functions. J. Nonparametr.Stat., 14(3):259–278, 2002.
C. Lacour. Adaptive estimation of the transition density of a markovchain. Ann. Inst. H. Poincare Probab. Statist., 43(5):571–597,2007.
M. Rosenblatt. Conditional probability density and regressionestimators. In Multivariate Analysis, II (Proc. Second Internat.Sympos., Dayton, Ohio, 1968), pages 25–31. Academic Press, NewYork, 1969.
M. Sklar. Fonctions de repartition a n dimensions et leurs marges.Publ. Inst. Statist. Univ. Paris, 8:229–231, 1959.
W. Stute. On almost sure convergence of conditional empiricaldistribution functions. Ann. Probab., 14(3):891–901, 1986.
Olivier P. Faugeras A quantile-copula approach to conditional density estimation.