On the weak convergence of the empirical conditional copula
under a simplifying assumption
François Portier∗

Johan Segers†

arXiv:1511.06544v2 [] 16 May 2017

May 17, 2017

Abstract. A common assumption in pair-copula constructions is that the copula of
the conditional distribution of two random variables given a covariate does not depend on
the value of that covariate. Two conflicting intuitions arise about the best possible rate of
convergence attainable by nonparametric estimators of that copula. On the one hand, the
best possible rates for estimating the marginal conditional distribution functions is slower
than the parametric one. On the other hand, the invariance of the conditional copula given
the value of the covariate suggests the possibility of parametric convergence rates. The more
optimistic intuition is shown to be correct, confirming a conjecture supported by extensive
Monte Carlo simulations by I. Hobaek Haff and J. Segers [Computational Statistics and Data
Analysis 84:1–13, 2015] and improving upon the nonparametric rate obtained theoretically
by I. Gijbels, M. Omelka and N. Veraverbeke [Scandinavian Journal of Statistics 42:1109–
1126, 2015]. The novelty of the proposed approach lies in a double smoothing procedure for
the estimator of the marginal conditional distribution functions. The copula estimator itself
is asymptotically equivalent to an oracle empirical copula, as if the marginal conditional
distribution functions were known.
Keywords: Donsker class; Empirical copula process; Local linear estimator; Pair-copula
construction; Partial copula; Smoothing; Weak convergence.

1

Introduction

Let (Y1 , Y2 ) be a pair of continuous random variables with joint distribution function H(y1 , y2 ) =
Pr(Y1 ≤ y1 , Y2 ≤ y2 ) and marginal distribution functions Fj (yj ) = Pr(Yj ≤ yj ), for yj ∈ R
and j ∈ {1, 2}. If H is continuous, then (F1 (Y1 ), F2 (Y2 )) is a pair of uniform (0, 1) random
variables. Their joint distribution function, D(u1 , u2 ) = Pr{F1 (Y1 ) ≤ u1 , F2 (Y2 ) ≤ u2 } for uj ∈
[0, 1], is therefore a copula. By Sklar’s celebrated theorem (Sklar, 1959), we have H(y1 , y2 ) =
D{F1 (y1 ), F2 (y2 )}. The copula D thus captures the dependence between the random variables
Y1 and Y2 .
Suppose there is a third random variable, X, and suppose the joint conditional distribution
of (Y1 , Y2 ) given X = x, with x ∈ R, is continuous. Then we can apply Sklar’s theorem to the
joint conditional distribution function H(y1 , y2 |x) = Pr(Y1 ≤ y1 , Y2 ≤ y2 | X = x). Writing
Fj (yj |x) = Pr(Yj ≤ yj | X = x), we have H(y1 , y2 |x) = C{F1 (y1 |x), F2 (y2 |x) | x}, where
C(u1 , u2 |x) = Pr{F1 (Y1 |x) ≤ u1 , F2 (Y2 |x) ≤ u2 | X = x} is the copula of the conditional
distribution of (Y1 , Y2 ) given X = x. This conditional copula thus captures the dependence
between Y1 and Y2 conditionally on X = x. Examples include exchange rates before and
after the introduction of the euro (Patton, 2006), diastolic versus systolic blood pressure when
∗

LTCI, CNRS, Télécom ParisTech, Université Paris-Saclay. E-mail: francois.portier@gmail.com
Université catholique de Louvain, Institut de Statistique, Biostatistique et Sciences Actuarielles, Voie du
Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. E-mail: johan.segers@uclouvain.be
†

1

controlling for cholesterol (Lambert, 2007), and life expectancies of males versus females for
different categories of countries (Veraverbeke et al., 2011).
Evidently, we can integrate out the joint and marginal conditional
R distributions to obtain
their unconditional versions: if X has density fX , then H(y1 , y2 ) = H(y1 , y2 |x) fX (x) dx and
similarly for Fj (yj ). For
R the copula, however, this relation does not hold: in general, D(u1 , u2 )
will be different from C(u1 , u2 |x) fX (x) dx.
Example 1.1. Suppose that Y1 and Y2 are conditionally independent given X: for all x, we
have H(y1 , y2 |x) = F1 (y1 |x) F2 (y2 |x). Then the copula of the conditional distribution is the
independence copula: C(u1 , u2 |x) = u1 u2 . Nevertheless, unconditionally, Y1 and Y2 need not
be independent, and the copula, D, of the unconditional distribution of (Y1 , Y2 ) can be different
from the independence copula.
In the previous example, the copula of the conditional distribution of (Y1 , Y2 ) given X = x
does not depend on the value of x. This invariance property is called the simplifying assumption.
The property does not hold in general but it is satisfied, for instance, for trivariate Gaussian
distributions. For further examples and counterexamples of distributions which do or do not
satisfy the simplifying assumption, see for instance Hobaek Haff et al. (2010) and Stöber et al.
(2013).
Example 1.2. Let (Y1 , Y2 , X) be trivariate Gaussian with means µ1 , µ2 , µX , standard deviations
σ1 , σ2 , σX > 0, and correlations ρ12 , ρ1X , ρ2X ∈ (−1, 1). The conditional distribution of (Y1 , Y2 )
given X = x is bivariate Gaussian with means µj − ρjX x and standard deviations σj (1 − ρ2jX )1/2
for j ∈ {1, 2}, while the correlation is given by the partial correlation of (Y1 , Y2 ) given X, i.e.,
ρ12|X = (ρ12 − ρ1X ρ2X )/{(1 − ρ21X )(1 − ρ22X )}1/2 . As a consequence, the conditional copula
of (Y1 , Y2 ) given X = x is the so-called Gaussian copula with correlation parameter ρ12|X ,
whatever the value of x, while the unconditional copula of (Y1 , Y2 ) is the Gaussian copula with
correlation parameter ρ12 .
Given an iid sample (Xi , Yi1 , Yi2 ), i = 1, . . . , n, we seek to make nonparametric inference
on the conditional dependence of (Y1 , Y2 ) given X under the simplifying assumption that there
exists a single copula, C, such that C(u1 , u2 |x) = C(u1 , u2 ) for all x and all (u1 , u2 ). As
noted in the two examples above, C is usually not equal to the copula, D, of the unconditional
distribution of (Y1 , Y2 ). The inference problem arises in so-called pair-copula constructions,
where a multivariate copula is broken down into multiple bivariate copulas through iterative
conditioning (Joe, 1996; Bedford and Cooke, 2002; Aas et al., 2009). If the pair copulas are
√
assumed to belong to parametric copula families, then likelihood-based inference yields 1/ nconsistent estimators of the copula parameters (Hobaek Haff, 2013). Here, we concentrate
instead on the nonparametric case. The mathematical analysis is difficult and our treatment is
therefore limited to a single conditioning variable.
Suppose we have no further structural information on the joint distribution of (X, Y1 , Y2 )
besides smoothness and the simplifying assumption. Then how well can we hope to estimate the
conditional copula C? The simplifying assumption implies that C is an object that combines
both local and global properties of the joint distribution of (X, Y1 , Y2 ): conditioning upon X = x
and integrating out, we find that C is equal to the partial copula of (Y1 , Y2 ) given X (Bergsma,
2011; Gijbels et al., 2015b),
C(u1 , u2 ) = Pr{F1 (Y1 |X) ≤ u1 , F2 (Y2 |X) ≤ u2 }.
On the one hand, the necessity to estimate the univariate conditional distribution functions
Fj (yj |x) suggests that the best convergence rate that can be achieved for estimating C will be
√
slower than 1/ n. On the other hand, since the outer probability is an unconditional one, we
√
may hope to achieve the parametric rate, 1/ n.
2

In Hobæk Haff and Segers (2015), evidence from extensive numerical experiments was pro√
vided to support the more optimistic conjecture that the parametric rate 1/ n can be achieved.
The estimator proposed was constructed as the empirical copula (Deheuvels, 1979; Rüschendorf,
1976) based on the pairs

F̂n,1 (Yi1 |Xi ), F̂n,2 (Yi2 |Xi ) ,
i = 1, . . . , n,
(1.1)
where the F̂n,j were nearest-neighbour estimators of the Fj . In Gijbels et al. (2015a), a highlevel theorem was provided saying that if F̂n,j are estimators of the univariate conditional
margins that satisfy a number of assumptions, then the empirical copula based on the pairs
√
(1.1) is consistent and asymptotically normal with convergence rate 1/ n. In addition, it was
proposed to estimate the univariate conditional margins Fj by local linear estimators. However,
attempts to prove that any of these estimators of Fj also satisfy the conditions of the cited
theorem have failed so far. The only positive results have been obtained under additional
structural assumptions on the conditional margins, exploitation of which leads to estimators
F̂n,j with the desired properties.
The contribution of our paper is then two-fold:
1. We provide an alternative to Theorem 2 in Gijbels et al. (2015a), imposing weaker conditions on the estimators F̂n,j of the univariate conditional margins, and concluding that
the empirical copula based on the pairs (1.1) is consistent and asymptotically normal with
√
rate 1/ n. The conclusion of our main theorem is a bit weaker than the one of the cited
theorem in that we can only prove weak convergence of stochastic processes on [γ, 1 − γ]2 ,
for arbitrary 0 < γ < 1/2, rather than on [0, 1]2 .
2. We provide nonparametric estimators F̂n,j of the conditional margins Fj that actually
satisfy the conditions of our theorem. The estimators are smoothed local linear estimators.
In contrast to Gijbels et al. (2015a), we also smooth in the yj -direction, ensuring that the
trajectories (yj , x) 7→ F̂n,j (yj |x) belong to a Donsker class with high probability.
To prove that the smoothed local linear estimators F̂n,j satisfy the requirements of our main
theorem, we prove a number of asymptotic results on those estimators that may be interesting
in their own right.
Although our estimator of Fj and thus of C is different from the ones of Hobæk Haff and
Segers (2015) or Gijbels et al. (2015a), we do not claim it to be superior. Our objective is rather
to provide the first construction of a nonparametric estimator of C that can be proven to achieve
√
the 1/ n convergence rate. This is why, in the numerical experiments, we limit ourselves to
illustrate the asymptotic theory but do not provide comparisons between estimators. For the
same reason, we do not address other important questions of practical interest such as how to
choose the bandwidths or how to test the hypothesis that the simplifying assumption holds.
For the latter question, we refer to Acar et al. (2012) and Derumigny and Fermanian (2016).
The paper is structured as follows. In Section 2, we state the main theorem giving conditions on the estimators F̂n,j for the empirical copula based on the pairs (1.1) to be consistent
√
and asymptotically normal with convergence rate 1/ n. In Section 3, we then show that the
smoothed local linear estimator satisfies the requirements. The theory is illustrated by numerical experiments. Auxiliary results of independent interest on the smoothed local linear
estimator are stated in Section 4. The proofs build upon empirical process theory and their
details are spelled out in the appendices.

3

2

Empirical conditional copula

The goal of this section is to present the mathematical framework of the paper (Section 2.1)
and to provide general conditions on the estimated margins that ensure the weak convergence
of the estimated conditional copula (Section 2.2).

2.1

Set-up and definitions

Let fX,Y be the density function of the random triple (X, Y ) = (X, Y1 , Y2 ). Let fX and
SX = {x ∈ R : fX (x) > 0} denote the density and the support of X, respectively. The
conditional cumulative distribution function of Y given X = x is given by
Z y1 Z y2
fX,Y (x, z1 , z2 )
H(y | x) = Pr(Y1 ≤ y1 , Y2 ≤ y2 | X = x) =
dz2 dz1 ,
fX (x)
−∞ −∞
for y = (y1 , y1 ) ∈ R2 and x ∈ SX . Since H( · |x) is a continuous bivariate cumulative distribution
function, its copula is given by the function
C(u | x) = Pr {F1 (Y1 |X) ≤ u1 , F2 (Y2 |X) ≤ u2 | X = x} ,
for u = (u1 , u2 ) ∈ [0, 1]2 and x ∈ SX , where F1 ( · |x) and F2 ( · |x) are the margins of H( · |x).
We make the simplifying assumption that the copula of H( · |x) does not depend on x, i.e.,
C( · |x) ≡ C( · ). This assumption is equivalent to the one that

F1 (Y1 |X), F2 (Y2 |X) is independent of X.
(2.1)
Under (2.1), the copula of H( · |x) is given by C for every x ∈ SX . It is worth mentioning that,
for any j ∈ {1, 2}, and even without the simplifying assumption,
Fj (Yj |X) is uniformly distributed on [0, 1] and independent of X.

(2.2)

Let (Xi , Yi1 , Yi2 ), for i ∈ {1, . . . , n}, be independent and identically distributed random
vectors, with common distribution equal to the one of (X, Y1 , Y2 ). Our aim is to estimate the
conditional copula C without any further structural or parametric assumptions on C or on
Fj ( · |x). A reasonable procedure is to estimate the conditional margins in some way, producing
random functions F̂n,j ( · |x), and then proceed with the pseudo-observations F̂n,j (Yij |Xi ) from
C. Exploiting the knowledge that C is a copula, we estimate it by the empirical copula, Ĉn ,
of those pseudo-observations. Formally, let Ĝn,j , for j ∈ {1, 2}, be the empirical distribution
function of the pseudo-observations F̂n,j (Yij |Xi ), i ∈ {1, . . . , n}, i.e.,
n

Ĝn,j (uj ) =

1X
1{F̂n,j (Yij |Xi ) ≤ uj } ,
n

uj ∈ [0, 1].

i=1

The generalized inverse of a univariate distribution function F is defined as
F − (u) = inf{y ∈ R : F (y) ≥ u},

u ∈ [0, 1].

(2.3)

Let Ĝ−
n,j be the generalized inverse of Ĝn,j . The empirical conditional copula, Ĉn , is defined by
n

Ĉn (u) =

1X
1{F̂n,1 (Yi1 |Xi ) ≤ Ĝ− (u1 )} 1{F̂n,2 (Yi2 |Xi ) ≤ Ĝ− (u2 )} ,
n,1
n,2
n
i=1

4

u ∈ [0, 1]2 ,

(2.4)

the empirical copula of the pseudo-observations (F̂n,1 (Yi1 |Xi ), F̂n,2 (Yi2 |Xi )), for i ∈ {1, . . . , n}.
We introduce an oracle copula estimator, defined as the empirical copula based on the
(or)
unobservable random pairs (F1 (Yi1 |Xi ), F1 (Yi2 |Xi )), i ∈ {1, . . . , n}. Let Ĝn,j be the empirical
distribution function of the uniform random variables Fj (Yij |Xi ), i ∈ {1, . . . , n}, i.e.,
n

(or)

Ĝn,j (uj ) =

1X
1{Fj (Yij |Xi ) ≤ uj } ,
n

uj ∈ [0, 1].

i=1

(or)−

Let Ĝn,j

be its generalized inverse, as in (2.3). The oracle empirical copula is defined as
n

Ĉn(or) (u) =

1X
1{F (Y |X ) ≤ Ĝ(or)− (u )} 1{F (Y |X ) ≤ Ĝ(or)− (u )} ,
2
1
2 i2
1 i1
i
i
n
n,2
n,1

u ∈ [0, 1]2 .

i=1

The oracle empircal copula is not computable in practice as it requires the knowledge of the
marginal conditional distributions F1 and F2 .
We rely on the following Hölder regularity class. Let d ∈ N \ {0}, 0 < δ ≤ 1, k ∈ N, and
M > 0 be scalars and let S ⊂ Rd be non-empty, open and convex. Let Ck+δ,M (S) be the space
of functions S → R that are k times differentiable and whose derivatives (including the zero-th
derivative, that is, the function itself) are uniformly bounded by M and such that every mixed
partial derivative of order l ≤ k, say f (l) , satisfies the Hölder condition
sup
z6=z̃

f (l) (z) − f (l) (z̃)
|z − z̃|δ

≤ M,

(2.5)

where | · | in the denominator denotes the Euclidean norm. In particular, C1,M (R) is the space
of Lipschitz functions R → R bounded by M and with Lipschitz constant bounded by M .

2.2

Asymptotic normality
(or)

We now give a set of sufficient conditions to ensure that Ĉn and Ĉn are asymptotically equivalent, i.e., their difference is oP (n−1/2 ). As a consequence, Ĉn is consistent and asymptotically
normal with convergence rate OP (n−1/2 ). In the same spirit as Theorem 2 in Gijbels et al.
(2015a), some of the assumptions are “ground-level conditions” and concern directly the distribution P , while others are “high-level conditions” and deal with the estimators F̂n,j of the
conditional margins. Given a choice for F̂n,j , the high-level conditions need to be verified, as
we will do in Section 3 for a specific proposal. Ground-level assumptions are denoted with the
letter G and high-level conditions are denoted with the letter H.
(G1) The law P admits a density fX,Y on SX × R2 such that SX is a nonempty, bounded,
open interval. For some M > 0 and δ > 0, the functions F1 ( · | · ) and F2 ( · | · ) belong to
C3+δ,M (R × SX ) and the function fX belongs to C2,M (SX ). There exists b > 0 such that
fX (x) ≥ b for every x ∈ SX . For any j ∈ {1, 2} and any γ ∈ (0, 1/2), there exists bγ > 0
such that, for every yj ∈ [Fj− (γ|x), Fj− (1 − γ|x)] and every x ∈ SX , we have fj (yj |x) ≥ bγ .
(G2) Let Ċj and C̈jk denote the first and second-order partial derivatives of C, where j, k ∈
{1, 2}. The copula C is twice continuously differentiable on the open unit square, (0, 1)2 .
There exists κ > 0 such that, for all u = (u1 , u2 ) ∈ (0, 1)2 and all j, k ∈ {1, 2},
C̈jk (u) ≤ κ {uj (1 − uj ) uk (1 − uk )}−1/2 .

5

(H1) For any j ∈ {1, 2} and any γ ∈ (0, 1/2), with probability going to 1, we have that for
every x ∈ SX , the function yj 7→ F̂n,j (yj |x) is continuous on R and strictly increasing on
[Fj− (γ|x), Fj− (1 − γ|x)].
(H2) For any j ∈ {1, 2}, we have
sup
x∈SX , yj ∈R

F̂n,j (yj |x) − Fj (yj |x) = oP (n−1/4 ).

(H3) For any j ∈ {1, 2} and any γ ∈ (0, 1/2), there exist positive numbers (δ1 , M1 ) such that,
with probability going to 1,
−
{x 7→ F̂n,j
(uj |x) : uj ∈ [γ, 1 − γ]} ⊂ C1+δ1 ,M1 (SX ).

The support SX is assumed to be open so that the derivatives of functions on SX are defined as
usual. Condition (G2) also appears as equation (9) in Omelka et al. (2009), where it is verified
for some popular copula models (Gaussian, Gumbel, Clayton, Student t).
Let `∞ (T ) denote the space of bounded real functions on the set T , the space being equipped
with the supremum distance, and let “ ” denote weak convergence in this space (van der Vaart
and Wellner, 1996). Let P denote the probability measure on the underlying probability space
associated to the whole sequence (Xi , Yi )i=1,2,... .
Theorem 2.1. Assume that (G1), (G2), (H1), (H2) and (H3) hold. If the simplifying assumption (2.1) holds, then for any γ ∈ (0, 1/2), we have

sup
n1/2 Ĉn (u) − Ĉn(or) (u) = oP (1),
n → ∞.
(2.6)
u∈[γ,1−γ]2


Moreover, n1/2 Ĉn (u) − C(u)
law as the random process

C in `∞ ([γ, 1 − γ]2 ), the limiting process C having the same

B(u) − Ċ1 (u) B(u1 , 1) − Ċ2 (u) B(1, u2 ),

u ∈ [γ, 1 − γ]2 ,

where B is a C-Brownian bridge, i.e., a centered Gaussian process with continuous trajectories
and with covarance function given by

2
cov B(u), B(v) = C(u1 ∧ v1 , u2 ∧ v2 ) − C(u) C(v),
(u, v) ∈ [γ, 1 − γ]2 .
(2.7)
The proof of Theorem 2.1 is given in Appendix A. The theorem bears resemblance with Theorem 2 in Gijbels et al. (2015a). Our approach is more specific because we consider a smoothness
approach through the spaces C1+δ,M (SX ) to obtain their Donsker property. Exploiting this context, we formulate in (H2) a condition on the rate of convergence of the estimator F̂n,j that
−
does not involve the inverse F̂n,j
, as expressed in their condition (Yn).

3

Smoothed local linear estimators of the conditional margins

The objective in this section is to provide estimators, F̂n,j , j ∈ {1, 2}, of the conditional marginal
distribution functions Fj (yj |x) = Pr(Yj ≤ yj | X = x) that satisfy conditions (H1), (H2) and
(H3) of Theorem 2.1 (Section 3.1). As a consequence, the empirical conditional copula based
on these estimators is a nonparametric estimator of the conditional copula in the simplifying
assumption which, by Theorem 2.1, is consistent and asymptotically normal with convergence
√
rate 1/ n (Section 3.2). Numerical experiments confirm that the asymptotic theory provides
a good approximation to the sampling distribution, at least for large samples (Section 3.3).
6

3.1

Definition of the smoothed local linear estimator

Let K : R → [0, ∞) and L : R → [0, ∞) be two kernel functions, i.e., nonnegative, symmetric
functions integrating to unity. Let (hn,1 )n≥1 and (hn,2 )n≥1 two bandwidth sequences that tend
to 0 as n → ∞. For (y, Y ) ∈ R2 and h > 0, put
Z y
Lh (t − Y ) dt.
(3.1)
Lh (y) = h−1 L(h−1 y),
ϕh (y, Y ) =
−∞

For j ∈ {1, 2}, we introduce the smoothed local linear estimator of Fj (yj |x) defined by
F̂n,j (yj |x) = ân,j ,

(3.2)

where ân,j is the first component of the random pair
(ân,j , b̂n,j ) = arg min

n
X

ϕhn,2 (yj , Yij ) − a − b(Xi − x)

2


K

(a,b)∈R2 i=1

x − Xi
hn,1


,

(3.3)

where ϕh in (3.1) serves to smooth the indicator function y 7→ 1{Y ≤y} . The kernels K and L do
not have the same role: L is concerned with “smoothing” over Y1 and Y2 whereas K “localises”
the variable X at x ∈ SX . For this reason, we purposefully use two different bandwidth
sequences (hn,1 )n≥1 and (hn,2 )n≥1 . We shall see that the conditions on the bandwidth hn,2
for the y-directions are weaker than the ones for the bandwidth hn,1 for the x-direction. The
assumptions related to the two kernels and bandwidth sequences are stated in (G3) and (G4)
below.
In the classical regression context, local linear estimators have been introduced in Stone
(1977) and are further studied for instance in Fan and Gijbels (1996). In Gijbels et al. (2015a),
local linear estimators for Fj (y|x) are considered too, but without smoothing in the y-variable,
so that condition (H3) does not hold; see Section 3.4 below.

3.2

Weak convergence of the empirical conditional copula

We derive the limit distribution of the empirical conditional copula Ĉn in (2.4) when the
marginal conditional distribution functions are estimated via the smoothed local linear estimators F̂n,j (yj |x) in (3.2). We need the following additional conditions.
(G3) The kernels K and L are bounded,
nonnegative,
symmetric functions on R, supported
R
R
on (−1, 1), and such that L(u) du = K(u) du = 1. The function L is continuously
differentiable on R and its derivative is a bounded real function of bounded variation.
The function K is twice continuously differentiable on R and its second-order derivative
is a bounded real function of bounded variation.
(G4) There exists α > 0 such that the bandwidth sequences hn,1 > 0 and hn,2 > 0 satisfy, as
n → ∞,
nh8n,1 → 0,
nh3+α
n,1
|log hn,1 |

→ ∞,

nh8n,2 → 0,

−1−α/2 2
hn,2

hn,1

nh1+α
n,1 hn,2
|log hn,1 hn,2 |

→ 0,

→ ∞.

Condition (G4) implies in particular that the bandwidth sequences are small enough to ensure
that the bias associated to the estimation of F1 and F2 does not affect the asymptotic distribution of the empirical conditional copula. An interesting situation occurs when the bandwidths
7

1/2+α/4

satisfy hn,2 /hn,1

→ 0 and h2n,1 /hn,2 → 0. Then the above condition becomes
nh8n,1 → 0,

nh8n,2 → 0,

nh3+α
n,1
|log hn,1 |

→ ∞.

Hence the conditions on the bandwidth h1,n are more restrictive than the ones on h2,n . Typically,
hn,2 might be chosen smaller than hn,1 as it does not need to satisfy nh3+α
n,1 /|log hn,1 | → ∞. This
might result in a smaller bias. By way of comparison, in the location-scale model, no smoothing
in the y-direction is required, but Gijbels et al. (2015a) still require the stronger condition that
nh5n,1 → 0 and nh3+α
n,1 / log n → ∞.
Theorem 3.1. Assume that (G1), (G2), (G3) and (G4) hold. Then (H1), (H2) and (H3) are
valid. If the simplifying assumption (2.1) also holds, then for any γ ∈ (0, 1/2), equation (2.6)
is satisfied and n1/2 (Ĉn − C)
C in `∞ ([γ, 1 − γ]2 ), where the limiting process C is defined in
the statement of Theorem 2.1.
The proof of Theorem 3.1 is given in Appendix B and relies on results on the smoothed
local linear estimator presented in Section 4.
Distinguishing between the bandwidth sequence hn,1 for the x-direction on the one hand
and the bandwidth sequence hn,2 for the y1 and y2 -directions on the other hand allows for
a weaker assumption than if both sequences would have been required to be the same. In
practice, one could even consider, for each j ∈ {1, 2}, the smoothed local linear estimator F̂n,j
(j)
(j)
based on a bandwidth sequence hn,1 for the x-direction and a bandwidth sequence hn,2 for
the yj -direction, yielding four bandwidth sequences in total. However, this would not really
lead to weaker assumptions in Theorem 3.1, since (G4) would then be required to hold for
(j)
(j)
each pair (hn,1 , hn,2 ). The same remark also applies to the kernels K and L, which could be
chosen differently for each margin j ∈ {1, 2}. The required modification of the formulation of
Theorem 3.1 is obvious.

3.3

Numerical illustrations

The assertion in (2.6) that the empirical conditional copula Ĉn is only oP (n−1/2 ) away from
(or)
the oracle empirical copula Ĉn is perhaps surprising, since the estimators of the conditional
margins converge at a rate slower, rather than faster, than OP (n−1/2 ). To support the claim,
we performed a number of numerical experiments based on independent random samples from
the copula of the trivariate Gaussian distribution (Example 1.2) with correlations given by
ρ1X = 0.4, ρ2X = −0.2, and ρ12 = 0.3689989. The conditional copula of (Y1 , Y2 ) given X is
then equal to the bivariate Gaussian copula with correlation parameter ρ12|X = 0.5. Estimation
target was the value of the copula at u = (0.5, 0.7).
To estimate the conditional margins, we used the smoothed local linear estimator (3.2)–
(3.3) based on the triweight kernel K(x) = (35/32)(1 − x2 )3 1[−1,1] (x) and the biweight kernel
L(y) = (15/16)(1 − y 2 )2 1[−1,1] (y), in accordance to assumption (G3). The bandwidths where
chosen as hn,1 = hn,2 = 0.5n−1/5 , with n the sample size. The experiments were performed
within the statistical software environment R (R Core Team, 2017). The algorithms for the
smoothed local linear estimator were implemented in C for greater speed.
(or)
Figure 1 illustrates the proximity between the estimator Ĉn (u) and the oracle Ĉn (u).
The left-hand and middle panels show scatterplots of 1 000 independent realizations of the
(or)
pairs (n1/2 {Ĉn (u) − C(u)}, n1/2 {Ĉn (u) − C(u)}) based on samples of size n = 500 and
n = 2 000. As n increases, the points are concentrated more strongly along the diagonal. The

8

sample size n = 2000

correlation between estimator and oracle
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Figure 1:
Scatterplots of 1 000 independent realizations of the pairs (n1/2 {Ĉn (u) −
(or)
C(u)}, n1/2 {Ĉn (u) − C(u)}) based on the trivariate Gaussian copula for sample sizes n = 500
(left) and n = 2 000 (middle). Right: correlation between the empirical conditional copula and
the oracle empirical copula as a function of the sample size n between 100 and 10 000, each
point being based on 5 000 samples of size n.

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200

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sample size

Figure 2: QQ-plot (left) and density plot (middle) based on 1 000 independent realizations of
n1/2 {Ĉn (u)−C(u)} for the trivariate Gaussian copula for sample size n = 500. Right: standard
deviation of n1/2 {Ĉn (u) − C(u)} as a function of n between 100 and 10 000 versus the limit
value σ(u) = 0.2080, each point being based on 5 000 samples of size n.
linear correlations between the estimator and the oracle for sample sizes between n = 100 and
n = 10 000 are plotted in the right-hand panel, each point being based on 5 000 samples.
The proximity of the sampling distribution of the estimation error n1/2 {Ĉn (u) − C(u)}
and the limiting normal distribution is illustrated in Figure 2. According to Theorems 2.1
and 3.1, the limiting distribution of the empirical conditional copula process is N (0, σ 2 (u)).
The asymptotic variance is σ 2 (u) = var{C(u)} = var{B(u) − Ċ1 (u) B(u1 , 1) − Ċ2 (u) B(1, u2 )}
and can be computed from (2.7) and the knowledge that C is the bivariate Gaussian copula
with correlation parameter ρ12|X . As the sample size increases, the sampling distribution moves
closer to the limiting normal distribution as confirmed in the QQ-plot and density plot in
Figure 2, both based on 1 000 samples of size n = 500. The realized standard deviations of the
empirical conditional copula process are given in the right-hand panel, again for sample sizes
between n = 100 and n = 10 000, each point being based on 5 000 samples of size n. The limit
value σ(u) = 0.2080 is represented as a horizontal line.

9

3.4

Further comments

Smoothing in two directions rather than one. In Gijbels et al. (2015a), local linear
estimators that involve smoothing in x but not in yj are employed to estimate the conditional
margins Fj (yj |x). Interestingly, the approach taken in our paper does not extend to those estimators. Indeed, smoothing in the y-direction is crucial to obtain that the random functions
−
x 7→ F̂n,j
(u|x), when u ranges over [γ, 1 − γ], are included in a Hölder regularity class (Proposition 4.4 and its proof). Our approach is based on the result that the class of functions formed
by the indicators (y, x) 7→ 1{y ≤ q(x)} , where q : SX → R belongs to C1+δ,M (SX ) with δ > 0,
M > 0 and SX convex, is Donsker (Dudley, 1999; van der Vaart and Wellner, 1996). When the
estimators of the margins are not smooth with respect to yj , the difficulty consists in finding
appropriate restrictions on q so that the class (y, x) 7→ 1{y ≤ q(x)} is Donsker and still contains
−
the functions corresponding to q(x) = F̂n,j
(u|x), u ∈ [γ, 1 − γ]. Negative results are available
and they shed some light on the difficulties that arise. For instance, when q is constrained to be
nonincreasing, the class of indicators is equivalent to the class of lower layers in R2 studied by
Dudley (1999), section 12.4, that fails to be Donsker. When q is restricted to C1,M (SX ) rather
than C1+δ,M (SX ), the corresponding class is not Donsker either (Dudley, 1999, section 12.4).
Local linear rather than Nadaraya–Watson smoothing. Rather than a local linear
smoother in the x-direction,
one may be tempted to consider a simpler Nadaray–Watson
R yj
f˜n,j (y|x) dy, where
smoother F̃n,j (yj |x) = −∞
f˜n,j (yj |x) =

Pn

i=1 Khn,1 (x − Xi ) Lhn,2 (yj
Pn
i=1 Khn,1 (x − Xi )

− Yij )

.

However, the Nadaraya–Watson density estimator f˜n,j behaves poorly at the boundary points
of the support SX , since the true marginal density fX of X is not continuous at those boundary
points in view of the assumption inf{fX (x) : x ∈ SX } ≥ b > 0 (Fan, 1992). In particular, the
uniform rates given in Proposition 4.2 are no longer available, making our approach incompatible
with such a Nadaraya–Watson type estimator. Some techniques to bound the metric entropy
of such estimators are investigated in Portier (2016).
Extensions. The extension of Theorem 3.1 to the whole square unit [0, 1]2 might be obtained
by verifying “high-level” conditions given in Theorem 2 and Corollary 1 in Gijbels et al. (2015a).
We believe this is not straightforward and needs further research.
The extension to multivariate covariates X is technical and perhaps even impossible as some
conflicts between the bias and the variance might arise when choosing the bandwidths hn,1 and
hn,2 in (G4). As noted by a Referee, using higher-order local polynomial estimators would result
in lower bias (under appropriate smoothness conditions) and would (in theory) allow to extend
the results presented here to higher-dimensional covariates. Still, completely nonparametric
estimation would remain infeasible in high dimensions due to the curse of dimensionality in
nonparametric smoothing.
The extension to vectors Y of arbitrary dimension is probably feasible, but we did not
pursue this in view of the motivation from pair-copula constructions.
As suggested by a Referee, an alternative approach to estimating the conditional marginal
quantiles could be to use (nonparametric) quantile regression directly instead of inverting conditional distribution functions.

10

4

Auxiliary results on the smoothed local linear estimator of
the conditional distribution function

This section contains asymptotic results on the smoothed local linear estimator of the conditional
distribution function introduced in Section 3.1.
The presentation is independent from the copula set-up introduced before, so that the index
j ∈ {1, 2} will be omitted in this section. Moreover, when possible, we provide weaker assumptions on the bandwidth sequences than the ones introduced in Section 3. The proofs of the
results are given in Appendix C.2.
The main difficulty is to show that the estimator (x, y) 7→ F̂n (y|x) introduced in (3.2) satisfies
(H3). Our approach relies on bounds on the uniform convergence rates of the estimator and its
partial derivatives. Exact rates of uniform strong consistency for Nadaraya–Watson estimators
of the conditional distribution function are given in Einmahl and Mason (2000) and Härdle
et al. (1988), among others. Strong consistency of derivatives of the Nadaraya–Watson and
Gasser–Müller estimators of the regression function are studied in Akritas and Van Keilegom
(2001) and Gasser and Müller (1984), respectively. References on strong uniform rates for local
linear estimators include Masry (1996) and Dony et al. (2006).
Assume that (X1 , Y1 ), . . . , (Xn , Yn ) are independent and identically distributed random vectors with common distribution equal to the law, P , of the random vector (X, Y ) valued in R2 .
Assume that P has a density fX,Y with respect to the Lebesgue measure. As before, let fX
and SX = {x ∈ R : fX (x) > 0} denote the density and the support of X, respectively. The
conditional distribution function of Y given X = x is given by
Z y
fX,Y (x, z)
F (y|x) =
dz,
y ∈ R, x ∈ SX .
fX (x)
−∞
(G1’) The law P of (X, Y ) verifies (G1), i.e., the function F ( · | · ) replaces Fj ( · | · ).
(G4’) The bandwidth sequences hn,1 > 0 and hn,2 > 0 satisfy, as n → ∞,
hn,1 → 0,

nh3n,1
→ ∞,
|log hn,1 |

hn,2 → 0,

nhn,1 hn,2
→ ∞.
|log hn,1 hn,2 |

The quantity F̂n (y|x) can be found as the solution of a linear system of equations derived
from (3.3). With probability going to 1, a closed formula is available for F̂n (y|x). Similar
expressions are available for the local linear estimator of the regression function (Fan and
Gijbels, 1996, page 55, equation (3.5)). For every k ∈ N, define
p̂n,k (x) = n

−1

Q̂n,k (y, x) = n−1

n
X
i=1
n
X

wk,hn,1 (x − Xi ),
ϕhn,2 (y, Yi ) wk,hn,1 (x − Xi ),

i=1

where wk,h ( · ) = h−1 wk ( · /h) and wk (u) = uk K(u). In what follows, the function wk plays the
role of a kernel for smoothing in the x-direction. Let P denote the probability measure on the
probability space carrying the sequence of random pairs (X1 , Y1 ), (X2 , Y2 ), . . ..
Lemma 4.1. Let (X1 , Y1 ), (X2 , Y2 ), . . . be independent random vectors with common law P .
Assume that (G1’), (G3) and (G4’) hold. For some c > 0 depending on K, we have

lim P p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2 ≥ b2 c for all x ∈ SX = 1.
(4.1)
n→∞

11

Consequently, with probability going to 1, we have
F̂n (y|x) =

Q̂n,0 (y, x) p̂n,2 (x) − Q̂n,1 (y, x) p̂n,1 (x)
,
p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2

y ∈ R, x ∈ SX .

(4.2)

Differentiating F̂n (y|x) in (4.2) with respect to y, we obtain fˆn (y|x), the estimated conditional density of Y given X = x. It is given by
q̂n,0 (y, x) p̂n,2 (x) − q̂n,1 (y, x) p̂n,1 (x)
,
fˆn (y|x) =
p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2

(4.3)

where, for every k ∈ N,
q̂n,k (y, x) = n

−1

n
X

Lhn,1 (y − Yi ) wk,hn,2 (x − Xi ).

i=1

To derive the uniform rates of convergence of the smoothed local linear estimator F̂n (y|x) and its
derivatives, formula (4.2) permits to work with the quantities Q̂n,k (y, x), k ∈ {0, 1}, and p̂n,k (x),
k ∈ {0, 1, 2}. The asymptotic behavior of the latter quantities is handled using empirical process
theory in Proposition C.1. For ease of writing, we abbreviate the partial differential operator
∂ l /∂ l x to ∂xl .
Proposition 4.2. Let (X1 , Y1 ), (X2 , Y2 ), . . . be independent random vectors with common law
P . Assume that (G1’), (G3) and (G4’) hold. We have, as n → ∞,
s
!
|log hn,1 |
2
2
sup
F̂n (y|x) − F (y|x) = OP
+ hn,1 + hn,2 ,
nhn,1
x∈SX , y∈R
!
s
|log hn,1 |
2
sup
∂x {F̂n (y|x) − F (y|x)} = OP
+ hn,1 + h−1
n,1 hn,2 ,
nh3n,1
x∈SX , y∈R
!
s
|log
h
|
n,1
2
sup
∂x2 {F̂n (y|x) − F (y|x)} = OP
+ 1 + h−2
n,1 hn,2 ,
nh5n,1
x∈SX , y∈R
s
!
|log hn,1 hn,2 |
2
2
∂y {F̂n (y|x) − F (y|x)} = OP
sup
+ hn,1 + hn,2 ,
nhn,1 hn,2
x∈SX , y∈R
!
s
|log
h
h
|
n,1
n,2
1+δ
sup
∂y2 {F̂n (y|x) − F (y|x)} = OP
,
+ h1+δ
n,1 + hn,2
nhn,1 h3n,2
x∈SX , y∈R
!
s
|log hn,1 hn,2 |
sup
∂y ∂x {F̂n (y|x) − F (y|x)} = OP
+ h2n,2 h−1
n,1 + hn,1 .
nh3n,1 hn,2
x∈SX , y∈R
For the sake of brevity, the previous proposition is stated under (G4’) although the first
assertion remains true under the weaker condition hn,1 + hn,2 → 0 and nhn,1 /|log hn,1 | → ∞.
Recall the generalized inverse in (2.3). Uniform convergence rates for the estimated conditional quantile function Fn− (u|x) are provided in the following proposition.
Proposition 4.3. Let (X1 , Y1 ), (X2 , Y2 ), . . . be independent random vectors with common law
P . Assume that (G1’), (G3), (G4’) hold. For any γ ∈ (0, 1/2), we have, as n → ∞,
s
!

|log
h
|
n,1
sup
F F̂n− (u|x)|x − u = OP
+ h2n,1 + h2n,2 ,
nhn,1
x∈SX , u∈[γ,1−γ]
s
!
|log hn,1 |
−
−
2
2
sup
F̂n (u|x) − F (u|x) = OP
+ hn,1 + hn,2 .
nhn,1
x∈SX , u∈[γ,1−γ]
12

The convergence rates in Propositions 4.2 and 4.3 serve to show that the estimated quantile
functions x 7→ F̂n− (u|x), as u varies in [γ, 1 − γ], belong to a certain Hölder regularity class. The
bandwidths hn,1 and hn,2 are required to be large enough.
(G4”) There exists α > 0 such that the bandwidth sequences hn,1 > 0 and hn,2 > 0 satisfy, as
n → ∞,
hn,1 → 0,
nh3+α
n,1
|log hn,1 |

hn,2 → 0,

−1−α/2 2
hn,2

hn,1

nh1+α
n,1 hn,2

→ ∞,

|log hn,1 hn,2 |

→ 0,

→ ∞.

Note that in the case that hn,1 = hn,2 , the previous condition becomes hn,1 → 0 and
nh3+α
n,1 / |log hn,1 | → ∞. Recall the function class Ck+δ,M (S) defined via the Hölder condition (2.5).
Proposition 4.4. Let (X1 , Y1 ), (X2 , Y2 ), . . . be independent random vectors with common law
P . Assume that (G1’), (G3) and (G4”) hold. For any γ ∈ (0, 1/2), we have
h
i
lim P x 7→ F̂n− (u|x) : u ∈ [γ, 1 − γ] ⊂ C1+δ1 ,M1 (SX ) = 1,
n→∞

where δ1 = min(α/2, δ) and where M1 > 0 depends only on bγ and M .

Appendix A

Proof of Theorem 2.1

Condition (G2) implies the existence and continuity of Ċj on {u ∈ [0, 1]2 : 0 < uj < 1}, for
(or)
j ∈ {1, 2}. As a consequence, the oracle empirical process n1/2 (Ĉn − C) converges weakly in
`∞ ([0, 1]2 ) to the tight Gaussian process C (Segers, 2012). The proof of Theorem 2.1 therefore
consists of showing equation (2.6).
P
We use notation from empirical process theory. Let Pn = n−1 ni=1 δ(Xi ,YiR) denote the
empirical measure. For a function f and a probability measure Q, write Qf = f dQ. The
empirical process is
Gn = n1/2 (Pn − P ).
For any pair of cumulative distribution functions F1 and F2 on R, put F (y) = (F1 (y1 ), F2 (y2 ))
for y = (y1 , y2 ) ∈ R2 and F − (u) = (F1− (u1 ), F2− (u2 )) for u = (u1 , u2 ) ∈ [0, 1]2 , the generalized
inverse being defined in (2.3).
Our proof follows from an application of Theorem 2.1 stated in van der Vaart and Wellner
(2007) and reported below; for a proof see for instance van der Vaart and Wellner (1996,
Lemma 3.3.5), noting that the conclusion of their proof is in fact stronger than what is claimed
in their statement. Let ξ1 , ξ2 , . . . be independent and identically distributed random elements
of a measurable space (X , A) and with common distribution equal to P . Let P denote the
probability measure on the probability space on which the sequence ξ1 , ξ2 , . . . is defined. Let
Gξ,n be the empirical process associated to the sample ξ1 , . . . , ξn . Let E and U be sets and let
{mu,η : u ∈ U, η ∈ E} be a collection of real-valued, measurable functions on X .
Theorem A.1 (Theorem 2.1 in van der Vaart and Wellner (2007)). Let η̂n be random elements
in E. Suppose there exist η0 ∈ E and E0 ⊂ E such that the following three conditions hold:
2
(i) supu∈U P mu,η̂n − mu,η0 = oP (1) as n → ∞;
13

(ii) P(η̂n ∈ E0 ) → 1 as n → ∞;
(iii) {mu,η − mu,η0 : u ∈ U, η ∈ E0 } is P -Donsker.
Then it holds that
sup |Gξ,n (mu,η̂n − mu,η0 )| = oP (1),

n → ∞.

u∈U

The empirical process notation allows us to write
Ĉn(or) (u) = Pn 1{F ≤ Ĝ(or)− (u)} .

,
Ĉn (u) = Pn 1{F̂n ≤ Ĝ−
n (u)}

n

To establish (2.6), we rely on the decomposition

n1/2 Ĉn (u) − Ĉn(or) (u)
o
o
n
n
1/2
−
1
−
1
+
n
P
1
= Gn 1{F̂n ≤ Ĝ−
−
(or)−
(or)−
{F̂n ≤ Ĝn (u)}
(u)}
(u)}
{F ≤ Ĝ
{F ≤ Ĝ
n (u)}
n

n

= Ân,1 (u) + Ân,2 (u).
Let γ ∈ (0, 1/2). The proof consists in showing that the empirical process term Ân,1 (u) goes to
zero, uniformly over u ∈ [γ, 1 − γ]2 , in probability (first step) and that the bias term Ân,2 (u)
goes to zero, uniformly over u ∈ [γ, 1 − γ]2 , in probability (second step). The simplifying
assumption (2.1) is crucial for treating the bias term in the second step. But before executing
this program, it is useful to obtain some results on F̂n,j and Ĝn,j , j = 1, 2 (preliminary steps).
Preliminary step: We establish some preliminary results on F̂n,j and Ĝn,j , j = 1, 2, that we
list as facts in the following.
Fact 1. For any j = 1, 2,
sup
uj ∈[γ,1−γ], x∈SX


−
Fj F̂n,j
(uj |x)|x − uj = oP (n−1/4 ).

Proof. Because of (H1), invoking Lemma D.3, it holds that with probability going to one,

−
uj = F̂nj F̂n,j
(uj |x)|x ,
for each x ∈ SX and uj ∈ [γ, 1 − γ]. It follows that



−
−
−
Fj F̂n,j
(uj |x)|x − uj = Fj F̂n,j
(uj |x)|x − F̂n,j F̂n,j
(uj |x)|x
≤

sup

Fj (yj |x) − F̂n,j (yj |x) ,

x∈SX , yj ∈R

which is oP (n−1/4 ) by (H2).
A consequence of Fact 1 is that
inf


−
Fj F̂n,j
(u|x)|x = oP (1) + γ,

sup


−
Fj F̂n,j
(u|x)|x = oP (1) + 1 − γ.

x∈SX , uj ∈[γ,1−γ]
x∈SX , uj ∈[γ,1−γ]

14

Hence, the sequence of events
En,j,1 = {∀x ∈ SX , ∀uj ∈ [γ, 1 − γ]


−
γ/2 ≤ Fj F̂n,j
(uj |x)|x ≤ 1 − γ/2},

:

has probability going to one. From (H2) we have that, with probability going to 1, for any
x ∈ SX and yj ≥ Fj− (1 − γ/2|x), it holds that F̂n,j (yj |x) ≥ 1 − γ. Hence the sequence of events
En,j,2 = {∀x ∈ SX

:

inf

yj ≥Fj− (1−γ/2|x)

F̂n,j (yj |x) ≥ 1 − γ},

has probability going to one as well.
Fact 2. On a sequence of events whose probabilities tend to one, it holds that for every uj ∈
[γ, 1 − γ] and every (yj , x) ∈ R × SX ,
−
F̂n,j (yj |x) ≤ uj ⇔ yj ≤ F̂n,j
(uj |x).

Proof. The sense ⇐ is an easy consequence of (H1): because of the continuity of yj 7→ F̂n,j (yj |x)
we can apply Lemma D.3 to obtain the implication. The converse direction “⇒” requires the
fact that F̂n,j ( · |x) is strictly increasing on [Fj− (γ|x), Fj− (1 − γ|x)], which is given by (H1).
−
Assume that uj ∈ [γ, 1 − γ] and (yj , x) ∈ R × SX are such that F̂n,j
(uj |x) < yj . Consider two
−
cases, according to whether yj is smaller than Fj (1 − γ/2|x) or not.
• On the one hand, suppose that yj < Fj− (1 − γ/2|x). We have, under En,j,1 , that
−
Fj− (γ/2|x) ≤ F̂n,j
(uj |x). Hence we get the chain of inequalities
−
Fj− (γ/2|x) ≤ F̂n,j
(uj |x) < yj < Fj− (1 − γ/2|x).

Using the monotonicity of F̂n,j (·|x) on [Fj− (γ/2|x), Fj− (1 − γ/2|x)] and Lemma D.3, we
find that uj < F̂n,j (yj |x).
• On the other hand, suppose that Fj− (1 − γ/2|x) ≤ yj , or equivalently that 1 − γ/2 ≤
Fj (yj |x). Under En,j,2 , it then holds that uj ≤ 1 − γ < F̂n,j (yj |x).
Fact 3. We have
sup
uj ∈[γ,1−γ]

(or)

Ĝn,j (uj ) − Ĝn,j (uj ) = oP (n−1/4 ).

Proof. From Fact 2, it holds that, on a sequence of events whose probabilities tend to one, with
a slight abuse of notation,
(or)

Ĝn,j (uj ) − Ĝn,j (uj )
n
= n−1/2 Gn 1{Y ≤ F̂ −

n,j

(uj |X)} − 1{Y

o
≤ Fj− (uj |X)}

n
o
+ P 1{F̂n,j ≤ uj } − 1{Fj ≤ uj } .

We apply Theorem A.1 with ξi = (Yi,j , Xi ), X = R × SX , U = [γ, 1 − γ] and E the space of
measurable functions valued in R and defined on [γ, 1 − γ] × SX . Moreover, the quantities η0 ,
η̂n , and the map muj ,η are given by, for every uj ∈ [γ, 1 − γ] and x ∈ SX ,
η0 (uj , x) = Fj− (uj |x),
−
η̂n (uj , x) = F̂n,j
(uj |x),

muj ,η (y, x) = 1{y ≤ η(uj ,x)} .
15

Finally, the space E0 is the collection of those elements η in E such that
{x 7→ η(uj , x) : uj ∈ [γ, 1 − γ]} ⊂ C1+δ1 ,M1 (SX ).
The verification of the three assumptions in Theorem A.1 is as follows:
• First, we show point (i). Recall that if the random variable U is uniformly distributed on
(0, 1), then E(1{U ≤u1 } − 1{U ≤u2 } )2 = |u1 − u2 |. We have
Z
2
1{y ≤ F̂ − (uj |x)} − 1{y ≤ F − (uj |x)} fX,Y (x, y) d(x, y)
j
n,j
Z

−
=
F F̂n,j
(uj |x)|x − uj fX (x) dx
≤

sup
uj ∈[γ,1−γ], x∈SX


−
F F̂n,j
(uj |x)|x − uj ,

which, by Fact 1, tends to zero in probability.
• Second, point (ii) is directly obtained invoking (H3).
• Third, point (iii) follows from the existence of δ2 > 0 and M2 > 0 such that
{x 7→ Fj− (uj |x) : uj ∈ [γ, 1 − γ]} ⊂ C1+δ2 ,M2 (SX ).

(A.1)

The inclusion is indeed implied by the formula
∂x Fj− (uj |x) = −

∂x Fj (yj |x)
fj (yj |x)

,
y=Fj− (uj |x)

which, by (G1), is bounded by M/bγ . Then, based on (G1), we easily obtain that the
function x 7→ ∂x Fj− (uj |x) is δ-Hölder with Hölder constant depending only on bγ and M
(for more details, the reader is invited to read the proof of Proposition 4.4). It remains
to note that under (G1), the set SX is bounded and convex, implying that for any δ > 0
and M > 0, the class of subgraphs of C1+δ,M (SX ) is Donsker (van der Vaart and Wellner,
1996, Corollary 2.7.5). As the difference of two Donsker classes remains Donsker (van der
Vaart and Wellner, 1996, Example 2.10.7), we obtain point (iii).
Consequently, we have shown that
Z n
o

(or)
−
Ĝn,j (uj ) − Ĝn,j (uj ) =
F F̂n,j
(uj |x)|x − uj fX (x) dx + oP (n−1/2 ).
Conclude invoking Fact 1.
Fact 4. We have
sup
uj ∈[γ,1−γ]

(or)−

−1/4
Ĝ−
).
n,j (uj ) − Ĝn,j (uj ) = oP (n

(or)

Proof. By Fact 3, the supremum distance n = supuj ∈[γ,1−γ] |Ĝn,j (uj ) − Ĝn,j (uj )| converges to
zero in probability. We work on the event {n < γ}, the probability of which tends to 1. By
Lemma D.4, we have, for uj ∈ [γ, 1 − γ],
(or)−

Ĝ−
n,j (uj ) − Ĝn,j (uj )
(or)−

(or)−

(or)−

(or)−

≤ Ĝn,j ((uj − n ) ∨ 0) − Ĝn,j (uj ) ∨ Ĝn,j ((uj + n ) ∧ 1) − Ĝn,j (uj ) ,
16

with a ∨ b = max(a, b). In terms of Wn (uj ) =
(or)−

(or)−

√

(or)−

n{Ĝn,j (uj ) − uj }, we have

(or)−

(or)−

Ĝn,j ((uj − n ) ∨ 0) − Ĝn,j (uj ) ∨ Ĝn,j ((uj + n ) ∧ 1) − Ĝn,j (uj )
≤ n + 2 sup |Wn (uj )|.
uj ∈[0,1]

From Fact 3 and Lemma D.2, we get the desired rate oP (n−1/4 ).
For uj ∈ [γ, 1 − γ], x ∈ SX , and j ∈ {1, 2}, define
 
ˆ n,j (uj |x) = Fj F̂ − Ĝ− (uj )| x | x − Ĝ(or)− (uj ).
∆
n,j
n,j
n,j

(A.2)

Fact 5. We have
sup
uj ∈[γ,1−γ], x∈SX

ˆ n,j (uj |x)| = oP (n−1/4 ).
|∆

Proof. Write
 
ˆ n (uj |x) = F F̂ − Ĝ− (uj )| x | x − Ĝ− (uj ) + Ĝ− (uj ) − Ĝ(or)− (uj ) .
∆
n
n,j
n,j
n,j
n,j
By Fact 4, we only need to treat the first term on the right-hand side. Using the fact that
(or)−
(or)−
supuj ∈[γ,1−γ] |Ĝ−
n,j (uj )− Ĝn,j (uj )| = oP (1) and the fact that supuj ∈(0,1] |Ĝn,j (uj )−uj | = oP (1),
which is a consequence of Lemma D.2, we know that Ĝ−
n,j (uj ) takes values in [γ/2, 1 − γ/2] with
ˆ n (uj |x) = oP (n−1/4 ).
probability going to 1. Then we use Fact 1 to conclude that ∆
First step:

We show that
sup
u∈[γ,1−γ]2

Ân,1 (u) = oP (1),

By Fact 2, it holds that (with a slight abuse of notation)
n
Ân,1 (u) = Gn 1{Y ≤ F̂n− (Ĝ−
− 1{Y
n (u)|X)}

n → ∞.

o
(or)−
≤ F − (Ĝn
(u)|X)}

.

Therefore we apply Theorem A.1 with ξi = (Xi , Yi ), X = SX × R2 , U = [γ, 1 − γ]2 and E
the space of measurable functions valued in R4 and defined on SX × [γ, 1 − γ]2 . Moreover, the
quantities η0 and η̂n are given by, for every u ∈ [γ, 1 − γ]2 and x ∈ SX ,

η0 (u, x) = F − (u|x), F − (u|x) ,

 − (or)−

η̂n (u, x) = F̂n− Ĝ−
Ĝn
(u)|x .
n (u)|x , F
Identifying u ∈ U with u ∈ [γ, 1 − γ]2 and η ∈ E with (η1 , η2 ), where ηj , j ∈ {1, 2}, are valued
in R2 , the map mu,η : R2 × SX → R is given by
mu,η (y, x) = 1{y ≤ η1 (u,x)} − 1{y ≤ η2 (u,x)} ,
Finally, the space E0 is the collection of those elements η = (η1 , η2 ) in E such that
2
{x 7→ η1 (u, x) : u ∈ [γ, 1 − γ]2 } ⊂ C1+δ1 ,M1 (SX ) ,
2
{x 7→ η2 (u, x) : u ∈ [γ, 1 − γ]2 } ⊂ C1+δ,M2 (SX ) ,
17

where M2 depends only on bγ and M . In the following we check each condition of Theorem A.1.
Verification of Condition (i) in Theorem A.1. Because the indicator function is bounded by
1, we have
Z
2
1{y ≤ F̂n− (Ĝ−
−
1
fX,Y (x, y) d(x, y)
(or)−
−
{y ≤ F (Ĝ
(u)|x)}
n (u)|x)}
n

≤

2
X

2

Z
sup

j=1 uj ∈[γ,1−γ]

1{yj ≤ F̂ −

−
n,j (Ĝn,j (uj )|x)}

− 1{y

(or)−
−
(uj )|x)}
j ≤ Fj (Ĝn,j

fX,Yj (x, yj ) d(x, yj ),

so that we can focus on each margin separately. Recall that if the random variable U is
uniformly distributed on (0, 1), then E(1{U ≤u1 } − 1{U ≤u2 } )2 = |u1 − u2 |. Writing ân,x (uj ) =

−
F̂n,j
Ĝ−
n,j (uj )|x , and using (2.1) and (2.2), we have
Z
2
1{yj ≤ ân,x (uj )} − 1{y ≤ F − (Ĝ(or)− (u )|x)} fX,Yj (x, yj ) d(x, yj )
j

j

j

n,j

2

Z
1{Fj (yj |x) ≤ Fj (ân,x (uj )|x)} − 1{F

=

(or)−
(uj )}
j (yj |x) ≤ Ĝn,j

Z
=
ZSX
=

fX,Yj (x, yj ) d(x, yj )

(or)−

Fj (ân,x (uj )|x) − Ĝn,j (uj ) fX (x) dx
ˆ n,j (uj |x) fX (x) dx,
∆

SX

ˆ n,j has been defined in (A.2). Fact 5, demonstrated during the preliminary step, permits
where ∆
to conclude.
Verification of Condition (ii) in Theorem A.1. We establish that, for each j = 1, 2,
n
o

−
x 7→ F̂n,j
Ĝ−
(u
)|x
:
u
∈
[γ,
1
−
γ]
⊂ C1+δ1 ,M1 (SX ),
j
n,j j
n
o

(or)−
x 7→ Fj− Ĝn,j (uj )|x : uj ∈ [γ, 1 − γ] ⊂ C1+δ,M2 (SX ),
with probability going to 1.
• For the first inclusion, we have already shown in the proof of Fact 5 that Ĝ−
n,j (uj ) ∈
[γ/2, 1 − γ/2], for every uj ∈ [γ, 1 − γ], with probability going to 1. On this set, we have



−
−
Ĝ−
x 7→ F̂n,j
n,j (uj )|x : uj ∈ [γ, 1 − γ] ⊂ x 7→ F̂n,j (uj |x) : uj ∈ [γ/2, 1 − γ/2] ,
which is included in C1+δ1 ,M1 (SX ) by (H3).
(or)−
• For the second inclusion, by Lemma D.2, for every uj ∈ [γ, 1−γ], it holds that Ĝn,j (uj ) ∈
[γ/2, 1 − γ/2] with probability going to one. It follows that



(or)−
x 7→ Fj− Ĝn,j (uj )|x : uj ∈ [γ, 1 − γ] ⊂ x 7→ Fj− (uj |x) : uj ∈ [γ/2, 1 − γ/2] ,
which is included in C1+δ,M2 (SX ) by (A.1), for some M2 > 0.
Verification of Condition (iii) in Theorem A.1. It is enough to show that the class of
functions
n
2
2 o
1{y ≤ g1 (x)} − 1{y ≤ g2 (x)} : (g1 , g2 ) ∈ C1+δ1 ,M1 (SX ) × C1+δ,M2 (SX )
is P -Donsker. Since the sum and the product of two bounded Donsker classes is Donsker
(van der Vaart and Wellner, 1996, Example 2.10.8), we can focus on the class

1{y ≤ g(x)} : g ∈ C1+δ,M (SX ) .
For any δ > 0 and M > 0, the latter is Donsker since the class of subgraphs of C1+δ,M (SX ),
under (G1), has a sufficiently small entropy (van der Vaart and Wellner, 1996, Corollary 2.7.5).
18

Second step:

We show that
sup
u∈[γ,1−γ]2

n → ∞.

Ân,2 (u) = oP (1),

Because of the simplifying assumption (2.1), we have, for every u ∈ [0, 1]2 , the formula
Z
Z
 
1{y ≤ F̂n− (Ĝ−
fX,Y (x, y) d(x, y) = C F F̂n− Ĝ−
n (u)|x | x fX (x) dx.
n (u)|x)}

(A.3)

(or)−

Next we use (G2) to expand C in the previous expression around Ĝn
(u) and then we
ˆ n,j (uj |x), j = 1, 2, established in Fact 5.
conclude by using the rates for the quantities ∆
In the light of the first step, we have
o
n
= oP (1).
−
1
sup
n1/2 {Ĉn (u) − Ĉn(or) (u)} − n1/2 P 1{F̂n ≤ Ĝ−
(or)−
(u)}
{F ≤ Ĝ
n (u)}
n

u∈[γ,1−γ]2



(or)−
(or)
(or)
Moreover, because Ĉn (u1 , 1) = Ĝn,1 Ĝ−
n,1 (u1 ) and Ĉn (u1 , 1) = Ĝn,1 Ĝn,1 (u1 ) , we find
sup

Ĉn (u1 , 1) − u1 = O(n−1 )

and

u1 ∈[γ,1−γ]

sup
u1 ∈[γ,1−γ]

Ĉn(or) (u1 , 1) − u1 = O(n−1 ),

which implies, by the triangle inequality, that
sup
u1 ∈[γ,1−γ]

n1/2 {Ĉn (u1 , 1) − Ĉn(or) (u1 , 1)} = o(1).

Similarly, we find
sup
u2 ∈[γ,1−γ]

n1/2 {Ĉn (1, u2 ) − Ĉn(or) (1, u2 )} = o(1).

Bringing all these facts together yields, for any j = 1, 2,

1/2
n P 1{F̂n,j ≤ Ĝ− (uj )} − 1{F
sup
uj ∈[γ,1−γ]

j

n,j


(or)−

≤ Ĝn,j

(uj )}

= oP (1).

ˆ n,j (uj |x) given in equation (A.2), it follows that, for any j = 1, 2,
Hence, from the definition of ∆
Z
ˆ n,j (uj |x) fX (x) dx = oP (1),
sup
n1/2 ∆
(A.4)
uj ∈[γ,1−γ]

Next define
ˆ n,1 (u1 |x) Ċ1 (u) + ∆
ˆ n,2 (u2 |x) Ċ2 (u),
Ŵn (u|x) = ∆
Z
Ŵn (u) = Ŵn (u|x) fX (x) dx.
√
By (A.4), it holds that supu∈[γ,1−γ]2 | n Ŵn (u)| = oP (1). Then because of (A.3) and the
simplifying assumption (2.1), we have
Ân,2 (u) − Ŵn (u)
Z n
o
 
 
1/2
−
(or)−
=n
H F̂n− Ĝ−
(u)|x
|x
−
H
F
Ĝ
(u)|x
|x
−
Ŵ
(u|X)
fX (x) dx
n
n
n
SX
Z n 


o
 
1/2
=n
C F F̂n− Ĝ−
− C Ĝ(or)−
(u) − Ŵn (u|x) fX (x) dx.
n (u)|x |x
n
SX

19

The term inside the second integral equals




ˆ n (u|x) − C Ĝ(or)− (u) − Ŵn (u|x),
C Ĝ(or)−
(u)
+
∆
n
n
ˆ n (u|x) = (∆
ˆ n,1 (u1 |x), ∆
ˆ n,2 (u2 |x)). Then, by Lemma D.1 [which we can apply since
with ∆
(or)−
Ĝn
(u) lies in the interior of the unit square], we have, for all u ∈ [γ, 1 − γ]2 and all x ∈ SX ,




2
ˆ n (u|x) − C Ĝ(or)− (u) − Ŵn (u|x) ≤ 4κ ∆
ˆ n (u|x) .
C Ĝ(or)−
(u)
+
∆
n
n
γ
Integrating out over x ∈ SX yields
Ân,2 (u) − Ŵn (u) ≤ n

1/2

≤ n1/2

4κ
γ
4κ
γ

Z

ˆ n (u|x)
∆

2

fX (x) dx

SX
2

sup

ˆ n (u|x) ,
∆

u∈[γ,1−γ]2 , x∈SX

which is oP (1) by Fact 5.

Appendix B

Proof of Theorem 3.1

The proof follows from an application of Theorem 2.1. We need to show that (H1), (H2)
and (H3) are valid. The last two assertions are direct consequences of Proposition 4.2 and
Proposition 4.4, respectively. Those propositions are stated in section 4.
Hence, we only need to show that (H1) holds. By Lemma 4.1, there exists c > 0 depending
on K such that the events



2
2
En,3 = inf p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x) ≥ b c ,
x∈SX

have probability going to 1. Under En,3 , equations (4.2) and (4.3) hold, i.e.,
Q̂n,j,0 (yj , x) p̂n,2 (x) − Q̂n,j,1 (yj , x) p̂n,1 (x)
,
p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2
q̂n,j,0 (yj , x) p̂n,2 (x) − q̂n,j,1 (yj , x) p̂n,1 (x)
,
fˆn,j (yj |x) =
p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2
F̂n,j (yj |x) =

with fˆn,j (yj |x) = ∂yj F̂n,j (yj |x), and, for every k ∈ N,
q̂n,j,k (yj , x) = n

−1

n
X

Lhn,2 (yj − Yij ) wk,hn,1 (x − Xi ),

i=1

Z

yj

Q̂n,j,k (yj , x) =

q̂n,j,k (t, x) dt.
−∞

Hence, on the event En,3 , yj 7→ F̂n,j (yj |x) is continuous and differentiable on R. By the uniform
convergence of fˆn,j , stated in Proposition 4.2, and using (G1), we get for all x ∈ SX and all
yj ∈ [Fj− (γ|x), Fj− (1 − γ|x)] that
fˆn,j (yj |x) ≥ fj (yj |x) −

sup
x∈SX , yj ∈R

≥ b − oP (1).
20

fˆn,j (yj |x) − fj (yj |x)

Hence the event
(
En,4 =

)
min

inf

inf

j∈{1,2} x∈SX yj ∈[F − (γ|x),F − (1−γ|x)]
j
j

fˆn,j (yj |x) ≥ b/2 .

has probability going to 1. Conclude by noting that under En,4 , for every x ∈ SX , the function
Fn,j ( · |x) is strictly increasing on [Fj− (γ|x), Fj− (1 − γ|x)].

Appendix C
C.1

Proofs of the auxiliary results on the smoothed
local linear estimator

Uniform convergence rates for kernel estimators

Our analysis of the smoothed local linear estimator relies on a result on the concentration around
their expectations of certain kernel estimators. The result follows from empirical process theory
and notably by the use of some version of Talagrand’s inequality (Talagrand, 1994), formulated
in Giné and Guillou (2002). For stronger results including exact almost-sure convergence rates,
see Einmahl and Mason (2000).
Proposition C.1. Let (Xi , Yi ), i = 1, 2, . . . , be independent copies of a bivariate random vector
(X, Y ). Assume that (X, Y ) has a bounded density fX,Y and that K and L are bounded real
functions of bounded variation that vanish outside [−1, 1]. For any sequences hn,1 → 0 and
hn,2 → 0 such that nhn,1 / |log hn,1 | → ∞ and nhn,1 hn,2 / |log hn,1 hn,2 | → ∞, we have, as n → ∞,
s
!
n
X

|log hn,1 |
−1
sup n
Khn,1 (x − Xi ) − E[Khn,1 (x − X)] = OP
,
nhn,1
x∈R
i=1
s
!
n
X

|log
h
h
|
n,1 n,2
sup n−1
Lhn,2 (y − Yi ) Khn,1 (x − Xi ) − E[Lhn,2 (y − Y ) Khn,1 (x − X)] = OP
,
nh
hn,2
2
n,1
(x,y)∈R
i=1
s
!
n
X


|log
h
h
|
n,1
n,2
,
sup n−1
Lhn,2 (y − Yi ) − E[Lhn,2 (y − Yi ) | Xi ] Khn,1 (x − Xi ) = OP
nhn,1 hn,2
(x,y)∈R2
i=1
s
!
n
X

|log hn,1 |
−1
sup n
,
ϕhn,2 (y, Yi ) Khn,1 (x − Xi ) − E[ϕhn,2 (y, Y ) Khn,1 (x − X)] = OP
nhn,1
(x,y)∈R2
i=1
s
!
n
X


|log
h
|
n,1
sup n−1
ϕhn,2 (y, Yi ) − E[ϕhn,2 (y, Yi ) | Xi ] Khn,1 (x − Xi ) = OP
,
nhn,1
(x,y)∈R2
i=1

where Kh ( · ) = h−1 K( · /h), Lh ( · ) = h−1 L( · /h), and ϕh (y, Yi ) =

Ry

−∞ Lh (t

− Yi ) dt.

Proof. We use the definition of bounded measurable VC classes given in Giné and Guillou
(2002). That is, call a class F of measurable functions a bounded measurable VC class if F is
separable or image admissible Suslin (Dudley, 1999, Section 5.3) and if there exist A > 0 and
v > 0 such that, for every probability measure Q, and every 0 <  < 1,
 v

A
,
N F, L2 (Q), kF kL2 (Q) ≤

where F is an envelope for F and N (T, d, ) denotes the -covering number of the metric space
(T, d) (van der Vaart and Wellner, 1996). [Other terminologies associated to the same concepts
21

are Euclidean classes (Nolan and Pollard, 1987) and Uniform entropy numbers (van der Vaart
and Wellner, 1996).] Using Nolan and Pollard (1987), Lemma 22, Assertion (ii), the classes
{z 7→ L(h−1 (y − z)) : y ∈ R, h > 0} and {z 7→ K(h−1 (x − z)) : x ∈ R, h > 0} are bounded
measurable VC [see Giné and Guillou (2002) for remarks and references on measurability issues
associated to the previous classes].
The first statement is a direct application of Theorem 2.1, Equation (2.2), in Giné and
Guillou (2002), with, in their notation,
n
o

Fn = z 7→ K h−1
(x
−
z)
:
x
∈
R
,
n,1
Z
σn2 = hn,1 kfX k∞ K(u)2 du,
U=

sup |K(u)|,
u∈[−1,1]

where fX denotes the density of X and k · k∞ is the supremum norm. The class Fn is included
in one of the bounded measurable VC classes given above. Moreover, we have
Z


2
Var K h−1
(x
−
X)
≤
K h−1
n,1
n,1 (x − z) fX (z) dz
Z
= hn,1 K(u)2 fX (x − hn,1 u) du
≤ σn2 .
The P
cited equation yields that the expectation of the supremum over x of the absolute value of
n
−1
n
i=1 Khn,1 (x − Xi ) − E[Khn,1 (x − X)] is bounded by some constant times

(nhn,1 )−1 |log(hn,1 )| + (nhn,1 )−1 |log(hn,1 )|

1/2

=O



(nhn,1 )−1 |log(hn,1 )|

1/2



.

Hence we have obtained the first assertion.
Preservation properties for VC classes (van der Vaart and Wellner, 1996, Lemma 2.6.18)
imply that the product class



−1
(z1 , z2 ) 7→ L h−1
2 (y − z2 ) K h1 (x − z1 ) : x ∈ R, y ∈ R, h1 > 0, h2 > 0
is bounded measurable VC. Then we can apply Theorem 2.1, Equation (2.2), in Giné and
Guillou (2002) in a similar fashion as before. The main difference lies in the variance term,
which follows from



−1
−1
Var L hn,2 (y − Y ) K hn,1 (x − X)
Z
2
2
−1
≤ L h−1
n,2 (y − z2 ) K hn,1 (x − z1 ) fX,Y (z1 , z2 ) d(z1 , z2 )
Z

= hn,1 hn,2 L(u2 )2 K(u1 )2 fX,Y x − hn,1 u1 , y − hn,2 u2 d(u1 , u2 )
Z
≤ hn,1 hn,2 kfX,Y k∞ L(u2 )2 K(u1 )2 d(u1 , u2 ) = σn2 .
Computing the bound leads to the second assertion.

22

Using the second statement and the triangle inequality, the third assertion is obtained whenever
sup

n−1

(x,y)∈R2

n
X


E[Lhn,2 (y − Yi ) | Xi ] Khn,1 (x − Xi ) − E[Lhn,2 (y − Y ) Khn,1 (x − X)]

i=1

s
= OP

|log hn,1 hn,2 |
nhn,1 hn,2

!
.

If the class {z 7→ E[L(h−1 (y − Y )) | X = z] : y ∈ R, h > 0} is a bounded measurable VC
−1
class of functions, then the class {z 7→ E[L(h−1
2 (y − Y )) | X = z] K(h1 (x − z)) : x ∈ R, y ∈
R, h1 > 0, h2 > 0} is still a bounded measurable VC class of functions (van der Vaart and
Wellner, 1996, Lemma 2.6.18). Consequently, we can apply Theorem 2.1, Equation (2.2) in
Giné and Guillou (2002), with the same σn2 as before, because, by Jensen’s inequality,
 
h 




2
2 i
−1
−1
−1
Var E L h−1
(y
−
Y
)
|
X
K
h
(x
−
X)
≤
E
E
L
h
(y
−
Y
)
|
X
K
h
(x
−
X)
n,2
n,1
n,2
n,1
h

2 i
2
−1
≤ E L h−1
(y
−
Y
)
K
h
(x
−
X)
n,2
n,1
≤ σn2 .
Now we show that L = {z 7→ E[L(h−1 (y − Y )) | X = z] : y ∈ R, h > 0} is a bounded
measurable VC class of functions. Let Q be a probability measure on SX . Define Q̃ as the
probability measure given by
Z
dQ̃(y) = f (y|x) dQ(x) dy.
Let f1 , . . . , fN denote the centers of an -covering of the class L0 = {z 7→ L(h−1 (y − z)) : y ∈
R, h > 0} with respect to the metric L2 (Q̃). For a function x 7→ E[f (Y ) | X = x], element of
the space F, there exists k ∈ {1, . . . , N } such that, by Jensen’s inequality and Fubini’s theorem,
Z
Z
 


 2


2
E f (Y ) | X = x − E fk (Y ) | X = x
dQ(x) ≤ E f (Y ) − fk (Y ) | X = x dQ(x)
Z Z

2
=
f (y) − fk (y) f (y|x) dy dQ(x)
Z

2
=
f (y) − fk (y) dQ̃(y) ≤ 2 .
Consequently,


N L, L2 (Q),  ≤ N L0 , L2 (Q̃),  .
Since the kernel L is bounded, there exists a positive constant c∞ that is an envelope for both
classes L and L0 . Note that kc∞ kL2 (Q) = c∞ for any probability measure Q. Using the fact
that L0 is bounded measurable VC (see the proof of the first assertion), it follows that
 v


A
,
N L, L2 (Q), c∞ ≤ N L0 , L2 (Q̃), c∞ ≤

i.e., the class L is bounded measurable VC.
In the same spirit, the fourth assertion holds true whenever the class {z 7→ ϕh (y, z) : y ∈
R, h > 0} is a bounded measurable VC class of functions. This is indeed true in virtue of
23

Lemma 22, Assertion (ii), in Nolan and Pollard (1987), because each function z 7→ ϕh (y, z) of
the class can be written as z 7→ L(h−1 (y − z)), where
Z t
Z t
Z t
+
L(t) =
L(u) du =
L (u) du +
L− (u) du,
−∞

−∞

−∞

with L = L+ + L− , L+ (u) ≥ 0, L− (u) < 0, which is indeed a bounded real function of
bounded variation (as the sum of an increasing function and a decreasing function is of bounded
variation). Applying Theorem 2.1, Equation (2.2) in Giné and Guillou (2002), with σn given
by the upper bound in the inequality chain
Z


2
Var ϕhn,2 (y, Y ) K h−1
(x
−
X)
≤
ϕhn,2 (y, z2 )2 K h−1
n,1
n,1 (x − z1 ) fX,Y (z1 , z2 ) d(z1 , z2 )
Z
≤ hn,1 sup {L(u)2 } kfX k∞ K(u)2 du = σn2 ,
u∈[−1,1]

leads to the same rate as in the first assertion.
Using the fourth statement and the triangle inequality, the fifth assertion is obtained whenever
sup
(x,y)∈R2

n−1

n
X


E[ϕhn,2 (y, Yi ) | Xi ] Khn,1 (x − Xi ) − E[ϕhn,2 (y, Y ) Khn,1 (x − X)]

i=1

s
= OP

|log hn,1 |
nhn,1

!
.

Following exactly the same lines as in the proof of the third statement [replacing L(h−1 (y − Y ))
by ϕh (y, Y )] we show that, since {z 7→ ϕh (y, z) : y ∈ R, h > 0} is a bounded measurable VC
class of functions (obtained in the proof of the fourth statement), the class {z 7→ E[ϕh (y, Y ) |
X = z] : y ∈ R, h > 0} is a bounded measurable VC class of functions. Then the class
{z 7→ E[ϕh2 (y, Y ) | X = z] K(h−1
1 (x − z)) : x ∈ R, y ∈ R, h1 > 0, h2 > 0} is still a bounded
measurable VC class of functions (van der Vaart and Wellner, 1996, Lemma 2.6.18). We conclude
by applying Theorem 2.1, Equation (2.2) in Giné and Guillou (2002), with the same σn2 as before;
note that Jensen’s inequality gives E[ϕhn,2 (y, Y ) | X = x]2 ≤ E[ϕhn,2 (y, Y )2 | X = x].

C.2

Proofs for Section 4

C.2.1

Proof of Lemma 4.1
P
Recall that p̂n,k (x) = n−1 ni=1 wk,hn,1 (x − Xi ) with wk,h ( · ) = h−1 wk ( · /h) and wk (u) =
uk K(u). We have
Z
E{p̂n,k (x)} =

fX (x − hn,1 u) wk (u) du.

By Condition (G3), the function K is a bounded function of bounded variation that vanishes
outside (−1, 1). Applying Proposition C.1, first statement, with kernel u 7→ wk (u) and using
Condition (G4), we have
sup |p̂n,k (x) − E{p̂n,k (x)}| = oP (1),

n → ∞.

x∈R

To obtain (4.1), we can rely on the expectations of all terms involved: it suffices to show that,
for some c > 0 and n sufficiently large, for all x ∈ SX ,
E{p̂n,0 (x)} E{p̂n,2 (x)} − [E{p̂n,1 (x)}]2 ≥ 2b2 c.
24

(C.1)

We have
Z

2

E{p̂n,0 (x)} E{p̂n,2 (x)} − E{p̂n,1 (x)} = E{p̂n,0 (x)}


fX (x − hn,1 u) u − an (x)

2

K(u) du,

with an (x) = E{p̂n,1 (x)}/ E{p̂n,0 (x)}. On the one hand, by (G3), as n is sufficiently large, we
have, for all x ∈ SX ,
Z
E{p̂n,0 (x)} = fX (x − hn,1 u) K(u) du
Z
≥b
K(u) du
x−hn,1 u ∈SX
Z 0

≥ b min

Z

1

K(u) du,
−1


K(u) du = b/2,

0

where the last equality is due to the symmetry of the function K. On the other hand, for n
large enough, for all x ∈ SX ,
Z

2
fX (x − hn,1 u) u − an (x) K(u) du
Z

2
≥b
u − an (x) K(u) du
x−hn,1 u ∈SX
Z 0


1


≥ min
u − an (x) K(u) du,
u − an (x) K(u) du
−1
0
Z 1

Z 1


2
2
= min
u + an (x) K(u) du,
u − an (x) K(u) du
0
0
Z 1

b cK
2
≥ b inf
u − a K(u) du =
,
a∈R 0
2
R1
R1
where cK = 2 0 (u − aK )2 K(u) du and aK = 2 0 u K(u) du. By taking c = cK /8 we obtain
(C.1). We have shown (4.1).
By differentiating, we see that (3.3) is equivalent to a linear system whose determinant is
equal to p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2 . Because the latter is strictly positive with probability going
to 1, we can invert the linear system to obtain (4.2).
C.2.2

Z

2

2

Proof of Proposition 4.2

Using Lemma 4.1, and since we are concerned with convergence in probability, we can restrict
2
2
attention
Pn to the event that p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x) ≥ b c for every x ∈ SX . Recall Q̂n,k (y, x) =
−1
n
i=1 ϕhn,2 (y, Yi ) wk,hn,1 (x−Xi ), where ϕh is defined in (3.1) and wk,h is defined right before
Lemma 4.1. It follows that
F̂n (y|x) =

Q̂n,0 (y, x) p̂n,2 (x) − Q̂n,1 (y, x) p̂n,1 (x)
p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2

= Q̂n,0 (y, x) ŝn,2 (x) − Q̂n,1 (y, x) ŝn,1 (x),
with ŝn,k (x) = p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2

−1

p̂n,k (x). Write

Q̂n,k (y, x) = β̂n,k (y, x) + b̂n,k (y, x) + γ̂n,k (y, x),

25

with
β̂n,k (y, x) = n

−1

b̂n,k (y, x) = n−1

n
X

i=1
n
X

ϕhn,2 (y, Yi ) − E[ϕhn,2 (y, Yi ) | Xi ] wk,hn,1 (x − Xi ),


E[ϕhn,2 (y, Yi ) | Xi ] − F (y|x) − (Xi − x) ∂x F (y|x) wk,hn,1 (x − Xi ),

i=1

γ̂n,k (y, x) = F (y|x) p̂n,k (x) − hn,1 p̂n,k+1 (x) ∂x F (y|x).
Elementary algebra yields
γ̂n,0 (y, x) ŝn,2 (x) − γ̂n,1 (x) ŝn,1 (x) = F (y|x).
It follows that
F̂n (y|x) − F (y|x) = {β̂n,0 (y, x) + b̂n,0 (y, x)} ŝn,2 (x) − {β̂n,1 (y, x) + b̂n,1 (y, x)} ŝn,1 (x).

(C.2)

In what follows, we apply Proposition C.1 to establish uniform rates of convergence for the
derivatives of order l ∈ {0, 1, 2} of β̂n,k and b̂n,k , for k ∈ {0, 1}, and of ŝn,k , for k ∈ {1, 2}. We
need to distinguish between many situations because differentiating with respect to x or y does
not impact the rates of convergence in the same way. In contrast, the index k has no effect.
Consequently, in what follows we fix k ∈ {0, 1, 2}. We define
(l)

r̂n,k (x) = n−1

n
X

(l)

(l)

(l)

wk,h ( · ) = h−1 wk ( · /h),

wk,hn,1 (x − Xi ) ,

i=1
(l)

where wk : u 7→ ∂ul wk (u) for l ∈ {0, 1, 2}. All asymptotic statements are for n → ∞, which is
omitted for brevity.
(l)

Rate of r̂n,k for l ∈ {0, 1, 2}. Invoking Condition (G3), the functions K, K 0 and K 00 are
bounded real functions of bounded variation that vanish outside (−1, 1). Consequently, the
(l)
functions wk , l ∈ {0, 1, 2}, are bounded functions of bounded variation that vanish outside
(−1, 1). Note that the absolute value of a function of bounded variation is of bounded variation
too, in view of the fact that |a| − |b| ≤ |a − b| for all a, b ∈ R. By Proposition C.1, first
(l)

assertion, with K equal to u 7→ |wk (u)|, we therefore have
s
sup
x∈SX

(l)
r̂n,k (x)

−

(l)
E{r̂n,k (x)}

= OP

|log hn,1 |
nhn,1

!
.

By (G1) it holds that
(l)

E{r̂n,k (x)} =

Z

(l)

fX (x − hn,1 u) |wk (u)| du ≤ M

Z

(l)

|wk (u)| du.

By the triangle inequality and Condition (G4), it follows that, for l ∈ {0, 1, 2},
s
!
|log hn,1 |
(l)
sup r̂n,k (x) = OP
+ 1 = OP (1).
nhn,1
x∈SX

26

(C.3)

Rate of ∂xl p̂n,k for l ∈ {0, 1, 2}. Differentiating under the expectation [apply the dominated
convergence theorem invoking (G3)], we get for any l ∈ {0, 1, 2},
Z

Z
(l)
−l
l
l
∂x {E p̂n,k (x)} = ∂x
fX (z) wk,hn,1 (x − z) dz = hn,1 fX (z) wk,h (x − z) dz.
(C.4)
Similarly [apply the dominated convergence theorem invoking (G1)], we have
Z
 Z
(l)
l
l
∂x {E p̂n,k (x)} = ∂x
fX (x − hn,1 u) wk (u) du = fX (x − hn,1 u) wk (u) du.

(C.5)

As a consequence of (C.4), we have
∂xl {p̂n,k (x)

− E p̂n,k (x)} =

(nhln,1 )−1

n n
o
X
(l)
(l)
wk,hn,1 (x − Xi ) − E[wk,hn,1 (x − X)] ,
i=1
(l)

and in view of Proposition C.1, first assertion, with K equal to u 7→ wk (u), it holds that
!
s
|log hn,1 |
l
.
sup ∂x {p̂n,k (x) − E p̂n,k (x)} = OP
nh1+2l
x∈SX
n,1
Using (C.5) and invoking (G1), we have
∂xl E{p̂n,k (x)}

Z
≤M

|wk (u)| du.

By the triangle inequality, it follows that, for l ∈ {0, 1, 2},
!
s
|log hn,1 |
l
+1 .
sup ∂x p̂n,k (x) = OP
nh1+2l
x∈SX
n,1

(C.6)

Rate of ∂xl ŝn,k for l ∈ {0, 1, 2}.

We use the quotient rule for derivatives to obtain that
!
s
|log
h
|
n,1
sup ∂xl ŝn,k (x) = OP
+1 .
nh1+2l
x∈SX
n,1

For l = 0, the previous formula follows from equation (C.6) because p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2 ≥
bc for every x ∈ SX . For l = 1, differentiating ŝn,k (x) gives a sum of terms which are all of the
form
∂x {p̂n,k1 (x)} p̂n,k2 (x) p̂n,k3 (x)
2 ,
p̂n,0 (x) p̂n,2 (x) − p̂n,1 (x)2
q
where k1 , k2 , k3 ∈ {0, 1, 2}. By (C.6), each term is of the order OP ( |log hn,1 |/(nh3n,1 ) + 1).
Finally, when l = 2, the order is given by differentiating the previous expression. We obtain a
sum of different terms. Putting dˆn (x) = p̂n,0 (x)p̂n,2 (x) − p̂n,1 (x)2 , these terms are of the form
∂xl1 {p̂n,k1 (x)} ∂xl2 {p̂n,k2 (x)} p̂n,k3 (x)
,
dˆn (x)2
and

with l1 + l2 = 2,

∂x {dˆn (x)} ∂x {p̂n,k1 (x)} p̂n,k2 (x) p̂n,k3 (x)
.
dˆn (x)3

By (C.6), the term with the highest order is
∂x2 {p̂n,k1 (x)} p̂n,k2 (x) p̂n,k3 (x)
= OP
dˆn (x)2
27

s

!
|log hn,1 |
+1 .
nh5n,1

(l)

Rate of ∂xl β̂n,k for l ∈ {0, 1, 2}.
sup
x∈SX , y∈R

Proposition C.1, fifth assertion, with K equal to wk , yields
!
s
|log
h
|
n,1
∂xl β̂n,k (y, x) = OP
.
nh1+2l
n,1

R
Rate of ∂xl b̂n,k for l ∈ {0, 1, 2}. Let Fn (y|x) = F (y − hn,2 u|x)L(u) du. Fubini’s theorem
gives that E[ϕhn,2 (y, Yi ) | Xi = x] = Fn (y|x). Hence we consider
(0)

(0)

b̂n,k (y, x) = B̂n,1 (y, x) + B̂n,2 (y, x),
with
(0)

n
X

(0)

n
X

B̂n,1 (y, x) = n−1

{Fn (y|Xi ) − F (y|Xi )} wk,hn,1 (x − Xi ),

i=1

B̂n,2 (y, x) = n−1

{F (y|Xi ) − F (y|x) − (Xi − x) ∂x F (y|x)} wk,hn,1 (x − Xi ).

i=1

Differentiating once with respect to x, we find
(1)

(1)

(1)

∂x b̂n,k (y, x) = B̂n,1 (y, x) + B̂n,2 (y, x) + B̂n,3 (y, x),
with
(1)

n
X

(1)

i=1
n
X

B̂n,1 (y, x) = n−1
B̂n,2 (y, x) = n−1

{Fn (y|Xi ) − F (y|Xi )} ∂x {wk,hn,1 (x − Xi )},
{F (y|Xi ) − F (y|x) − (Xi − x) ∂x F (y|x)} ∂x {wk,hn,1 (x − Xi )},

i=1
(1)
B̂n,3 (y, x)

= hn,1 ∂x2 F (y|x) p̂n,k+1 (x).

Differentiating twice with respect to x, we find
(2)

(2)

(2)

(2)

∂x2 b̂n,k (y, x) = B̂n,1 (y, x) + B̂n,2 (y, x) + B̂n,3 (y, x) + B̂n,4 (y, x),
with
(2)

n
X

(2)

i=1
n
X

B̂n,1 (y, x) = n−1
B̂n,2 (y, x) = n−1

{Fn (y|Xi ) − F (y|Xi )} ∂x2 {wk,hn,1 (x − Xi )},
{F (y|Xi ) − F (y|x) − (Xi − x) ∂x F (y|x)} ∂x2 {wk,hn,1 (x − Xi )},

i=1
(2)
B̂n,3 (y, x)
(2)
B̂n,4 (y, x)

= hn,1 p̂n,k+1 (x) ∂x3 F (y|x),
=

hn,1 ∂x2 F (y|x)

n

−1

n
X

!
{(x − Xi )/hn,1 } ∂x {wk,hn,1 (x − Xi )}

i=1

+ hn,1 ∂x {p̂n,k+1 (x)} ∂x2 F (y|x).
All this results in the formula
(l)

(l)

(l)

(l)

∂xl b̂n,k (y, x) = B̂n,1 (y, x) + B̂n,2 (y, x) + B̂n,3 (y, x) + B̂n,4 (y, x),
28

with, for l ∈ {0, 1, 2},
(l)

n
X

(l)

n
X

B̂n,1 (y, x) = n−1

{Fn (y|Xi ) − F (y|Xi )} ∂xl {wk,hn,1 (x − Xi )},

i=1

B̂n,2 (y, x) = n−1

{F (y|Xi ) − F (y|x) − (Xi − x) ∂x F (y|x)} ∂xl {wk,hn,1 (x − Xi )},

i=1
(0)

(l)

(0)

whereas B̂n,3 (y, x) = 0 and B̂n,3 (y, x) = hn,1 p̂n,k+1 (x) ∂xl+1 F (y|x) for l ∈ {1, 2}, and B̂n,4 (y, x) =
(1)

B̂n,4 (y, x) = 0. Assumption (G1) implies that
|F (y + u|x) − F (y|x) − u ∂y F (y|x)| ≤ M
R

By (G3) and (C.3), since

x∈SX , y∈R

B̂n,1 (y, x)
(l)

≤ h−l
n,1 sup {r̂n,k (x)}
x∈SX

=

u L(u) du = 0, we have

(l)

sup

h−l
n,1

sup
x∈SX

(l)
{r̂n,k (x)}

Z

x∈SX

2
h−l
n,1 hn,2

{F (y − hn,2 u|x) − F (y|x) + hn,2 u ∂y F (y|x)} L(u) du

sup
x∈SX , y∈R

(l)

s

|Fn (y|x) − F (y|x)|

sup
x∈SX , y∈R

2
≤ h−l
n,1 hn,2 sup {r̂n,k (x)}

= OP

u2
.
2

M
2

Z

u2 L(u) du

|log hn,1 |
2
+ h−l
n,1 hn,2
nhn,1

!



2
= OP h−l
n,1 hn,2 .
By (G1), one has
|F (y|x + u) − F (y|x) − u ∂x F (y|x)| ≤ M

u2
.
2

Using (C.3), it follows that
(l)

sup
x∈SX , y∈R

B̂n,2 (y, x)
(l)

≤ h−l
n,1 sup {r̂n,k (x)}
x∈SX

s
= OP

h2−l
n,1

|F (y|x̃) − F (y|x) − (x̃ − x) ∂x F (y|x)|

sup
|x−x̃|<hn,1 , y∈R

|log hn,1 |
+ h2−l
n,1
nhn,1

!



= OP h2−l
n,1 .

29

p
(l)
Next, by (C.6), we have B̂n,3 (y, x) = OP (hn,1 |log hn,1 | /(nhn,1 ) + hn,1 ) = OP (hn,1 ) for l ∈
{1, 2}. Finally, because ∂u {uk+1 K(u)} = uk K(u) + u ∂u {uk K(u)}, we have
!
n
X
(2)
2
−1
Bn,4 (y, x) = hn,1 ∂x {F (y|x)} n
{(x − Xi )/hn,1 } ∂x {wk,hn,1 (x − Xi )} + ∂x {p̂n,k+1 (x)}
i=1

=

hn,1 ∂x2 {F (y|x)}


2∂x {p̂n,k+1 (x)} − p̂n,k (x)
!
s
|log hn,1 |
+ hn,1
hn,1
nh3n,1

= OP

= OP (hn,1 ).
Putting all this together yields
sup
x∈SX , y∈R



−l 2
∂xl b̂n,k (y, x) = OP h2−l
n,1 + hn,1 hn,2 ,

Rate of ∂yl1 ∂xl2 β̂n,k for (l1 , l2 ) ∈ {(1, 0), (1, 1), (2, 0)}.
pectation to obtain
∂yl1 ∂xl2 β̂n,k = n−1

n 
X

l ∈ {0, 1, 2}.

Start by differentiating under the ex-


∂yl1 {ϕhn,2 (y, Yi )} − E[∂yl1 {ϕhn,2 (y, Yi )} | Xi ] ∂xl2 {wk,hn,1 (x − Xi )}

i=1
2
1 −1 −1
= (nhln,1
hln,2
)

n 
X


(l1 −1)
(l1 −1)
(l2 )
Lhn,2
(y − Yi ) − E[Lhn,2
(y − Yi ) | Xi ] wk,h
(x − Xi ),
n,1

i=1
(l)

with Lh (u) = h−1 L(l) (u/h). By Condition (G3), the functions L and L(1) are bounded functions
of bounded variation. Applying Proposition C.1, third assertion, with L equal to L(l1 −1) and
(l )
K equal to wk 2 , for (l1 , l2 ) ∈ {(1, 0), (1, 1), (2, 0)}, gives

v
u
u
|log
h
h
|
n,1 n,2
.
∂yl1 ∂xl2 β̂n,k (y, x) = OP t
sup
1+2(l −1)
1+2l
2
x∈SX , y∈R
nhn,1 hn,2 1
Rate of ∂yl1 ∂xl2 b̂n,k for (l1 , l2 ) ∈ {(1, 0), (1, 1), (2, 0)}.

We mimic here the approach taken when

(l ,l )
B̂n 1 2 (y, x)

treating ∂xl b̂n,k . In the following, terms denoted by
are related to the derivatives
of b̂n,k of order l1 (resp. l2 ) with respect to y (resp. x). Differentiating l1 times with respect to
y produces
(l ,0)

(l ,0)

∂yl1 b̂n,k (y, x) = B̂n,11 (y, x) + B̂n,21 (y, x),
with
(l ,0)

n
X


(l ,0)

n
X


B̂n,11 (y, x) = n−1

∂yl1 Fn (y|Xi ) − ∂yl1 F (y|Xi ) wk,hn,1 (x − Xi ),

i=1

B̂n,21 (y, x) = n−1

∂yl1 F (y|Xi ) − ∂yl1 F (y|x) − (Xi − x) ∂yl1 ∂x F (y|x) wk,hn,1 (x − Xi ).

i=1

Differentiating with respect to x gives
(1,1)

(1,1)

(1,1)

∂x ∂y b̂n,k (y, x) = B̂n,1 (y, x) + B̂n,2 (y, x) + B̂n,3 (y, x),
30

with
(1,1)
B̂n,1 (y, x)

=n

−1

n
X

∂y Fn (y|Xi ) − ∂y F (y|Xi ) ∂x {wk,hn,1 (x − Xi )},
i=1

n
X

(1,1)
−1
B̂n,2 (y, x) = n
∂y F (y|Xi ) − ∂y F (y|x) − (Xi − x) ∂y ∂x F (y|x) ∂x {wk,hn,1 (x − Xi )},
i=1
(1,1)
B̂n,3 (y)

= hn,1 p̂n,k+1 (x) ∂y ∂x2 F (y|x).

By (G1) we have
∂y F (y + u|x) − ∂y F (y|x) − u ∂y2 F (y|x) ≤
By (G3) and (C.3), since

x∈SX , y∈R

B̂n,1 2 (y, x)
(l )

2
2
≤ h−l
n,1 sup {r̂n,k (x)}

x∈SX

=
≤

sup
x∈SX

2
h2n,2 h−l
n,1

(l2 )
{r̂n,k
(x)}

sup
x∈SX

sup
x∈SX , y∈R

2
h2n,2 h−l
n,1

∂y Fn (y|x) − ∂y F (y|x)
Z

sup
x∈SX , y∈R

(l2 )
{r̂n,k
(x)}

s
= OP

u L(u) du = 0, we have

(1,l )

sup

2
h−l
n,1

R

M u2
.
2

M
2

Z

{∂y F (y − hn,2 u|x) − ∂y F (y|x) + hn,2 u ∂y2 F (y|x)} L(u) du

u2 L(u) du

|log hn,1 |
2
+ h2n,2 h−l
n,1
nhn,1

!



−l2
= OP h2n,2 hn,1
.
By (G1) we have
∂y2 F (y + u|x) − ∂y2 F (y|x) − u ∂y3 F (y|x) ≤ M

|u|1+δ
.
1+δ

Then, by (C.3), it holds that
(2,0)

sup
x∈SX , y∈R

B̂n,1 (y, x)

≤ sup {r̂n,k (x)}
x∈SX

sup
x∈SX , y∈R

∂y2 Fn (y|x) − ∂y2 F (y|x)
Z

= sup {r̂n,k (x)}
x∈SX

sup
x∈SX , y∈R

≤ h1+δ
n,2 sup {r̂n,k (x)}
x∈SX

s
= OP

h1+δ
n,2

M
1+δ

Z

{∂y2 F (y − hn,2 u|x) − ∂y2 F (y|x) + hn,2 u ∂y3 F (y|x)} L(u) du
|u|1+δ L(u) du

|log hn,1 |
+ h1+δ
n,2
nhn,1

!



= OP h1+δ
.
n,2

31

By (G1), one has
∂y F (y|x + u) − ∂y F (y|x) − u ∂y ∂x F (y|x) ≤ M

u2
.
2

Using (C.3), it follows that
(1,l )

sup
x∈SX , y∈R

B̂n,2 2 (y, x)
(l )

−l2
2
sup {r̂n,k
(x)}
≤ hn,1
x∈SX

s
= OP

2
h2−l
n,1

|∂y F (y|x̃) − ∂y F (y|x) − (x̃ − x) ∂y ∂x F (y|x)|

sup
|x−x̃|<hn,1 , y∈R

|log hn,1 |
2
+ h2−l
n,1
nhn,1

!



2
= OP h2−l
.
n,1
By (G1), one has
∂y2 F (y|x + u) − ∂y2 F (y|x) − u ∂y2 ∂x F (y|x) ≤ M

|u|1+δ
.
1+δ

Using (C.3), it then follows that
(2,0)

sup
x∈SX , y∈R

B̂n,2 (y, x)

≤ sup {r̂n,k (x)}
x∈SX

s
= OP

h1+δ
n,1

sup
|x−x̃|<hn,1 , y∈R

|log hn,1 |
+ h1+δ
n,1
nhn,1

∂y2 F (y|x̃) − ∂y2 F (y|x) − (x̃ − x) ∂y2 ∂x F (y|x)
!



.
= OP h1+δ
n,1
(1,1)

Finally, by (C.6), we have B̂n,3 (y) = OP (hn,1
all this together yields
sup
x∈SX , y∈R

sup
x∈SX , y∈R

sup
x∈SX , y∈R

p
|log hn,1 |/(nhn,1 ) + hn,1 ) = OP (hn,1 ). Putting


∂y b̂n,k (y, x) = OP h2n,1 + h2n,2 ,


1+δ
,
∂y2 b̂n,k (y, x) = OP h1+δ
+
h
n,2
n,1


∂y ∂x b̂n,k (y, x) = OP h2n,2 h−1
+
h
n,1 .
n,1

Back to equation (C.2). So far, we have obtained uniform convergence rates in probability
for β̂n,k , b̂n,k and ŝn,k (with k ∈ {0, 1} for β̂n,k and b̂n,k and k ∈ {1, 2} for ŝn,k ), and their first and
second-order partial derivatives. In combination with equation (C.2) and hypothesis (H3), these
yield the convergence rates for F̂n (y|x) − F (y|x) stated in the proposition. For the benefit of
the reader, we provide here the details. For l ∈ {0, 1, 2}, we have, since |log hn,1 | /(nh3n,1 ) = o(1)

32

by (G4),


if l ∈ {0, 1},
OP (1)r


l
sup ∂x ŝn,k (x) =
|log hn1 |

+1
if l = 2,
x∈SX
OP
nh5n1
s
!
|log hn1 |
l
∂x β̂n,k (y, x) = OP
sup
,
nh1+2l
x∈SX ,y∈R
n1


−l 2
∂xl b̂n,k (y, x) = OP h2−l
+
h
h
sup
n,1
n,1 n,2 .
x∈SX ,y∈R

Moreover,
sup
x∈SX , y∈R

sup
x∈SX , y∈R

sup
x∈SX , y∈R

∂y b̂n,k (x, y) = OP (h2n,1 + h2n,2 ),


1+δ
∂y2 b̂n,k (x, y) = OP h1+δ
,
n,1 + hn,2



∂y ∂x b̂n,k (x, y) = OP h2n,2 h−1
n,1 + hn,1 .

Finally, for (l1 , l2 ) equal to (1, 0), (1, 1) or (2, 0),
sup
x∈SX , y∈R

v
u
u
l1 l2

∂y ∂x β̂n,k (y, x) = OP t

|log hn,1 hn,2 |
2 1+2(l1 −1)
nh1+2l
n,1 hn,2


.

Using equation (C.2), we find the following uniform convergence rates for F̂n (y|x) − F (y|x) and
its derivatives. First, differentiating both sides of equation (C.2) l ∈ {0, 1, 2} times with respect
to x, we have, using |log hn,1 | /(nh3n,1 ) = o(1),
s
sup
x∈SX , y∈R

∂xl F̂n (y|x) − ∂xl F (y|x) = OP

|log hn,1 |
nh1+2l
n,1

!
−l 2
+ h2−l
n,1 + hn,1 hn,2

.

Second, differentiating once or twice with respect to y, we find
s
!
|log hn,1 hn,2 |
sup
∂y F̂n (y|x) − ∂y F (y|x) = OP
+ h2n,1 + h2n,2 ,
nhn,1 hn,2
x∈SX , y∈R
!
s
|log hn,1 hn,2 |
1+δ
1+δ
2
2
+ hn,1 + hn,2 .
sup
∂y F̂n (y|x) − ∂y F (y|x) = OP
nhn,1 h3n,2
x∈SX , y∈R
Finally, the rate for the mixed second-order partial derivative is
!
s
|log hn,1 hn,2 |
sup
∂y ∂x F̂n (y|x) − ∂y ∂x F (y|x) = OP
+ h2n,2 h−1
n,1 + hn,1 .
nh3n,1 hn,2
x∈SX , y∈R
This completes the proof of the proposition.

33

C.2.3

Proof of Proposition 4.3

The first statement is a direct consequence of Fact 1. On the event corresponding to En,j,1 , by
the mean-value theorem, we find
 
F̂n− (u|x) − F − (u|x) = F − F F̂n− (u|x)|x | x − F − (u|x)
≤


1
F F̂n− (u|x)|x − u .
b

Conclude
using the first statement of the proposition to obtain a uniform rate of convergence
p
of |log hn,1 |/nhn,1 + h2n,1 + h2n,2 .
C.2.4

Proof of Proposition 4.4

Recall that δ1 = min(α/2, δ). We establish the occurrence with high probability of the following
uniform bounds: for certain constants M10 and M100 , to be determined later,
∂x F̂n− (u|x) ≤ M10 ,

sup

(C.7)

x∈SX , u∈[γ,1−γ]

∂x F̂n− (u|x) − ∂x F̂n− (u|x0 )

sup
x6=x0 , u∈[γ,1−γ]

|x − x0 |δ1

≤ M100 .

(C.8)

The constant M1 of the statement will be taken equal to the maximum between M10 and M100 .
By Lemma D.3, we have, almost surely, F̂n F̂n− (u|x)|x = u for u ∈ [γ, 1−γ]. Differentiating
both sides of this equality with respect to x produces the identity
∂x F̂n− (u|x) = −

∂x F̂n (y|x)
fˆn (y|x)

.
y=F̂n− (u|x)

Using (G1), we have


fˆn F̂n− (u|x)|x − f F − (u|x)|x


≤ fˆn F̂n− (u|x)|x − f F̂n− (u|x)|x + M F̂n− (u|x) − F − (u|x)
≤

sup
x0 ∈SX , y∈R

fˆn (y|x0 ) − f (y|x0 ) + M

sup
x0 ∈SX , u∈[γ,1−γ]

F̂n− (u|x0 ) − F − (u|x0 ) .

Invoking Proposition 4.2 (using nh2n,1 / |log hn,1 | → ∞) and Proposition 4.3, we get
sup
x∈SX , u∈[γ,1−γ]



fˆn F̂n− (u|x)|x − f F − (u|x)|x = oP (1).

(C.9)

Now we introduce ĝn (y, x) = ∂x F̂n (y|x) and g(y, x) = ∂x F (y|x). In a similar way, invoking
2
Proposition 4.2 (using nh3n,1 / |log hn,1 | → ∞ and h−1
n,1 hn,2 → 0) and Proposition 4.3, we find
sup
x∈SX , u∈[γ,1−γ]



ĝn F̂n− (u|x), x − g F − (u|x), x = oP (1).

It follows that, with probability going to 1,
sup

∂x F̂n− (u|x) ≤

x∈SX , u∈[γ,1−γ]

34

2M
.
bγ /2

(C.10)

Hence we have shown (C.7) with M10 = 2M/(bγ /2). To show (C.8), we start by proving that
(C.8) holds true whenever the maps (y, x) 7→ ∂x F̂n (y|x) and (y, x) 7→ fˆn (y|x) are both δ1 -Hölder
with constant 2M , i.e., when
∂x F̂n (y|x) − ∂x F̂n (y 0 |x0 )
sup

|(y − y 0 , x − x0 )|δ1

x6=x0 , y6=y 0

fˆn (y|x) − fˆn (y 0 |x0 )
sup
x6=x0 , y6=y 0

≤ 2M,

(C.11)

≤ 2M.

|(y − y 0 , x − x0 )|δ1

(C.12)

Abbreviate F̂n− (u|x) to ξu,x , and write
∂x F̂n− (u|x) − ∂x F̂n− (u|x0 )
=


g(ξu,x0 , x0 ) − g(ξu,x , x)
g(ξu,x , x)
+
fˆn (ξu,x |x) − fˆn (ξu,x0 |x0 ) .
fˆn (ξu,x0 |x0 )
fˆn (ξu,x0 |x0 )fˆn (ξu,x |x)

Use (C.9) and (C.10) to obtain, with probability going to 1,
∂x F̂n− (u|x) − ∂x F̂n− (u|x0 ) ≤

2M
1
g(ξu,x0 , x0 ) − g(ξu,x , x) +
fˆn (ξu,x |x) − fˆn (ξu,x0 |x0 ) .
bγ /2
(bγ /2)2

Use the Hölder properties (C.11) and (C.12), then the equivalence between lp norms, p > 0,
and finally (C.7), to get
2M
|(ξu,x0
bγ /2

2M
1+
=
bγ /2

2M
≤
1+
bγ /2

2M
1+
≤
bγ /2

∂x F̂n− (u|x) − ∂x F̂n− (u|x0 ) ≤

(2M )2
− ξu,x , x0 − x)|δ1 +
|(ξu,x − ξu,x0 , x − x0 )|δ1
(bγ /2)2

2M
|(ξu,x0 − ξu,x , x0 − x)|δ1
bγ /2



2M
Mδ1 |ξu,x0 − ξu,x |δ1 + |x0 − x|δ1
bγ /2

2M 2
Mδ1 |x0 − x|δ1 ,
bγ /2

for some positive constant Mδ1 .
To terminate the proof, we just need to show (C.11) and (C.12). We only focus on the
former because the treatment of the latter results in the same approach involving slightly weaker
conditions. Because the function spaces Cs,M (SX ), s ∈ R, are decreasing sets in s, (G1) still
holds with δ1 = min(α/2, δ) in place of δ. We shall apply Proposition 4.2 with δ1 in place of δ.
Write m̂n (y, x) = ∂x {F̂n (y|x) − F (y|x)}. We distinguish between the case that |(y − y 0 , x − x0 )|
is smaller than hn,1 or not.
• First, suppose |(y − y 0 , x − x0 )| ≤ hn,1 . By the mean-value theorem and because of the
rates associated to ∂x2 and ∂x ∂y in Proposition 4.2, we have
|m̂n (y, x) − m̂n (y 0 , x0 )|
|(y − y 0 , x − x0 )|δ1
≤

|∇{m̂n (y, x)}| |(y − y 0 , x − x0 )|1−δ1

sup
x∈SX , y∈R

≤

sup
x∈SX , y∈R

s
= OP

1
|∇{m̂n (y, x)}| h1−δ
n,1

| log hn,1 |
1
nh3+2δ
n,1

s
−1−δ1 2
1
+ h1−δ
hn,2 +
n,1 + hn,1

35

| log hn,1 hn,2 |
1
nh1+2δ
n,1 hn,2

!
1−δ1 2
1
+ h2−δ
n,1 + hn,1 hn,2

.

Because δ1 = min(α/2, δ), (G4”) still holds with 2δ1 in place of α. This implies that the
previous bound goes to 0.
• Second, suppose |(y−y 0 , x−x0 )| > hn,1 . Using the rates associated to ∂x in Proposition 4.2,
we find
|m̂n (y, x) − m̂n (y 0 , x0 )|
1
≤ 2 sup |m̂n (y, x)| h−δ
n,1
|(y − y 0 , x − x0 )|δ1
x∈SX , y∈R
!
s
| log hn,1 |
−1−δ1 2
1−δ1
= OP
+ hn,1 + hn,1 hn,2 ,
1
nh3+2δ
n,1
which also goes to 0 by (G4”).
As a consequence of (G4’), the previous bounds convergence to 0. Now because ∂x F belongs
to Cδ1 ,M (R × SX ), we have, with probability going to 1,
∂x F̂n (y|x) − ∂x F̂n (y 0 |x0 )
|(y − y 0 , x − x0 )|δ1

|m̂n (y|x) − m̂n (y 0 |x0 )| |∂x F (y|x) − ∂x F (y 0 |x0 )|
+
|(y − y 0 , x − x0 )|δ1
|(y − y 0 , x − x0 )|δ1
≤ 2M,

≤

as required.

Appendix D

Analytical results

Lemma D.1. If C is a bivariate copula satisfying (G2), then for all γ ∈ (0, 1/2), all u ∈
[γ, 1 − γ]2 and all v ∈ [0, 1]2 ,
C(v) − C(u) −

2
2
X
4κ X
(vj − uj )2 .
(vj − uj ) Ċj (u) ≤
γ
j=1

j=1

In the lemma, the point v is allowed to lie anywhere in [0, 1]2 , even on the boundary or
at the corner. The ‘anchor point’ u, however, must be at a distance at least γ away from the
boundary.
Proof. Fix γ ∈ (0, 1/2), u ∈ [γ, 1 − γ]2 and v ∈ [0, 1]2 . For t ∈ [0, 1], put
w(t) = u + t(v − u).
Note that w(t) ∈ (0, 1)2 for t ∈ [0, 1). The function t 7→ C(w(t)) is continuous on [0, 1] and is
continuously differentiable on (0, 1). By the fundamental theorem of calculus,
Z
C(v) − C(u) = C(w(1)) − C(w(0)) =
0

1

dC(w(t))
dt =
dt

Z

2
1X

(vj − uj ) Ċj (w(t)) dt.

0 j=1

It follows that
Z 1
2
2
X
X
C(v) − C(u) −
(vj − uj ) Ċj (u) =
(vj − uj )
{Ċj (w(t)) − Ċj (u)} dt.
j=1

j=1

36

0

Fix t ∈ [0, 1) and j ∈ {1, 2}. Note that
u + s(w(t) − u) = u + st(v − u) = w(st).
The function s 7→ Ċj (w(st)) is continuous on [0, 1] and continuously differentiable on (0, 1). By
the fundamental theorem of calculus,
1

Z
Ċj (w(t)) − Ċj (u) =
0

2

X
dĊj (w(st))
ds =
t(vk − uk )
ds
k=1

1

Z

C̈jk (w(st)) ds.
0

We obtain
Z
2
2 X
2
X
X
C(v) − C(u) −
(vj − uj ) Ċj (u) =
(vj − uj )(vk − uk )
j=1

1Z 1

0

j=1 k=1

C̈jk (w(st)) t dt ds.
0

Suppose we find a positive constant L such that, for all u ∈ [γ, 1−γ]2 , v ∈ [0, 1]2 and j, k ∈ {1, 2},
Z

1Z 1

C̈jk (w(st)) t dt ds ≤ L.
0

(D.1)

0

Then
C(v) − C(u) −

2
X

(vj − uj ) Ċj (u) ≤ L

j=1

2 X
2
X

|vj − uj | |vk − uk |

j=1 k=1

2
= L |v1 − u1 | + |v2 − u2 |

≤ 2L (v1 − u1 )2 + (v2 − u2 )2 ,
which is the inequality stated in the lemma with 2L in place of 4κ/γ. It remains to show (D.1).
By Condition (G2),
C̈jk (w(st)) ≤ κ {wj (1 − wj ) wk (1 − wk )}−1/2 ,
where wj = wj (st) is the j-th coordinate of w(st). Now
wj (1 − wj ) = {uj + st(vj − uj )} {1 − uj − st(vj − uj )}.
For fixed st ∈ [0, 1), this expression is concave as a function of (uj , vj ) ∈ [γ, 1 − γ] × [0, 1]. Hence
it attains its minimum for (uj , vj ) ∈ {γ, 1 − γ} × {0, 1}. In each of the four possible cases, we
find
1
wj (1 − wj ) ≥ (1 − st)γ.
2
We obtain
2κ
C̈jk (w(st)) ≤
.
(1 − st)γ
As a consequence,
Z
0

1Z 1
0

2κ
C̈jk (w(st)) t dt ds ≤
γ

by direct calculation of the double integral.
37

Z
0

1Z 1
0

t
2κ
dt ds =
,
1 − st
γ

We quote Lemma 4.3 in Segers (2015), which is a variant of “Vervaat’s lemma”, i.e., the
functional delta method for the mapping sending a monotone function to its inverse:
Lemma D.2. Let F : R → [0, 1] be a continuous distribution function. Let 0 < rn → ∞ and
let F̂n be a sequence of random distribution functions such that, in `∞ (R),
rn (F̂n − F )

β ◦ F,

n → ∞,

where β is a random element of `∞ ([0, 1]) with continuous trajectories. Then β(0) = β(1) = 0
almost surely and
sup rn {F (F̂n− (u)) − u} + rn {F̂n (F − (u)) − u} = oP (1).

u∈[0,1]

As a consequence, in `∞ ([0, 1]),
rn {F (F̂n− (u)) − u}


u∈[0,1]

−β,

n → ∞.

Lemma D.3. Let F be a continuous function such that limy→−∞ F (y) = 0 and limy→+∞ F (y) =
1, then for any u ∈ (0, 1), we have F (F − (u)) = u.
Proof. Let y0 = F − (u). By assumption, the set {y ∈ R : F (y) ≥ u} is non empty and
therefore −∞ < y0 < +∞. By definition of the quantile transformation, for any y < y0 , it holds
F (y) < u. Now using the continuity of F gives F (y0 ) ≤ u. Conclude by noting that we always
have F (y0 ) ≥ u.
Lemma D.4. Let F and G be cumulative distribution functions. If there exists ε > 0 such that
|F (y) − G(y)| ≤  for every y ∈ R, then
G− ((u − ) ∨ 0) ≤ F − (u),
−

−

F (u) ≤ G (u + ε),

u ∈ [0, 1],

(D.2)

u ∈ [0, 1 − ε].

(D.3)

Proof. We first show (D.2). If F − (u) = ∞, there is nothing to show, while if F − (u) = −∞, then
u ≤ F (F − (u)) = F (−∞) = 0, so G− ((u−ε)∨0) = G− (0) = −∞ too. Hence we can suppose that
F − (u) is finite. Since G(y) ≥ F (y) −  for all y ∈ R, we have G(F − (u)) ≥ F (F − (u)) −  ≥ u − .
Trivially, also G(F − (u)) ≥ 0. Together, we find G(F − (u)) ≥ (u − ) ∨ 0. As a consequence,
F − (u) ≥ G− ((u − ) ∨ 0).
Next we show (D.3). Let u ∈ [0, 1 − ε]. By (D.2) with the roles of F and G interchanged
and applied to u + ε rather than to u, we find F − (u) = F − (((u + ε) − ε) ∨ 0) ≤ G− (u + ε).

Acknowledgments
The research by F. Portier was supported by the Fonds de la Recherche Scientifique Belgium
(F.R.S.-FNRS) A4/5 FC 2779/2014-2017 No. 22342320, and the research by J. Segers was
supported by the contract “Projet d’Actions de Recherche Concertées” No. 12/17-045 of the
“Communauté française de Belgique” and by IAP research network Grant P7/06 of the Belgian
government (Belgian Science Policy).
The authors greatfully acknowledge valuable comments by the reviewers and the editors, in
particular the helpful suggestions related to the structure of the theory and the presentation of
the paper.

38

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