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On copula-based conditional quantile estimators

Bruno R´emillarda, Bouchra Nasrib,∗, Taouﬁk Bouezmarnic

aGERAD, CRM, and Department of Decision Sciences, HEC Montr´eal, 3000 chemin de la Cˆote Sainte-Catherine,

Montr´eal(Qc), H3T 2A7. Canada

bDepartment of Water Sciences, Institut national de recherche scientiﬁque, Eau-Terre-Environnement, 490 rue de la

couronne, Qu´ebec,G1K 9A9. Canada

cDepartment of Statistics, Universit´e de Sherbrooke, 2500 boul. de l’Universit´e, Sherbrooke, J1K 2R1. Canada

Abstract

Recently, two diﬀerent copula-based approaches have been proposed to estimate the conditional quan-

tile function of a variable Ywith respect to a vector of covariates X: the ﬁrst estimator is related to

quantile regression weighted by the conditional copula density, while the second estimator is based on

the inverse of the conditional distribution function written in terms of margins and the copula. Using

empirical processes, we show that even if the two estimators look quite diﬀerent, their estimation

errors have the same limiting distribution. Also, we propose a bootstrap procedure for the limiting

process in order to construct uniform conﬁdence bands around the conditional quantile function.

Keywords: Conditional quantile function, copula, quantile regression, bootstrap

1. Introduction

Copulas, or dependence functions, are very popular to model the dependence between variables,

because one can remove the eﬀect of marginal distributions, provided the latter are continuous. This

is why dependence measures based on the copula are so robust, compared to the traditional Pearson

correlation coeﬃcient. Copulas also enter naturally when computing the conditional distribution

function of a random variable Ygiven covariates X= (X1, . . . , Xd). See, e.g., Bouy´e and Salmon [5]

when d= 1. This relation between the conditional distribution of Ygiven X=xand the associated

∗Corresponding author

Preprint submitted to Statistics and Probability Letters

copula was used recently to propose conditional quantile estimators, as alternative to the quantile

regression methods [11] or the parameter approach [6,15,16].

A ﬁrst copula-based estimator of the conditional quantile was proposed by Noh et al. [19] and is

based on a weighted quantile regression method. The asymptotic limiting distribution was proved to

be Gaussian. More recently, a more intuitive estimator of the plug-in type was proposed in Kraus and

Czado [12], Nasri and Bouezmarni [14], who compared the estimated MISE of various competitors,

including the estimator proposed by Noh et al. [19]. From the simulations performed in Kraus and

Czado [12], Nasri and Bouezmarni [14], it seems that the plug-in estimator performs better than the

other copula-based estimator. However the asymptotic behavior of this estimator was not discussed.

In Section 2, we describe the estimators of Noh et al. [19] and Kraus and Czado [12] and we discuss

their implementation. Another closely related parametric estimator proposed in Nasri and Bouezmarni

[14] is also discussed. In Section 3, we study the asymptotic limiting distribution of the estimators

viewed as stochastic processes over (0,1) and we show that the two semi-parametric estimators have

the same limiting distribution. We also propose a bootstrapping method for constructing uniform

conﬁdence bands for the conditional quantile functions.

2. Estimation of conditional quantiles

One way to model the dependence between a variable of interest Yand covariates X= (X1, . . . , Xd)

is to use dependence functions called copulas; see, e.g., Nelsen [17]. More precisely, suppose that

(Y1,X1),...,(Yn,Xn) are i.i.d. observations of (Y, X) with (unconditional) continuous margins F0,

F1, . . . , Fd, and copula Cwith density c. Set F(x)=(F1(x1), . . . , Fd(xd)).

By deﬁnition, a copula is a joint distribution function of uniform random variables. According to

Sklar’s theorem [17], since the margins are assumed to be continuous, there exists a unique copula C

so that the joint distribution function of (Y, X) can be written in terms of the copula and the margins

viz.

P(Y≤y, X≤x) = C{F0(y),F(x)}, y ∈R,x∈Rd.(1)

Note that the copula Cis the cdf of (U, V), where U=F0(Y) and V=F(X).

2

2.1. Copula-based conditional quantiles

Denote by H(y, x) the conditional distribution function of Ygiven X=x. The expression of the

conditional distribution function Hin terms of the copula function and the marginal distributions

appeared explicitly ﬁrst in a preliminary version of Bouy´e and Salmon [5] in the case d= 1. However,

it is easy to extend it to any d≥1, and one can easily show that

H(y, x) = P(Y≤y|X=x) = C{F0(y),F(x)}, y ∈R,x∈Rd,(2)

where C(u, v) is the conditional distribution function of Ugiven V≡F(X) = v≡F(x). In fact,

according to R´emillard [20, Proposition 8.6.2], for u∈[0,1] and v= (v1, . . . , vd)∈(0,1)d,

C(u, v) = ∂v1· ··∂vdC(u, v1, . . . , vd)

∂v1· ··∂vdC(1, v1, . . . , vd),

and ∂uC(u, v) = c(u, v)/R1

0c(z, v)dz, so C(u, v) = Ru

0c(z, v)dz/ R1

0c(z, v)dz.

Now, the associated conditional quantile function Q(α, x), α∈(0,1), is given by

Q(α, x) = inf{y∈R:H(y, x)≥α}.(3)

Using (2), we get that Qdepends only on the margins F0,Fand the copula Cviz.

Q(α, x) = F−1

0[Γ{α, F(x)}],(4)

where Γ(α, v) is the quantile of order αof the distribution function C(u, v), u∈[0,1], with v∈(0,1)d

ﬁxed. Note that (4) is the basic equation for deﬁning the plug-in estimator.

Next, using (2), one gets that Q(α, x) is also a solution of

arg min

aE [c{F0(Y),F(x)}ρα(Y−a)] ,(5)

where c(u, v) = ∂uC(u, v), ρα(y) = y{α−I(y < 0)}= (1 −α)|y|I(y < 0) + αyI(y≥0), y∈R, and I

is the indicator function. The latter equation is used by Noh et al. [19] to construct an estimator of

Q(α, x).

3

2.2. Estimation of the copula and the margins

To estimate the conditional quantile using copulas, one needs to estimate the copula Cassociated

with (Y, X) or (U, V), and the margins F0,F. First, one can assume that Yi=F−1

0(Ui) and Xij =

F−1

j(Vij), where (U1,V1),...,(Un,Vn) are i.i.d. observations from copula C.

2.2.1. Estimation of the copula

For sake of simplicity, we assume that the copula belongs to a parametric family {Cθ:θ∈ O}, so

the estimation of the copula is given as Cθn, where θnis a rank-based consistent estimator [7] of the

true parameter θ0. One can use the pseudo-MLE method proposed by Genest et al. [8]. Consequently,

the quantile function Γ(α, v)≡Γθ(α, v) can be estimated by Γθn(α, v), α∈(0,1), v∈(0,1)d. The

parametric family approach is also what Noh et al. [19] and Kraus and Czado [12] considered. In

fact, in the case of several covariates, Kraus and Czado [12] used a particular case of a parametric

copula family, namely a D-vine model [2,1], which is a construction of a copula using a given set of

parametric bivariate copula families. Note that instead of considering a parametric family of copulas,

one could estimate the density of the copula non-parametrically, so that all the conditional quantile

estimators discussed here could also be computed. However the convergence is slower and it often

suﬀers from the curse of dimensionality [4,9], with the possible exception of pair-copula construction

[13]. The next step is to estimate the margins.

2.2.2. Estimation of the margins

Motivated by the two-step inference function for margins (IFM) method [10], one could use para-

metric families to estimate each of the margins. This would make sense in several applications. For

copula-based quantile estimators, this approach was suggested in Nasri and Bouezmarni [14], where

a parametric copula-based estimator was proposed. Note that as discussed in Noh et al. [18], if the

estimation of the margins is incorrect, the estimation of the copula parameter θcan be biased. One

can also consider non-parametric estimators, namely for any y∈Rand any x= (x1, . . . , xd)∈Rd,

Fn0(y) = 1

n+ 1

n

X

i=1

I(Yi≤y), Fnj(xj) = 1

n+ 1

n

X

i=1

I(Xij ≤xj), j ∈ {1, . . . , d},(6)

4

and set Fn(x) = (Fn1(x1), . . . , Fnd(xd)). Further note that Fn0(y) = Dn◦F0(y), where Dnis the

empirical distribution function of the Ui’s and Fn(x) = Bn◦F(x), where Bnis the vector of empirical

marginal distribution functions of V1,...,Vd. Noh et al. [19] propose a kernel-based estimator ˆ

Fn0

for F0such that n1/2supy|ˆ

Fn0(y)−Fn0(y)|P r

→0 as n→ ∞. This was also used in Kraus and Czado

[12]. Even if ˆ

Fn0is continuous, the precision of the estimation might not be better and there is always

the question of the choice of the bandwidth. This is why we will use the estimators given by (6). For

the rest of the section, let xbe given and set v=F(x). It then follows that Fn(x) = Bn(v). For sake

of simplicity, xor vmight be omitted. We present the copula-based estimators we will study.

2.3. Weighted quantile regression estimator

Surprisingly, the natural plug-in estimator did not appear ﬁrst in the literature. In fact, Noh et al.

[19] proposed a copula-based model mixed with a quantile regression approach using (5) viz.

Qn,wqr (α, x) = arg min

a"n

X

i=1

ρα(Yi−a)cθn{Fn0(Yi),Fn(x)}#,(7)

even if the solution is not necessarily unique. In fact they take cθn(u, v) instead of taking cθn(u, v)

but it leads to the same estimator; see, e.g., (8). However, a unique way to deﬁne a solution to (7) is

by using the empirical weighted distribution function Hndeﬁned for any y∈Rby

Hn(y, x) =

n

X

i=1

I(Yi≤y)wi,n =Gn{F0(y),v},with Gn(u, v) =

n

X

i=1

I(Ui≤u)wi,n,

where, for any i∈ {1, . . . , n},

wi,n =cθn{Fn0(Yi),Fn(x)}

Pn

j=1 cθn{Fn0(Yj),Fn(x)}=cθn{Fn0(Yi),Fn(x)}

Pn

j=1 cθn{Fn0(Yj),Fn(x)}=cθn{Dn(Ui),Bn(v)}

Pn

j=1 cθn{Dn(Uj),Bn(v)}.(8)

The estimator Qn,wqr (α, x) is then deﬁned as the quantile of level αof Hn, i.e.,

Qn,wqr (α, x) = H−1

n(α, x) = F−1

0◦G−1

n(α, v), α ∈(0,1).(9)

If ˆa= arg mina[Pn

i=1 ρα(Yi−a)cθn{Fn0(Yi),Fn(x)}], then Hn(ˆa, x)≥α≥Hn(ˆa−,x). Hence

H−1

n(α, x) satisﬁes (7).

5

It is easy to show that Hnis a consistent estimator of the distribution function H(y, x) =

C{F0(y),v},y∈R. Also Gnis a consistent and asymptotically unbiased estimator of the distri-

bution function C(u, v), u∈[0,1].

2.4. Plug-in estimators

Expression (4) provides a natural way for estimating the conditional quantile. We now describe

both parametric and semi-parametric estimators of Q(α, x).

2.4.1. Parametric estimator

In the parametric approach, we assume that the marginal distributions F0and Fbelong to para-

metric families denoted by F0β0(·) and Fβ(·) respectively. If βn0and βnare consistent estimators of

β0and β, and if Cθ,F0β0(·) and Fβ(·) are continuous functions of the parameters, then for any y∈R,

ˇ

Hn(y, x) = Cθn{F0βn0(y),Fβn(x)}is clearly a consistent estimator of H(y, x), yielding

Qn,p(α, x) = ˇ

H−1

n(α, x) = F−1

0βn0[Γθn{α, Fβn(x)}], α ∈(0,1).(10)

2.4.2. Semiparametric estimator

Here, the marginal distributions are estimated using (6). Next, H(y, x) is estimated by

˜

Hn(y, x) = Cθn{Fn0(y),Fn(x)}=˜

Gn{F0(y),v}, y ∈R,(11)

where ˜

Gn(u, v) = Cθn{Dn(u),Bn(v)}, which is a consistent estimate of Cθ0(u, v), u∈[0,1]. As a

result, the estimation of Q(α, x) is deﬁned for any α∈(0,1) by

Qn,sp(α, x) = ˜

H−1

n(α, x) = F−1

n0[Γθn{α, Fn(x)}] = F−1

0◦˜

G−1

n(α, v).(12)

3. Asymptotic behavior of the copula-based estimators

In this section we ﬁnd the asymptotic distribution of the conditional quantile functions for the

proposed estimators, extending the results of Noh et al. [19]. As a result, we obtain that the estimation

error of the plug-in estimator and the weighted quantile regression estimator converge to the same

6

limiting distribution. We also propose, in Section 3.4, a diﬀerent bootstrap algorithm that can be

used to construct uniform conﬁdence bands about the conditional quantile function.

As before, xis ﬁxed and v=F(x). Throughout this section, we assume that the density f0=F0

0

exists and is positive everywhere. If the support is not R, just transform Yaccordingly. This way

F0(y)∈(0,1) for any y∈R. Also suppose that the density cof the (d+ 1)-dimensional copula Cis

positive on (0,1)d+1. Then H(·,x) is continuously diﬀerentiable with density hsatisfying h(y, x) =

f0(y)c(u, v)>0, for any y∈R. Further set Q(u, x) = H−1(u, x) and Γ(u, v) = C−1(u, v), u∈(0,1).

3.1. Convergence of the parametric estimator

In what follows, ∇β0F0β0(y) is a p0-dimensional column vector, ∇βFβis a p×dmatrix, ∇vCθ(u, v)

is a d-dimensional column vector, ∇θCθ(u, v) = ˙

Cθ(u, v) is a q-dimensional column vector which rep-

resent the partial derivatives with respect to β0,β,vand θof F0β0,Fβ,Cθand Cθrespectively.

Throughout this section, we assume that these derivatives are continuous, and that cθ(u, v) is contin-

uously diﬀerentiable with respect to u∈(0,1).

Set Bn0=n1/2(βn0−β0), Bn=n1/2(βn−β), and Θn=n1/2(θn−θ0). Finally, deﬁne ˇ

Hn(y) =

n1/2ˇ

Hn(y, x)− C(y, x)for any y∈R, and Qn,p(u) = n1/2{Qn,p (u, x)−Q(u, x)},u∈(0,1). The

proof of the following theorem, giving the asymptotic behavior of the parametric quantile process,

follows readily from the Delta method [21]. To simplify notations, set ˙

C(u, v) = ∇θCθ(u, v)|θ=θ0and

∇vCθ0(u, v) = ∇vC(u, v).

Theorem 1. Assume that (Bn0,Bn,Θn) converges in law to a centered Gaussian vector (B0,B,Θ).1

Then, as n→ ∞,ˇ

Hnconverges in D(R)2to a continuous centered Gaussian process ˇ

H, denoted

ˇ

Hn ˇ

H=ˇ

G◦F0β0, where

ˇ

G(u) = Θ>˙

C(u, v) + B>∇βFβnF−1

β(v)o∇vC(u, v) + cθ0(u, v)B>

0∇β0F0β0nF−1

0β0(u)o, u ∈[0,1].

1See, e.g. Joe [10] for suﬃcient regularity conditions.

2Convergence in D(I) means that for any close interval [a, b]⊂I, the process converges in law in the Skorokhod

topology on D([a, b]). In particular, continuous functions of the process converges in law. See, e.g., Billingsley [3].

7

Furthermore, Qn,p Qpin D(0,1), where Qp(u) = −ˇ

H{Q(u,x)}

h{Q(u,x),x},u∈(0,1). In particular, for any

[a, b]⊂(0,1), n1/2sup

u∈[a,b]|Qn,p(u, x)−Q(u, x)|converges in law to sup

u∈[a,b]

ˇ

H{Q(u, x)}

h{Q(u, x),x}

.

3.2. Convergence of the semiparametric estimator

We now study the convergence of the process Qn,sp(u) = n1/2{Qn,sp (u, x)−Q(u, x)},u∈(0,1).

Before stating the theorem, deﬁne Dn(u) = n1/2{Dn(u)−u}, and B

B

Bn(v) = n1/2(Bn(v)−v), u∈[0,1],

v∈(0,1)d. The proof of this theorem follows from the Delta method [21].

Theorem 2. Assume that (Dn,B

B

Bn,Θn) converges in D[0,1]1+d×Rqto (D,B

B

B,Θ), where Band B

B

Bare

centered Gaussian processes and Θis a centered random vector.3Then, as n→ ∞,˜

Gnconverges in

D([0,1]) to ˜

G=H+Dcθ0(·,v), where H(u) = Θ>˙

C(u, v) + B

B

B(v)>∇vC(u, v), u∈[0,1]. Furthermore,

Qn,sp Qsp in D(0,1), where Qsp(u) = −˜

G{Γ(u,v)}

h{Q(u,x),x},u∈(0,1). In particular, for any [a, b]⊂(0,1),

n1/2sup

u∈[a,b]|Qn,sp(u, x)−Q(u, x)|converges in law to sup

u∈[a,b]

˜

G{Γ(u, v)}

h{Q(u, x),x}

.

3.3. Convergence of the weighted quantile regression estimator

We now study the convergence of the process Qn,wqr (u) = n1/2{Qn,wqr(u, x)−Q(u, x)}. It ex-

tends the results in Noh et al. [19], where only the convergence at a single value was proven. In

order to formulate the result, we need to deﬁne another sequence of stochastic processes, namely

◦

Gn(u) = n−1/2Pn

i=1 {I(Ui≤u)cθ0(Ui,v)− Cθ0(u, v)},u∈[0,1]. It follows from the theory of stochas-

tic processes [21] that (Dn,B

B

Bn,◦

Gn) converges in D[0,1]2+dto centered Gaussian processes (D,B

B

B,◦

G).

We can now show that the two estimators have the same limiting distribution.

Theorem 3. Assume that (Dn,B

B

Bn,◦

Gn,Θn) converges in D[0,1]2+d×Rqto centered Gaussian

processes (D,B

B

B,◦

G,Θ). Then, as n→ ∞,Gnconverges in D([0,1]) to G=˜

G. Furthermore, Qn,wgr

Qn,wqr in D(0,1), where Qwqr(u) = −G{Γ(u,v)}

h{Q(u,x),x},u∈(0,1). In particular, for any [a, b]⊂(0,1),

n1/2sup

u∈[a,b]|Qn,wqr (u, x)−Q(u, x)|converges in law to sup

u∈[a,b]

G{Γ(u, v)}

h{Q(u, x),x}

.

3This assumption is satisﬁed for most well-behaved rank-based estimator of θ. See, e.g., Genest and R´emillard [7].

8

Proof. Set ˙

c(u, v) = ∇θcθ(u, v)|θ=θ0,∇vc(u, v) = ∇vcθ0(u, v). It suﬃces to prove the convergence

of Gn(u) = √n(Gn(u)− Cθ0(u, v)). Write Gn(u) = 1

nPn

i=1 I(Ui≤u)cθn{Dn(Ui),Bn(v)}/sn, where

sn=1

nPn

i=1 cθn{Dn(Ui),Bn(v)}.

Set rn(u) = cθn{Dn(u),B

B

Bn(v)}−cθ0(u, v)−{Θ>

n˙

c(u,v)+∂ucθ0(u,v)Dn(u)+∇vc(u,v)>B

B

Bn(v)}

n1/2,u∈[0,1]. By

hypothesis, as n→ ∞,n1/2sup

u∈[0,1] |rn(u)|converges in probability to 0. It follows that

Gn(u) = n−1/2

sn

n

X

i=1

I(Ui≤u){cθn{Dn(Ui),Bn(v)} − cθ0(Ui,v)}+◦

Gn(u)/sn− Cθ0(u, v)n1/2(1 −1/sn)

={Ln(u) + ◦

Gn(u)− Cθ0(u, v)Ln(1) − Cθ0(u, v)◦

Gn(1)}/sn,

where Ln(u) = n−1/2Pn

i=1 I(Ui≤u){cθn{Dn(Ui),Bn(v)} − cθ0(Ui,v)}. Now,

Ln(u) = Θ>

n(1

nX

i=1

I(Ui≤u)˙

c(Ui,v))+1

n

n

X

i=1

I(Ui≤u)Dn(Ui)∂uc(Ui,v)

+B

B

Bn(v)>(1

n

n

X

i=1

I(Ui≤u)∇vc(Ui,v))+oP(1)

=Θ>

n˙

C(u, v) + Zu

0

Dn(z)∂uc(z, v)dz +B

B

Bn(v)>∇vC(u, v) + oP(1).

Next, assuming that ucθ0(u, v)→0 as u→0, we have

Zu

0

Dn(z)∂zcθ0(z, v)dz =n−1/2

n

X

i=1 Zu

0

∂zcθ0(z, v){I(Ui≤z)−z}dz

=n−1/2

n

X

i=1

I(Ui≤u){cθ0(u, v)−cθ0(Ui,v)} − n1/2{ucθ0(u, v)− Cθ0(u, v)}

=cθ0(u, v)Dn(u)−◦

Gn(u).

As a result, Ln H+Dc(·,v)−◦

G=˜

G−◦

G. so, Gn G=˜

Gin D([0,1]).

Remark 1.Note that Theorems 2and 3are still valid if we choose the kernel distribution for marginals

instead of the empirical distribution functions since their asymptotic behavior is the same. This is

also true for the bootstrapping procedure deﬁned next.

9

3.4. Bootstrapping

Algorithm 1 (Bootstrapping ˜

G).First, estimate θusing a regular rank-based estimator θnof the

form θn=Tn(U1,n,V1,n, . . . , Un,n,Vn,n) in the sense of Genest and R´emillard [7], and set vn=Fn(x).

Then, for each k∈ {1, . . . , N}, repeat the following steps:

•Generate (U?

i,V?

i)∼Cθn,i∈ {1, . . . , n}, and compute the empirical margins D?

n,F?

n;

•Calculate the pseudo-observations U?

i,n =D?

n(U?

i), V?

i,n =F?

n(V?

i), i∈ {1, . . . , n};

•Estimate θ?

n=TnU?

1,n,V?

1,n, . . . , U ?

n,n,V?

n,n;

•Set ˜

G(k)

n(u) = n1/2Cθ?

n{D?

n(u),B?

n(vn)}−Cθn(u, vn),u∈[0,1].

The next theorem shows the consistency of the proposed bootstrap.

Theorem 4. Under the conditions of Theorem 2, as n→ ∞,˜

G(1)

n,..., ˜

G(N)

nconverge to independent

copies of ˜

G.

Proof. From Genest and R´emillard [7], (D?

n,B

B

Bn,B

B

B?

n,Θn,Θ?

n) D⊥,B

B

B,B

B

B⊥,Θ,Θ+Θ⊥, where

D⊥,B

B

B⊥,Θ⊥is an independent copy of (D,B

B

B,Θ). Hence, since n1/2{B?

n(vn)−v}=B

B

B?

n(vn) + B

B

Bn(v),

it follows from the Delta Method and Theorem 2that

˜

G(k)

n(u) = ˙

C(u, v)>Θ?

n+∇vC(u, v){B

B

B?

n(vn) + B

B

Bn(v)}+cθ0(u, v)D?

n(u) + oP(1)

˙

C(u, v)>Θ⊥+Θ+∇vC(u, v){B

B

B⊥(v) + B

B

B(v)}+cθ0(u, v)D⊥(u)

=˙

C(u, v)>Θ⊥+∇vC(u, v)B

B

B⊥(v) + cθ0(u, v)D⊥(u) + H(u) = ˜

G⊥(u) + H(u),

where ˜

G⊥is an independent copy of ˜

G, while n1/2{Cθn(u, vn)− Cθ0(u, v)} H. As a result,

˜

G(1)

n,..., ˜

G(N)

nconverge to independent copies of ˜

G.

Remark 2.Note that as shown in Genest and R´emillard [7], most interesting estimators are regular.

In particular, estimators of the class R1: this means that there exists a continuously diﬀerentiable

function Jso that E[J(U, V)] = 0 and Θn=n−1/2Pn

i=1 J{Dn(Ui), Bn(Vi)}+oP(1). For example,

pseudo-maximum likelihood estimators, as deﬁned in Genest et al. [8], belong to this class.

10

3.4.1. Construction of the uniform 100(1 −α)% conﬁdence band for Q

To construct the uniform conﬁdence band on [a, b]⊂(0,1), we generate Nprocesses ˜

G(k),k∈

{1, . . . , N }and they are evaluated at u∈ A ={a+j(b−a)/m;j= 0, . . . , m}, where mis ﬁxed but

large enough (say m= 1000). The density f0is estimated with a Gaussian kernel estimator fn0, so

h(u) = h{Q(u, x),x}is estimated by hn(u) = fn0◦Qn,sp(u, x)cθn(u, vn), when vn=Fn(x). One then

computes bk,n = maxu∈A

˜

G(k)(u)/hn(u), k∈ {1, . . . , N }, and let bn(α) be the associated quantile of

order 1−α. The uniform conﬁdence band about Q(·,x) is given by Qn,sp(u, x)±n−1/2bn(α), u∈[a, b].

A 95% conﬁdence interval about a single point Q(u, x) is given by Qn,sp(u, x)±n−1/21.96ˆσ/hn(u) where

ˆσ2is the sample variance of the values ˜

G(k)(u), k∈ {1, . . . , N }.

Remark 3.Using our notations, the bootstrap algorithm proposed in Noh et al. [19] yields values

Q(k)

n,wqr ,k∈ {1, . . . , N }, so that Q(k)

n,wqr =n1/2nQ(k)

n,wqr −Qoconverges to Q(k)

wqr +Qwq r, where Q(k)

wqr is

an independent copy of Qwqr. It then follows that their algorithm works for estimating the asymptotic

variance σ2

α, in the sense that what they call ˆσ2

boot satisﬁes ˆσ2

boot ≈σ2

α

nif nand Nare large. However,

their procedure is slower than the one we propose since we do not need to compute Y?

i=F−1

n0(U?

i)

and X?

i=F−1

n(V?

i), i∈ {1, . . . , n}. In addition, computing ˜

Hnis faster than computing Hn.

4. Conclusion

We have shown that two seemingly diﬀerent estimators for the conditional quantile function have in

fact the same limiting distribution. However, the plug-in estimator is easier and faster to implement,

in addition to being more accurate for small samples, as shown by simulations in Kraus and Czado

[12] and Nasri and Bouezmarni [14]. Therefore, this is the one we recommend.

Acknowledgements

Funding in support of this work was provided by the Natural Sciences and Engineering Research

Council of Canada and the Fonds de recherche du Qu´ebec – Nature et technologies. The authors

thank the editor and three anonymous reviewers for their constructive comments, which helped them

to improve the manuscript.

11

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