On the strong consistency of the kernel estimator of extreme conditional quantiles

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On the strong consistency of the kernel estimator of extreme conditional quantiles

Stéphane Girard, Sana Louhichi

To cite this version:

Stéphane Girard, Sana Louhichi. On the strong consistency of the kernel estimator of extreme con-

ditional quantiles. Elias Ould Said. Functional Statistics and Applications, Springer, pp.59–77, 2015,

Contributions to Statistics, 978-3-319-22475-6. �10.1007/978-3-319-22476-3_4�. �hal-00956351v2�

On the strong consistency of the kernel estimator of extreme conditional quantiles

St´ephane Girard

and Sana Louhichi

(1)Mistis, Inria Grenoble Rhˆone-Alpes, France.

(2)Laboratoire Jean Kuntzmann, France.

Abstract

Nonparametric regression quantiles can be obtained by inverting a kernel esti- mator of the conditional distribution. The asymptotic properties of this estimator are well-known in the case of ordinary quantiles of fixed order. The goal of this paper is to establish the strong consistency of the estimator in case of extreme con- ditional quantiles. In such a case, the probability of exceeding the quantile tends to zero as the sample size increases, and the extreme conditional quantile is thus located in the distribution tails.

1 Introduction

Quantile regression plays a central role in various statistical studies. In particu- lar, nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution function are extensively investigated in the sample case [3, 27, 29, 30]. Extensions to random fields [1], time series [14], functional data [11] and truncated data [25] are also available. However, all these papers are restricted to conditional quantiles having a fixed orderα∈(0,1). In the follow- ing,αdenotes the conditional probability to be larger than the conditional quan- tile. Consequently, the above mentioned asymptotic theories do not apply in the distribution tails, i.e whenα=αn→0 orαn→1 as the sample size n goes to in- finity. Motivating applications include for instance environmental studies [16, 28], finance [31], assurance [4] and image analysis [26].

The asymptotics of extreme conditional quantile estimators have been estab- lished in a number of regression models. Chernozhukov [6] and Jureckov´a [22]

considered the extreme quantiles in the linear regression model and derived their asymptotic distributions under various error distributions. Other parametric mod- els are considered in [10, 28]. A semi-parametric approach to modeling trends in extremes has been introduced in [9] basing on local polynomial fitting of the Generalized extreme-value distribution. Hall and Tajvidi [21] suggested a non- parametric estimation of the temporal trend when fitting parametric models to extreme-values. Another semi-parametric method has been developed in [2] us- ing a conditional Pareto-type distribution for the response. Fully nonparametric estimators of extreme conditional quantiles have been discussed in [2, 5] including

local polynomial maximum likelihood estimation, and spline fitting via maximum penalized likelihood. Recently, [15, 18] proposed, respectively, a moving-window based estimator for the tail index and extreme quantiles of heavy-tailed conditional distributions, and they established their asymptotic properties.

In the kernel-smoothing case, the asymptotic theory for quantile regression in the tails is still in full development. [19, 20] have analyzed the caseαn=1/n in the particular situation where the response Y given X=x is uniformly distributed.

The asymptotic distribution of the kernel estimator of extreme conditional quantile is established by [7, 17] for heavy-tailed conditional distributions. This result is extended to all types of tails in [8].

Here, we focus on the strong consistency of the kernel estimator for extreme conditional quantiles. Our main result is established in Section 2. Some illustrative examples are provided in Section 3. The proofs of the main results are given in Section 4 and the proofs of the auxiliary results are postponed to the Appendix.

2 Main results

Let(Xi,Yi)1≤i≤nbe independent copies of a random pair(X,Y)∈R^d×R⁺with density f_(X,Y). Let g be the density of X that we suppose strictly positive. The conditional survival function of Y given X=x is denoted by,

F(y¯ |x) =P(Y>y|X=x) = 1 g(x)

Z+∞ y

f_(X,Y)(x,z)dz The kernel estimator of ¯F(y|x)is, for x such that∑ⁿi=1Kh(x−Xi)6=0,

F¯n(y|x) =∑ⁿi=1K_h(x−X_i)1IYi>y

∑ⁿi=1K_h(x−Xi) ,

where h=hn→0 as n→∞and K_h(u) = _h¹_dK(^u_h), the kernel K is a measurable function which satisfies the conditions:

(K₁)K is a continuous probability density.

(K₂)K is with compact support: ∃R>0 such that K(u) =0 for anykuk ≤R.

Defineκ:=kKk∞=sup_x∈RdK(x)<∞.

Recall that, for a class of functionG,N(ε,G,d_Q)denotes the minimal number of balls{g,dQ(g,g^′)<ε}of dQ-radiusεneeded to coverG and dQis the L2(Q)- metric. LetK be the set of functionsK ={K((x− ·)/h),h>0,x∈R^d}and N(ε,K) =sup_QN(εkKk∞,K,dQ)where the supremum is taken over all the probability measure Q onR^d×R. Suppose that,

(K₃)for some C,ν>1,N(ε,K)≤Cε^−νfor anyε∈]0,1[.

A number of sufficient conditions for which(K₃)holds are discussed in [13] and the references therein. Finally suppose that

(A₁)for allα∈(0,1)there exists an unique q(α|x)∈Rsuch that ¯F(q(α|x)|x) =α.

The conditional quantile q(α|x)is then the inverse of the function ¯F(·|x)at the pointα. Let(αn)be a fixed sequence of levels with values in[0,1]. For any x∈R^d, define q(αn|x)as the unique solution of the equation,

αn=F(q(α¯ n|x)|x), (1)

whose existence is guaranteed by Assumption(A₁). The kernel estimator of the conditional quantiles q(α|x)is:

ˆ

qn(α|x) =inf{y∈R,F¯n(y|x)≤α}. (2) Finally, denote by ˆgn(x)the kernel density estimator of the probability density g i.e.

ˆ gn(x) =1

n

∑

K_h(x−X_i).

Our main result is the following.

Theorem 1 Let(X_i,Y_i)_1≤i≤n be independent copies of a random pair(X,Y)∈ R^d×R⁺with density f_(X,Y). Let g be the density of X that we suppose bounded and strictly positive. Suppose that assumption(A₁)is satisfied. Let(αn) be a sequence of levels in[0,1]for which

lim sup

n→∞ sup

x∈Rd

ˆ qn(αn|x)

q(αn|x) ≤Cst, (3)

almost surely. Suppose that the kernel K satisfies Conditions(K₁),(K₂),(K₃).

Define

A(y,z,x,hn) = sup

u:d(u,x)≤Rhn

¯

¯

¯

¯ F(y¯ |u) F¯(z|x)−1

¯

¯

¯

¯ .

Suppose that for some fixed positiveε0and for z∈ {q(αn|x),(1+ε)q(αn|x)} lim sup

n→∞ sup

x∈Rd,|ε|≤ε0

A((1+ε)q(αn|x),z,x,hn)≤C<∞. (4) If, moreover,

nlim→∞nh^d_nαn=∞and lim

n→∞

ln(αnh^d_n∧αn²) nh^d_nαn

=0, (5)

then there exists a positive constant C, not dependent on x, such that one has for n sufficiently large

¯

¯

¯

¯

1−F(¯ qˆn(αn|x)|x) F(q(α¯ n|x)|x)

¯

¯

¯

¯ ≤ C sup

|ε|≤ε0

A((1+ε)q(αn|x),(1+ε)q(αn|x),x,hn)

+ C

ˆ gn(x)

s

ln(αn⁻¹h⁻n^d)∨ln ln n

nαnh^d_n ,almost surely.

The first term of the bound can be interpreted as a bias term due to the kernel smoothing. The second term can be seen as a variance term, nαnh^d_n being the effective number of points used in the estimation. The following proposition gives conditions under which (3) is satisfied.

Proposition 1 Suppose that g is Lipschitzian and bounded above by gmax. Let vd=^R_kvk≤1dv be the volume of the unit sphere. If A(q(αn|x),q(αn|x),x,0,h)→0 and there existε>0 such that

∑

nh^dαnexp{−v_dgmaxnh_dαn(1+ε)}=∞, (6) then,

lim sup

n→∞ sup

x∈Rd

ˆ qn(αn|x)

q(αn|x) ≤1, almost surely.

Some examples of distributions satisfying condition (4) are provided in the next section.

3 Examples

Let us first focus on a conditional Pareto distribution defined as

F(y¯ |x) =y^−θ(x),for all y>0. (7) Here,θ(x)>0 can be read as the inverse of the conditional extreme-value index.

The above distribution belongs to the so-called Fr´echet maximum domain of at- traction which encompasses all distributions with heavy tails. As a consequence of Theorem 1, we have:

Corollary 1 Let us consider a conditional Pareto distribution (7) such that 0<

θmin≤θ(x)≤θmax for all x∈R^d. Assume thatθ is Lipschitzian. If the se- quences(αn)and(hn)are such that hnlogαn→0 as n→∞and (5), (6) hold, then ˆqn(αn|x)/q(αn|x)→1 almost surely as n→∞.

Let us now consider a conditional exponential distribution defined as

F¯(y|x) =exp(−θ(x)y),for all y>0, (8) whereθ(x)>0 is the inverse of the conditional expectation of Y given X=x. This distribution belongs to the Gumbel maximum domain of attraction which collects all distributions with a null conditional extreme-value index. These distributions are often referred to as light-tailed distributions. In such a case, Theorem 1 yields a stronger convergence result than in the heavy-tail framework:

Corollary 2 Let us consider a conditional exponential distribution (8) with 0<

θmin≤θ(x)≤θmax for all x∈R^d. Assume thatθ is Lipschitzian. If the se- quences(αn)and(hn)are such that hnlogαn→0 as n→∞and (5), (6) hold, then(qˆn(αn|x)−q(αn|x))→0 almost surely as n→∞.

4 Proofs

4.1 Proof of Theorem 1

Clearly, by (1),

|F¯(q(αn|x)|x)−F¯(qˆn(αn|x)|x)|

F(q(α¯ n|x)|x) ≤ |αn−F¯n(qˆn(αn|x)|x)| F(q(α¯ n|x)|x)

+ |F¯n(qˆn(αn|x)|x)−F(¯ qˆn(αn|x)|x)| F(q(α¯ n|x)|x) . First, from (2),|αn−F¯n(qˆn(αn|x)|x)|is bounded above by the maximal jump of F¯n(y|x)at some observation point(Xj,Yj):

|αn−F¯n(qˆn(αn|x)|x)| ≤max_j=1,...,nK_h(x−X_j)

∑ⁿi=1K_h(x−X_i) . It follows from(K₂)that

|αn−F¯n(qˆn(αn|x)|x)|

F¯(q(αn|x)|x) ≤ κ nh^dαngˆn(x).

Let us then focus on the second term:

|F¯n(qˆn(αn|x)|x)−F(¯ qˆn(αn|x)|x)|

F¯(q(αn|x)|x) =F¯(qˆn(αn|x)|x) F¯(q(αn|x)|x)

¯

¯

¯

¯

F¯n(qˆn(αn|x)|x) F¯(qˆn(αn|x)|x) −1

¯

¯

¯

¯

. (9)

We write ˆqn(αn|x) = (1+ε)q(αn|x), withε= ^qˆ_q(α^n(αn|x)

n|x) −1. Condition (3) allows to deduce that there exists, for n sufficiently large, ε0 not dependent on x such that|ε| ≤ε0. Consequently and taking into account (9) there exists a positive constantε0not dependent on x and n such that (for the sake of simplicity, we write q=q(αn|x))

¯

¯

¯

¯

F(¯ qˆn(αn|x)|x) F(q(α¯ n|x)|x) −1

¯

¯

¯

¯ ≤ κ

nh^dαngˆn(x)

+ sup

|ε|≤ε0

µF(q(1¯ +ε))|x) F(q¯ |x)

¯

¯

¯

¯

F¯n(q(1+ε)|x) F¯(q(1+ε))|x)−1

¯

¯

¯

¯

¶ . (10)

Our purpose now is to control the term

¯

¯

¯

F¯n(q(1+ε)|x) F¯(q(1+ε)|x)−1

¯

¯

¯of (10). For this, write F¯n(y|x) =∑ⁿi=1K_h(x−X_i)1IYi>y

∑ⁿi=1K_h(x−Xi) =:ψˆn(y,x) ˆ gn(x) , with

ψˆn(y,x) =1 n

∑

Kh(x−Xi)1IYi>y, gˆn(x) =1 n

∑

Kh(x−Xi).

We need the following lemma.

Lemma 1 Suppose that Condition (K₂) holds. Then, for each n, one has for any y∈Rand x∈R^dfor which ¯F(y|x)gˆn(x)6=0,

¯

¯

¯

¯ F¯n(y|x)

F(y¯ |x) −1

¯

¯

¯

¯

≤A(y,y,x,hn) + (1+A(y,y,x,hn))|gˆn(x)−Egˆn(x)| ˆ

gn(x) +|ψˆn(y,x)−E(ψˆn(y,x))| F(y¯ |x)gˆn(x) . The proof is postponed to the Appendix. According to Lemma 1, we have to control the two quantitiesE|gˆn(x)−Egˆn(x)|and ^|ψˆn(y,x)_F⁻_¯(y^{E(ψˆn(y,x))|}

|x) . This is the purpose of Propositions 2 and 3 below.

Proposition 2 Einmahl-Mason (2005).

Suppose that g is a bounded density onR^d, and that the assumptions(K.i),···,(K.iv) of [13] are all satisfied. Then, for any c>0,

lim sup

n→∞ sup

{c ln n/n≤h^d_n≤1}

s

nh^d_n

ln(h⁻n^d)∨ln ln n sup

x∈Rd

|gˆn(x)−Egˆn(x)|=: K(c)<∞, almost surely.

Our task now is to control^|ψˆn(y,x)_F(y⁻_¯^{E(ψˆn(y,x))|}

|x) . Letεbe a fixed real in[−ε0,ε0]for some arbitrary positiveε0. The following proposition evaluates the almost sure asymptotic behaviour of

|ψˆn(q(1+ε),x)−E(ψˆn(q(1+ε),x))|

F¯(q|x) .

Proposition 3 Let(αn)be a sequence in[0,1]and for x∈R^d, q=q(αn|x)be the conditional quantile as defined by (1). Define the set of functionsFby,

F=

½

(u,v)7−→K µx−u

h

¶

Iv>q(1+ε),n∈N,h>0,x∈R^d,|ε| ≤ε0

¾ (11) and suppose thatN(ε,F)≤Cε^−ν,for some C,ν>1 and allε∈]0,1[. Suppose also that Condition (4) is satisfied. If nh^d_nαn→∞and ^ln(α_nh^nh_d^dn∧αn2)

nαn →0 as n→∞, then there exists a positive constant C₁such that

lim sup

n→∞

s

nαnh^d_n

ln(αn⁻¹h⁻_n^d)∨ln ln n sup

x∈Rd,|ε|≤ε0

|ψˆn(q(1+ε),x)−E(ψˆn(q(1+ε),x))| F(q¯ |x) ≤C₁, almost surely.

Proof of Proposition 3. We have, 1

F¯(q|x)(ψˆn(q(1+ε),x)−E(ψˆn(q(1+ε),x)))

= 1

n ¯F(q|x)

∑

[Kh(x−Xi)1I_Yi>q(1+ε)−E(Kh(x−Xi)1I_Yi>q(1+ε))]

= 1

nh^d_n

∑

(vh,x,ε(Xi,Yi)−E(vh,x,ε(Xi,Yi))) = 1 h^d_n√

nβn(vh,x,ε), where,

v_h,x,ε(u,v) =K(x−u

h )1Iv>q(1+ε)

F¯(q|x) , βn(g) = 1

√n

∑

(g(Xi,Yi)−E(g(Xi,Yi))) Define the class of functions:

G :=G_n,h={v_h,x,ε,x∈R^d,ε∈[−ε0,ε0]} (12) and letkβnk^G =sup_g∈G|βn(g)|and

Θn= sup

x∈Rd,ε∈[−ε0,ε0]

|ψˆn(q(1+ε),x)−E(ψˆn(q(1+ε),x))|

F(q¯ |x) .

Consequently, for anyγ>0, P(Θn>γ)≤P³√

nkβnkG >γnh^d´

≤P µ

max

1≤m≤n

√mkβmkG>γnh^d

¶ (13) We have then to evaluate max_1≤m≤n√

mkβmkG. By Talagrand Inequality, (see A.1. in [12]), we have for any t>0 and suitable finite constants A1,A2>0,

P Ã

1≤maxm≤n

√mkβmk^G >A1

à Ek

∑

εig(Xi,Yi)k^G+t

!!

≤2 exp(−A2t²/nσ²) +2 exp(−A2t/M),

where(εi)iis a sequence of independent Rademacher random variables indepen- dent of the random vectors(Xi,Yi)1≤i≤nand

sup

g∈Gkgk∞≤M, sup

g∈G

Var(g(X,Y))≤σ².

Herekgk∞≤^kK_α^k_n^∞ =: M, and

Var(v²_h,x,ε(X,Y))≤E(v²_h,x,ε(X,Y)) = 1

F¯²(q|x)E(K²(x−X

h )1I_Y>q(1+ε))

= 1

F¯(q|x) Z

K²(x−u

h )F(q(1¯ +ε)|u) F¯(q|x) g(u)du

≤ h^d_n αn

sup

x∈Rd,ε∈[−ε0,ε0]

(1+A(q(1+ε),q,x,h))kKk²2kgk∞= h^d_n αn

L=:σ²,(14) for some positive constant L, since for n sufficiently large

sup

x∈Rd,ε∈[−ε0,ε0]

A(q(1+ε),q,x,h)≤cst.

We obtain combining this with the above Talagrand’s Inequality, P

Ã

1max≤m≤n

√mkβmk∞>A1

à Ek

∑

εig(Xi,Yi)k^G+t

!!

≤2 exp(−A₂t² αn

nh^d_nL) +2 exp(−A₂t αn

kKk∞). (15) The last bound together with (13) gives

P Ã

Θn>A1n⁻¹h⁻_n^d Ã

Ek

∑

εig(Xi,Yi)k^G+t

!!

≤2 exp(−A₂t² αn

nh^d_nL) +2 exp(−A₂t αn

kKk∞). (16) We have now to evaluateEk∑ⁿi=1εig(Xi,Yi)k^G.We will argue as for the proof of Proposition A.1. in [12]. We have by (6.9) of Proposition 6.8 in [24],

Ek

∑

εig(Xi,Yi)k^G ≤6tn+6E(max

i≤nkεig(Xi,Yi)k^G)≤6tn+6kKk∞

αn

, (17)

where we have defined tn=inf

( t>0,P

à k

∑

εig(Xi,Yi)k^G>t

!

≤ 1 24

) .

Our purpose is then to control tn. Define the event Fn=

( n⁻¹sup

g∈G

∑

g²(X_j,Y_j)≤64σ² )

,

whereσ²is as in (14). Let g0be a fixed element ofG. We have, E|

∑

εig0(Xi,Yi)1IF_n| ≤8σ√ n.

By (A8) of [12], we have now to controlN(ε,G,dn,2). Recall thatN(ε,G,d_Q) is the minimal number of balls{g,dQ(g,g^′)<ε}of dQ-radiusεneeded to cover G, d_Qis the L₂(Q)-metric and

d_n,2(f,g) =d_Qn(f,g) = Z

(f(x)−g(x))²dQn(x),

with Qn=¹_n∑ⁿi=1δ_(Xi,Yi). In other words d_n,2(f,g) =1

n

∑

(f(Xi,Yi)−g(Xi,Yi))². We note first that, on the event Fn,

N(ε,G,dn,2) =1, wheneverε>16σ.

We will suppose thenε≤16σ. We have

N(ε,G,d_n,2) =N(ε,G,d_Qn)≤N(εαn,F,d_Qn),

whereFis as defined by (11). Recall thatN(ε,F) =sup_QN(εkKk∞,F,d_Q) where the supremum is taken over all the probability measure Q onR^d×R. We have supposed that,

N(ε,F)≤Cε^−ν. for some C,ν>1 and allε∈]0,1[. Consequently,

N(ε,G,d_n,2)≤C( αnε

kKk∞)^−ν, (18) as soon asαnε<kKk∞. Hence, we have almost surely on the event Fn,

Z_∞ 0

q

ln(N(ε,G,dn,2))dε = Z_16σ

0

q

ln(N(ε,G,dn,2))dε

≤

∑

i=0 Z₂−i16σ

2⁻ⁱ⁻¹16σ

s

ln(C) +νln(kKk∞

αnε )dε

≤

∑

i=0

2⁻ⁱ⁻¹16σ s

ln(C) +νln(2ⁱ⁺¹kKk∞

αn16σ )

≤ C2

∑

√i+1 2ⁱ⁺¹

s

σ²max(ln(C1),ln( 1 αn²σ²)), for some positive constants C1,C2that depend only on C,νandkKk∞. We con- clude, using (A8) of [12],

E Ã

k

∑

εig(Xi,Yi)k^G1IFn

!

≤C₂^′ s

nσ²max(ln(C1),ln( 1

αn²σ²)). (19) We now use Inequality A2 in [12] (which is due to Gin´e and Zinn), with t= 64√nσ². We obtain, for m≥1, since for any g∈G,kgk∞≤ ^kK_α^k_n^∞,

P(F_n^c) =P Ã

n⁻¹sup

g∈G

∑

g²(Xi,Yi)>64σ²

!

≤4P(N(ρn^−1/4,G,d_n,2)≥m) +8m exp(−nσ²αn²/kKk²_∞) where n^−1/4ρ=n^−1/4min(σn^1/4,n^1/4) =min(σ,1). Hence by (18),

N(ρn^−1/4,G,d_n,2)≤C(αnmin(σ,1) kKk∞ )^−ν,

Consequently, we have for m= [2C(^αnmin(σ,1)

kKk∞ )^−ν], P(F_n^c)≤16C(αnmin(σ,1)

kKk∞ )^−νexp(−nσ²αn²/kKk²_∞) The last bound together with (19) gives,

P Ã

k

∑

εig(Xi,Yi)k^G>t

!

≤ P(F_n^c) +1 tE

à k

∑

εig(Xi,Yi)k^G1IF_n

!

≤ 16C( kKk∞

αnmin(σ,1))^νexp(−nσ²αn²/kKk²_∞) + C₂^′

t s

nσ²max(ln(C1),ln( 1

αn²σ²)). (20) Let us control the second term in (20). Recall that by (14),σ²=L_α^hdn

n and thus αn²σ²=Lh^d_nαnfor some positive constant L. This fact together with h^d_nαn→0 as n→∞allows to deduce that for n sufficiently large,

C₂^′ t

s

nσ²max(ln(C1),ln( 1

αn²σ²))≤Cst t

s nh^d_n

αn

ln( 1

αnh^d_n). (21) Our task now is to control the first term in (20). We have,

16C( kKk∞

αnmin(σ,1))^νexp(−nσ²αn²/kKk²_∞)

≤L^ν₀^/2exp µ

−nh^d_nαn(L₁+ ν 2nh^d_nαn

ln(αnh^d_n∧αn²))

¶ , which tends to 0 as n→∞as soon as nh^d_nαn→0 and ^ln(α_nh^nh_d^dn∧αn2)

nαn →0. Hence for t≥48Cst

r

nh^d_n αn ln(_α¹

nh^d_n) =: tn, Inequalities (20) and (21) give for n sufficiently large and for t≥t_n

P Ã

k

∑

εig(Xi,Yi)k^G >t

!

≤ 1 24. We conclude using this fact together with Inequality (17), Ek

∑

εig(Xi,Yi)k^G ≤c Ã1

αn

+ s

nh^d_n αn

ln( 1 αnh^d_n)

!

=O Ãsnh^d_n

αn

ln( 1 αnh^d_n)

!

. (22)

Recalling that

Θn= sup

x∈Rd,ε∈[−ε0,ε0]

|ψˆn(q(1+ε),x)−E(ψˆn(q(1+ε),x))|

F(q¯ |x) ,

and collecting (22), (16), yields P

Ã

Θn>A1n⁻¹h⁻_n^d Ã

Ek

∑

εig(Xi,Yi)k^G+t

!!

≤2

· exp

µ

−A2˜L²ln(ln(n)) L

¶ +exp

µ

− A2˜L kKk∞

q

nh^d_nαnln(ln n)

¶¸

,(23)

for any t≥ µ

Cst r

nh^d_n αn ln(_α¹

nh^d_n)

¶

∨˜Lq

nh^d_n

αn ln(ln n)and some ˜L>0. Now, ln n

nαnh^d_n+ln(αnh^d_n∧αn²)

nαnh^d_n ≤ln(nαnh^d_n) nαnh^d_n ,

which tends to 0 as n tends to infinity, since limn→∞nαnh^d_n=∞.Hence,

nlim→∞

ln n nαnh^d_n =0,

which proves that for n sufficiently large nh^d_nαn≥ln n. We conclude then from (23) that,

P Ã

Θn>A1n⁻¹h⁻_n^d Ã

Ek

∑

εig(Xi,Yi)k^G+t

!!

≤4(ln n)^−ρ,

since we have nh^d_nαn≥ln n. We choose ˜L in such a way thatρ>1. Proposition 3 is proved thanks to Borel-Cantelli lemma.

We continue the proof of Theorem 1. Inequality (10), together with Lemma 1 and Condition (4), gives for some universal positive constant C

¯

¯

¯

¯

F(¯ qˆn(αn|x)|x) F(q(α¯ n|x)|x) −1

¯

¯

¯

¯≤ κ

nh^dαngˆn(x) +C sup

|ε|≤ε0

A(q(1+ε),q(1+ε),x,hn) +C sup

|ε|≤ε0

(1+A(q(1+ε),q(1+ε),x,hn))|gˆn(x)−Egˆn(x)| ˆ

gn(x) +C sup

|ε|≤ε0

|ψˆn(q(1+ε),x)−E(ψˆn(q(1+ε),x))|

F(q¯ |x)gˆn(x) . (24) We first use Einmahl and Mason’s result (cf. Proposition 2 above). All the require- ments of Einmahl and Mason result are satisfied from that of Theorem 1. This gives that, for c ln n/n≤h^d_n≤1,

s

nh^d_n

ln(h⁻n^d)∨ln ln n sup

x∈Rd

|gˆn(x)−Egˆn(x)|<C, (25) almost surely. Our task now is to apply Proposition 3. We first claim that Lemma 2 Under Condition(K₃), the class of functionFdefined by(11)satisfies N(ε,F)≤Cε^−ν, for C>0,ε>0,ν>1.

Proof of Lemma 2. Define the set of functionF=K I, where the set of func- tionsK isK =©

u7−→K¡_x−u

h

¢,x∈R^d,h>0ª

,andI={v7−→1Iv>q(1+ε),x∈ R^d,n∈N,|ε| ≤ε0}.The proof of Lemma 2 follows from Lemma A.1 of [12] since N(ε,{v7−→1I_v>y,y∈R})≤Cε^−ν˜ with ˜ν>0 and C>0.

Consequently, all the requirements of Proposition 3 are satisfied from that of The- orem 1. The conclusion of Proposition 3 together with (25), (24) and the facts that

s

ln(h⁻n^d)∨ln ln n

nh^d_n ≤

sln(αn⁻¹h⁻n^d)∨ln ln n nh^d_nαn

, 1

nh^d_nαn ≤

sln(αn⁻¹h⁻n^d)∨ln ln n nαnh^d_n complete the proof of Theorem 1.

4.2 Proof of Proposition 1

Let us introduce Z_i⁽ⁿ⁾(x)for i=1, . . . ,n a triangular array of i.i.d. random variables defined by Z⁽ⁿ⁾_i (x) =YiI_kx−Xik≤h. Their common survival distribution function can be expanded as:

Ψ¯n(t,x) = P(Z₁⁽ⁿ⁾(x)>t) = Z

kx−uk≤h

F¯(t|u)g(u)du

= h^d Z

kvk≤1

F(t¯ |x−hv)g(x−hv)dv, or equivalently,

Ψ¯n(t,x) h^dF(t¯ |x)g(x) =

Z

kvk≤1dv +

Z

kvk≤1

µF¯(t|x−hv) F(t¯ |x) −1

¶g(x−hv) g(x) dv

+ Z

kvk≤1

µg(x−hv) g(x) −1

¶ dv.

Letting v_d=^R_kvk≤1dv the volume of the unit sphere, and assuming that g is Lips- chitzian, it follows,

Ψ¯n(t,x)

h^dF(t¯ |x)g(x)=v_d+o(1) +O(A(t,t,x,0,h)) and introducingβn(x) =n ¯Ψn(q(αn|x)|x), we obtain

βn(x) =vdg(x)nhdαn(1+o(1))

under condition A(q(αn|x),q(αn|x),x,0,h)→0 as n→∞. We now need the fol- lowing lemma (also available to triangular arrays).

Lemma 3 (Klass, 1985) [23] Let Z,Z1,Z2, . . . be a sequence of i.i.d. random vectors and define Mn=max{Z₁, . . . ,Zn}. Suppose that(bn)is nondecreasing, P(Z>bn)→0 and nP(Z>bn)→∞as n→∞. If, moreover,

∑

P(Z>bn)exp{−nP(Z>bn)}=∞, then lim sup_n→∞Mn/bn<1 a.s.

From Lemma 3, a sufficient condition for lim sup

n→∞

max_1≤i≤nZ_i⁽ⁿ⁾(x)

q(αn|x) <1 a.s. (26)

is ∞

n=1

∑

βn(x)

n exp{−βn(x)}=∞, which is fulfilled under (6). Finally,

ˆ qn(αn|x)

q(αn|x) ≤max1≤i≤nZ⁽ⁿ⁾_i (x) q(αn|x) and the conclusion follows from (26).

4.3 Proofs of corollaries

Proof of Corollary 1. For allτ∈[0,1]and(u,x)such that d(u,x)≤Rhn, we have F(q(α¯ n|x)(1+ε)|u)

F(q(¯ αn|x)(1+ε)^τ|x) = q(αn|x)^{θ(x)−θ(u)}(1+ε)^{τθ(x)−θ(u)}

= exp

½θ(u)−θ(x)

θ(x) logαn+ (τθ(x)−θ(u))log(1+ε)

¾

= exp{O(hnlogαn) +O(τθ(x)−θ(u))}

= (1+O(hnlogαn))exp{O(τθ(x)−θ(u))}. (27) Assuming that hnlogαn→0 as n→∞, it follows that (27) is bounded above for all τ∈[0,1]and(u,x)such that d(u,x)≤Rhnand therefore condition (4) is fulfilled.

If, moreover,τ=1 then O(θ(x)−θ(u)) =O(hn)and thus (27) tends to zero as n goes to infinity. Proposition 1 then entails that assumption (3) holds. Theorem 1 implies that

¯

¯

¯

¯

1−F(¯ qˆn(αn|x)|x) F(q(α¯ n|x)|x)

¯

¯

¯

¯

=

¯

¯

¯

¯

¯ 1−

µqˆn(αn|x) q(αn|x)

¶−θ(x)¯

¯

¯

¯

¯

→0 almost surely as n→∞. The conclusion follows.

Proof of Corollary 2. For allτ∈[0,1]and(u,x)such that d(u,x)≤Rhn, we have F(q(α¯ n|x)(1+ε)|u)

F(q(α¯ n|x)(1+ε)^τ|x) = exp

½

(1+ε)log(αn)

µθ(u)−θ(x)

θ(x) +1−(1+ε)^τ−1¶¾

= exp{O(hnlogαn)}expn

(1+ε)log(αn)³

1−(1+ε)^τ−1´o

= (1+O(hnlogαn)o(1) =o(1). (28)

Assuming that hnlogαn→0 as n→∞, it follows that (28) tends to zero as n goes to infinity. Assumptions (3) and (4) both hold. Theorem 1 implies that

¯

¯

¯

¯

1−F¯(qˆn(αn|x)|x) F¯(q(αn|x)|x)

¯

¯

¯

¯

=|1−exp((q(αn|x)−qˆn(αn|x))θ(x))| →0 almost surely as n→∞. The conclusion follows.

Appendix: proof of auxiliary results

Proof of Lemma 1. Clearly,

¯

¯

¯

¯ F¯n(y|x)

F(y¯ |x) −1

¯

¯

¯

¯≤

¯

¯

¯

¯ F¯n(y|x)

F(y¯ |x) − E(ψˆn(y,x)) F(y¯ |x)E(gˆn(x))

¯

¯

¯

¯ +

¯

¯

¯

¯

E(ψˆn(y,x)) F(y¯ |x)E(gˆn(x))−1

¯

¯

¯

¯ (29) We have,

E(ψˆn(y,x)) =1 n

∑

E(Kh(x−Xi)1IY_i>y) =E(Kh(x−X1)1IY₁>y)

=E(K_h(x−X₁)P(Y₁>y|X₁)) = Z

K_h(x−z)P(Y₁>y|X₁=z)g(z)dz

= Z

K_h(x−z)F(y¯ |z)g(z)dz,

andE(gˆn(x)) =¹_n∑ⁿi=1E(Kh(x−Xi)) =E(Kh(x−X1)). Consequently, E(ψˆn(y,x))

F¯(y|x)E(gˆn(x))−1=

= 1

E(Kh(x−X1)) µZ

K_h(x−z)

·F¯(y|z) F(y¯ |x)−1

¸ g(z)dz

¶

= 1

E(K_h(x−X₁)) µZ

K_h(u)

·F¯(y|x−u) F(y¯ |x) −1

¸

g(x−u)du

¶ .

We conclude, since the kernel K is compactly supported, that for some R>0,

¯

¯

¯

¯

E(ψˆn(y,x)) F¯(y|x)E(gˆn(x))−1

¯

¯

¯

¯≤ sup

{x′,d(x,x′)≤hR}

¯

¯

¯

¯ F(y¯ |x^′) F¯(y|x)−1

¯

¯

¯

¯

=A(y,y,x,h). (30) Now,

¯

¯

¯

¯ F¯n(y|x)

F(y¯ |x) − E(ψˆn(y,x)) F(y¯ |x)E(gˆn(x))

¯

¯

¯

¯

≤|ψˆn(y,x)−E(ψˆn(y,x))|

F(y¯ |x)gˆn(x) +E(ψˆn(y,x))|gˆn(x)−Egˆn(x)| F¯(y|x)gˆn(x)Egˆn(x)

≤|ψˆn(y,x)−E(ψˆn(y,x))|

F(y¯ |x)gˆn(x) + (1+A(y,y,x,h))E|gˆn(x)−Egˆn(x)| ˆ

gn(x) , by (30). The last bound together with (30) and (29) prove Lemma 1.

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