Uniform asymptotic properties of a nonparametric regression estimator of conditional tails

www.imstat.org/aihp 2015, Vol. 51, No. 3, 1190–1213

DOI:10.1214/14-AIHP624

© Association des Publications de l’Institut Henri Poincaré, 2015

Uniform asymptotic properties of a nonparametric regression estimator of conditional tails

Yuri Goegebeur

Armelle Guillou

and Gilles Stupfler

aDepartment of Mathematics & Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark.

E-mail:[email protected]

bUniversité de Strasbourg & CNRS, IRMA, UMR 7501, 7 rue René Descartes, 67084 Strasbourg Cedex, France.

E-mail:[email protected]

cCERGAM, Aix Marseille Université, EA 4225, 13540 Puyricard, France. E-mail:[email protected] Received 22 November 2013; accepted 27 April 2014

Abstract. We consider a nonparametric regression estimator of conditional tails introduced by Goegebeur, Y., Guillou, A., Schor- gen, G. (2013).Nonparametric regression estimation of conditional tails – the random covariate case. It is shown that this estimator is uniformly strongly consistent on compact sets and its rate of convergence is given.

Résumé. Nous considérons l’estimateur à noyau de l’indice des valeurs extrêmes conditionnel présenté dans Goegebeur, Y., Guillou, A., Schorgen, G. (2013).Nonparametric regression estimation of conditional tails – the random covariate case. Nous montrons la consistance uniforme presque sûre de cet estimateur sur les compacts et nous calculons sa vitesse de convergence presque sûre.

MSC:62G05; 62G20; 62G32

Keywords:Tail-index; Kernel estimation; Strong uniform consistency

1. Introduction

Extreme value analysis has attracted considerable attention in many fields of application, such as hydrology, biology and finance, for instance. The main result of extreme value theory asserts that the asymptotic distribution of the – prop- erly rescaled – maximum of a sequence(Y₁, . . . , Y_n)of independent copies of a random variableY with distribution functionF is a distribution having the form

G_γ(x)=exp

−(1+γ x)⁻₊^1/γ

, wherey₊=max(0, y)

for someγ∈R, withG₀(x)=exp(−e^−x). The distribution functionFis then said to belong to the maximum domain of attraction ofG_γ and the parameterγ is called the extreme value index. Many applications in the areas of finance, insurance and geology, to name a few, can be found in the case whenγ >0, whereF is a heavy-tailed distribution i.e. the associated survival functionF:=1−F satisfiesF (x)=x^−1/γL(x), whereγ shall now be referred to as the tail-index andLis a slowly varying function at infinity: namely,Lsatisfies, for allλ >0,L(λx)/L(x)→1 asxgoes to infinity. In this case, the parameterγ clearly drives the tail behavior ofF; its estimation is in general a first step of extreme value analysis. For instance, if the idea is to estimate extreme quantiles – namely, quantiles with order α_n>1−1/n, wherenis the sample size – then one has to extrapolate beyond the available data using an extreme value model which depends on the tail-index. For this reason, the problem of estimatingγhas been extensively studied in the literature. Recent overviews on univariate tail-index estimation can be found in the monographs of [2] and [17].

In practice, it is often useful to link the variable of interest Y to a covariate X. In this situation, the tail-index depends on the observed value x of the covariateX and shall be referred to, in the following, as the conditional tail-index. Its estimation has been addressed in the recent extreme value literature, albeit mostly when the covariates are nonrandom. In [31] and [10] a parametric regression model was considered while [19] used a semi-parametric approach to estimate the conditional tail-index. Fully nonparametric methods have been considered using splines (see [4]), local polynomials (see [9]), a moving window approach (see [12]), or a nearest neighbor approach (see [13]), among others.

Less attention though has been paid to the random covariate case, despite its practical interest. One can recall the works of [33], based on a maximum likelihood approach in the Hall class of distribution functions (see [18]), [7] who use a fixed number of nonparametric conditional quantile estimators to estimate the conditional tail-index, later generalized in [6] to a regression context with response distributions belonging to the general max-domain of attraction, and [16] and [14] who both provide adaptations of Hill’s estimator, [22], the latter also studying an average of Hill-type statistics to improve the finite sample performance of the method.

In this paper, we focus on a nonparametric regression estimator of conditional tails introduced by [16]. The par- ticular structure of this estimator makes it possible to study its uniform properties. Note that uniform properties of estimators of the conditional tail-index are seldom considered in the literature. One can think of the work of [14], who study the uniform weak consistency of their estimator. Outside the field of conditional tail-index estimation, uniform convergence of the Parzen–Rosenblatt density estimator ([28] and [29]) was first considered by [27]. His results were then improved by [30] and [32], the latter proving a law of the iterated logarithm in this context. Analogous results on kernel regression estimators were obtained by, among others, [26], [20] and [11]. Uniform consistency of isotonized versions of order-αquantile estimators introduced in [1] was shown in [8]. The case of estimators of the left-truncated quantiles is considered in [25]. Finally, the uniform strong consistency of a frontier estimator using kernel regression on high order moments was shown in [15].

The paper is organised as follows. Our main results are stated in Section2. The estimator is shown to be uniformly strongly consistent on compact sets in a semiparametric framework. The rate of convergence is provided when a further condition on the bias is satisfied. The rate of uniform convergence is closely linked to the rate of pointwise convergence in distribution established in [16]. The proofs of the main results are given in Section3. Auxiliary results are postponed to theAppendix.

2. Main results

We assume that the covariateXtakes its values inR^dfor somed≥1. We shall work in the following semiparametric framework:

(SP) Xhas a probability density functionf with supportS⊂R^d having nonempty interior and the conditional survival function ofY givenX=xis such that

∀x∈S,∀y≥1, F (y|x)=y^{−1/γ (x)}L(y|x),

whereγ (x) >0 andL(·|x)is a slowly varying function at infinity.

The estimator of the conditional tail-index we shall study in this paper is defined as

γ_n(x):=

_n

i=1K_h(x−_nX_i)(logY_i−logω_n,x)₊1_{Yi>ωn,x}

i=1K_h(x−X_i)1_{Yi>ωn,x} . (1)

Here K_h(u):=h^−dK(u/ h) whereK is a probability density function on R^d andh:=h_n is a positive sequence tending to 0 while for all x,(ωn,x)is a positive sequence tending to infinity. Note that γn(x)=Tn⁽¹⁾(x)/Tn⁽⁰⁾(x) where, for allt≥0,

T_n^(t)(x):=1 n

n

i=1

Kh(x−Xi)(logYi−logωn,x)^t₊1_{Y_i>ω_n,x}.

The estimator (1) is an element of the family of estimators introduced in [16], which can be seen as an adaptation of the classical Hill estimator of the tail-index for univariate distributions (see [22]). Note that the thresholdω_n,xis local, i.e. it depends on the pointxwhere the estimation is to be made, while the bandwidthhis global.

We first wish to state the uniform strong consistency of our estimator on an arbitrary compact subsetΩ of R^d contained in the interior ofS. To this end, we first assume that for everyx ∈S the slowly varying functionL(·|x) appearing inF (·|x)is normalised (see [3]):

(A1) For allx∈Sandy≥1, L(y|x)=c_L(x)exp

_y

1

α(v|x) v dv ,

wherec_L(x) >0 andα(·|x)is a function converging to 0 at infinity.

Let · be a norm onR^d and forr >0, letΩ^r be the set of those points inR^d whose distance toΩ is not more thanr:

Ω^r=

x∈R^d|∃x ∈Ω,x−x≤r .

Remark that sinceΩ is contained in the interior of the closed setS, the distance ofΩ to the boundary ofSmust be positive. As a consequence, the setΩ^ris contained inSfor allr >0 small enough. We can therefore introduce some classical regularity assumptions:

(A2) For somer >0, onΩ^r, the functionsf andγ are positive Hölder continuous functions, logcLis a Hölder continuous function andα(y|·)is a Hölder continuous function uniformly iny≥1: for allx, x ∈Ω^r,

f (x)−f

x≤Mfx−x^ηf, γ (x)−γ

x ≤Mγx−x ^ηγ, logcL(x)−logcL

x ≤Mc_Lx−x^ηcL, sup

y≥1

α(y|x)−α

y|x≤Mαx−x ^ηα.

Let moreoverη:=η_γ∧η_cL ∧η_α. We introduce the oscillation of x→logω_n,x at a pointx∈R^d over the ball B(x, ε):

∀ε >0, (logω_n,x)(ε):= sup

z∈B(x,ε)

|logω_n,x−logω_n,z|

and the quantityα(y|x):=sup_t≥y|α(t|x)| for ally≥1. Our results are established under the following classical regularity condition on the kernel:

(K) K is a probability density function which is Hölder continuous with Hölder exponent η_K >0: for all x, x ∈R^d,

K(x)−K

x ≤M_Kx−x^ηK

and its support is included in the unit ballBofR^d.

Especially, if(K)holds thenKis bounded with compact support. Let vn(x):=

nh^d

lognF (ωn,x|x) and introduce the hypothesis

(C) For someb >0, it holds that lim sup_n→∞sup_x∈Ωvn(x) (logωn,x)(n^−b) <∞.

Our uniform strong consistency result may now be stated:

Theorem 1. Assume that(SP),(K),(A₁)and(A₂)hold and that:

• infx∈Ωv_n(x)→ ∞;

• infx∈Ωωn,x→ ∞;

• h^ηsup_x∈Ωlogω_n,x→0;

• sup_x∈Ω (logω_n,x)(h)→0;

• sup_x∈Ωα(y|x)→0asy→ ∞.

Assume moreover that condition(C)is satisfied.Then it holds that

sup

x∈Ω

γ_n(x)−γ (x)→0 almost surely asn→ ∞.

Note that the hypotheses infx∈Ωω_n,x→ ∞and sup_x∈Ωα(y|x)→0 asy→ ∞imply the convergence sup

x∈Ω

α(ωn,x|x)→0

which shall frequently be used in the proofs of our results. Besides, using the mean value theorem, it holds that|e^u− 1| ≤2|u|foru∈Rsuch that|u|is sufficiently small. As a consequence, using the condition sup_x∈Ω (logωn,x)(h)→ 0, this inequality implies that fornlarge enough

sup

x∈Ω

sup

z∈B(x,h)

ω_n,x ω_n,z −1

≤2 sup

x∈Ω

(logω_n,x)(h)→0. (2)

Finally, the conditions sup

x∈Ω

(logω_n,x)(h)→0 and lim sup

n→∞ sup

x∈Ω

v_n(x) (logω_n,x) n^−b

<∞

are satisfied if for instance ωn,x =n^g(x) whereg:S→R is a positive Hölder continuous function whose Hölder exponent is not less thanη. In other words, Theorem1requires that a continuity property onx→logω_n,xbe satisfied.

Our second aim is to compute the rate of uniform strong consistency of the estimator (1):

Theorem 2. Assume that the conditions of Theorem1are satisfied.If moreover lim sup

n→∞

sup

x∈Ω

v_n(x)

α(ω_n,x|x)∨h^ηf ∨h^ηlogω_n,x∨ (logω_n,x)(h)

<∞ (3)

then it holds that sup

x∈Ω

vn(x)γn(x)−γ (x)=O(1) almost surely asn→ ∞.

Let us highlight that condition (3) controls the bias of the estimatorγ_n. The termsh^ηf andh^ηlogω_n,xcorrespond to the bias which stems from the use of a kernel regression, while the presence of the other terms is due to the particular structure of the semiparametric model(SP). Besides, as pointed out in [16], the rate of pointwise convergence ofγ_n(x) toγ (x)is[nh^dF (ω_n,x|x)]^1/2. Up to the term[logn]^1/2, the rate of uniform convergence ofγ_ntoγ is therefore the infimum (overΩ) of the rate of pointwise convergence ofγn(x)toγ (x).

3. Proofs of the main results

Before starting the proof of Theorem1, let us note that assuming that(SP),(A₁)and(A₂)hold then it is easy to show that there exists a positive constantM_F such that the function(x, y)→logF (y|x)has the following property: for all x, x ∈Ω^r such thatx−x ≤1 andy, y ≥e,

log F (y|x) F (y|x)

≤M_Fx−x^ηlogy+ 1

γ (x)+α

y∧y|x logy−logy. (4)

Moreover, if(A₂)holds then one may take a positive numberrsuch that the four conditions of the hypothesis hold onΩ^2r. SinceΩ^r is compact,f :=sup_Ω^rf <∞ andf :=infΩ^rf >0. As a consequence, the uniform relative oscillation off over the ballB(x, h)can be controlled as

sup

x∈Ω^r

sup

z∈B(x,h)

f (z) f (x)−1

=O h^ηf

→0. (5)

Second,γ:=sup_Ωrγ <∞andγ:=infΩ^rγ >0 and we thus have sup

x∈Ω^r

sup

z∈B(x,h)

γ (z) γ (x)−1

=O h^ηγ

→0. (6)

Third, we can write for allx, x ∈Ω^r andt≥1 α(t|x)≤α

t|x

+α(t|x)−α t|x

and the roles ofx andx are symmetric in the above inequality, so that taking the supremum overt≥yon both sides yields

∀y≥1, α(y|x)−α

y|x≤Mαx−x^ηα. (7)

We may now prove the key result for the proof of Theorem1, which is a uniform law of large numbers forT_n⁽⁰⁾(x) andT_n⁽¹⁾(x). In what follows, we letμ^(t)_n (x):=E(T_n^(t)(x)).

Proposition 1. Assume that the conditions of Theorem1are satisfied.Then for everyt∈ {0,1}it holds that sup

x∈Ω

v_n(x) Tn^(t)(x)

μ^(t)_n (x)−1

=O(1) almost surely asn→ ∞.

Proof. The proof is based on that of Lemma 1 in [21]: we shall in fact show complete convergence in the sense of [24]. SinceΩ is a compact subset ofR^d, we may, for everyn∈N\ {0}, find a finite subsetΩ_nofΩsuch that:

∀x∈Ω,∃χ (x)∈Ωn, x−χ (x)≤n^−b and ∃c >0, |Ωn| =O n^c

,

whereb, which we may take to be not less than 1/d+1/2ηK, is given by condition(C) and|Ω_n| stands for the cardinality ofΩ_n. Notice that, sincenh^d→ ∞, one hasn^−b/ h→0, so that one can assume that eventuallyχ (x)∈ B(x, h)for allx∈Ω. Next, remark thatx−χ (x) ≤n^−b≤h≤1 and that sincen^−b≤hthe convergences

n^−bηsup

x∈Ω

logωn,x≤h^ηsup

x∈Ω

logωn,x→0 and sup

x∈Ω

(logωn,x) n^−b

≤sup

x∈Ω

(logωn,x)(h)→0 hold. Consequently, Lemma1entails

sup

x∈Ω

v_n(x) v_n(χ (x))−1

=sup

x∈Ω

F (ω_n,x|x) F (ω_{n,χ (x)}|χ (x))−1

→0. (8)

Pickε >0 and an arbitrary sequence of positive numbers(δ_n)converging to 0; using together (8) and the triangular inequality thus shows that fornlarge enough

P

δnsup

x∈Ω

vn(x) T_n^(t)(x)

μ^(t)_n (x)−1

> ε ≤R1,n+R2,n,

where

R_1,n:=

z∈Ω_n

P

δ_nv_n(z) T_n^(t)(z)

μ^(t)_n (z)−1 >ε

4 and

R_2,n:=P

δ_nsup

x∈Ω

v_n(x) T_n^(t)(x)

μ^(t)_n (x)−T_n^(t)(χ (x)) μ^(t)_n (χ (x))

>ε 2 . The goal of the proof is now to show that the series

nR1,nand

nR2,nconverge. The result of Proposition1shall then be an easy consequence of Borel–Cantelli’s lemma and Lemma6.

We start by controllingR_1,n. To this end, apply Lemma3to get that there exists a positive constantκsuch that for nlarge enough,

∀z∈Ω_n, P

δ_nv_n(z) T_n^(t)(z)

μ^(t)_n (z)−1 >ε

4 ≤2 exp

−κ

16ε²nh^dF (ω_n,z|z) δ²_nv_n²(z)

.

Use now the definition ofv_n(z)to get R1,n=O

n^cexp

−κ

16ε²logn δ²_n . Hence

nR_1,nconverges.

We now turn toR2,n. Using the triangular inequality gives R2,n≤P

δnsup

x∈Ω

vn(x)S1,n(x) > ε 4 +P

δnsup

x∈Ω

vn(x)S2,n(x) >ε

4 =:R3,n+R4,n, where

S1,n(x):=1 n

n

i=1

K_h(x−X_i)

μ^(t)_n (x) −K_h(χ (x)−X_i) μ^(t)_n (χ (x))

(logYi−logω_{n,χ (x)})^t₊1_{Y_i>ω_{n,χ (x)}},

S_2,n(x):=1 n

n

i=1

K_h(x−X_i) μ^(t)_n (x)

(logY_i−logω_n,x)^t₊1_{Y_i>ωn,x}−(logY_i−logω_{n,χ (x)})^t₊1_{Y_i>ω_{n,χ (x)}}, and it is enough to show that the series

nR_3,nand

nR_4,nconverge.

To deal with

nR3,nuse once again the triangular inequality to obtain μ^(t)_n χ (x)Kh(x−Xi)

μ^(t)_n (x) −Kh(χ (x)−Xi) μ^(t)_n (χ (x))

≤K_h(x−X_i)−K_h

χ (x)−X_i +

μ^(t)_n (χ (x)) μ^(t)_n (x) −1

K_h(x−X_i).

Using hypothesis(K)and Lemma4, there exists a positive constantMsuch that fornlarge enough:

∀x∈Ω, μ^(t)_n χ (x)Kh(x−Xi)

μ^(t)_n (x) −Kh(χ (x)−Xi) μ^(t)_n (χ (x))

≤ M h^d

n^−b h

η_K

∨ (logω_n,x) n^−b

.

Besides

m^(t)_n (z):=1 n

n

i=1

K2h(z−Xi)(logYi−logωn,z)^t₊1_{Y_i>ω_n,z}

is the empirical analogue ofm^(t)n (z)defined before Lemma4; since the support of the random variableKh(x−Xi)is included inB(χ (x),2h), one has fornlarge enough

∀x∈Ω, v_n(x)S1,n(x)≤2^dVMv_n(x) n^−b

h ηK

∨ (logω_n,x)

n^−b m^(t)_n (χ (x)) μ^(t)_n (χ (x)) .

Moreover, sincem^(t)_n (z)is a kernel estimator ofm^(t)_n (z, z)for which the conditions of Lemma2are satisfied, we get fornlarge enough:

∀z∈Ω_n, δ_nv_n(z)m^(t)_n (z)

μ^(t)_n (z) ≤2δnv_n(z)

1+ m^(t)_n (z)

m^(t)_n (z)−1

.

The fact thatb≥1/d+1/2ηKgives sup

z∈Ωn

v_n(z) n^−b

h η_K

≤√ n

n^−b h

η_K

≤ 1

nh^d η_K/d

→0.

Using first this convergence together with hypothesis(C)and (8) and then Lemma3entails fornlarge enough:

R_3,n≤

z∈Ωn

P

δ_nv_n(z) m^(t)n (z)

m^(t)_n (z)−1

> ε =O

n^cexp

−κε²logn δ²_n , whereκ is a positive constant. Hence

nR3,nconverges.

To control

nR_4,nfirst use Lemmas2(iv) and4to get, fornlarge enough sup

x∈Ω

m^(t)n (χ (x)) μ^(t)_n (x) =sup

x∈Ω

m^(t)n (χ (x)) μ^(t)_n (χ (x))

μ^(t)n (χ (x)) μ^(t)_n (x)

≤2.

Therefore, since the support of the random variableK_h(x−X_i)is included inB(χ (x),2h), one has fornlarge enough and allx∈Ω

S_2,n(x)≤2^d+1VK_∞S_3,n(x), whereK_∞:=sup_BKand

S3,n(x):=1 n

n

i=1

K2h(χ (x)−X_i) m^(t)n (χ (x))

(logYi−logωn,x)^t₊1_{Y_i>ω_n,x}−(logYi−logωn,χ (x))^t₊1_{Y_i>ω_{n,χ (x)}}.

We then get R4,n≤P

δnsup

x∈Ω

vn(x)S3,n(x) > ε

2^d+3VK∞ =:R5,n

and it is enough to control

nR_5,n. We start by considering the caset=0. In this case,S_3,n(x)reduces to S_3,n(x)=1

n n

i=1

K2h(χ (x)−X_i)

m⁽⁰⁾_n (χ (x)) 1_{ωn,x∧ωn,χ (x)<Yi≤ωn,x∨ωn,χ (x)}.

Lettingρ_n,x:=2 (logω_n,x)(n^−b)and using (2), we have sup_x∈Ωρ_n,x→0 and fornlarge enough

∀x∈Ω, (1−ρ_{n,χ (x)})ω_{n,χ (x)}≤ω_n,x≤(1+ρ_{n,χ (x)})ω_{n,χ (x)}. As a consequence, fornlarge enough it holds that

∀x∈Ω, S_3,n(x)≤ 1 n

n

i=1

K2h(χ (x)−Xi)

m⁽⁰⁾_n (χ (x)) 1_{(1−ρ_{n,χ (x)})ω_{n,χ (x)}<Yi≤(1+ρ_{n,χ (x)})ω_{n,χ (x)}}. Similarly to Lemma5, let

Mn(x):=E

K2h(x−X)1_{(1−ρ_n,x)ω_n,x<Y <(1+ρ_n,x)ω_n,x} and

U_n(x):=1 n

n

i=1

K2h(x−X_i)1_{(1−ρ_n,x)ω_n,x<Y_i<(1+ρ_n,x)ω_n,x}.

Write

∀x∈Ω, δ_nv_n(x)S_3,n(x)≤δ_nv_n(x)M_n(χ (x)) m⁽⁰⁾_n (χ (x))

1+

U_n(χ (x)) M_n(χ (x))−1

.

Use together Lemmas2(iv) and5along with (8) to get fornlarge enough

∀x∈Ω, δnvn(x)S3,n(x)≤ 4

γ (χ (x))δnvn

χ (x) ρn,χ (x)

1+

U_n(χ (x)) M_n(χ (x))−1

.

Recall thatρn,x=2 (logωn,x)(n^−b)and that condition(C)is satisfied to obtain δ_n sup

z∈Ωn

v_n(z)ρ_n,z→0.

Therefore, since 0< γ ≤γ (χ (x)), the triangular inequality implies that R5,n≤

z∈Ωn

P

δnvn(z)ρn,z

U_n(z) M_n(z)−1

> εγ 2^d+6VK∞

fornlarge enough. Lemma5now makes it clear that R_5,n=O

n^c sup

z∈Ωn

exp

−κ εγ

2^d+6VK_∞v_n(z)logn δn

=o

n^cexp

−κ εlogn δn

which proves that

nR_5,nconverges in this case.

If nowt=1, we recall (45) in the proof of Lemma4to get fornlarge enough and for allx∈Ω S3,n(x)=

log ω_n,x ω_{n,χ (x)}

m⁽⁰⁾_n (χ (x)) m⁽¹⁾n (χ (x)) 1 n

n

i=1

K2h(χ (x)−X_i) m⁽⁰⁾n (χ (x))

1_{Y_i>ω_n,x∧ω_{n,χ (x)}}.

Use (2) and Lemma2(iv) to get fornlarge enough

∀x∈Ω, S_3,n(x)≤ 2

γ (logω_n,x) n^−b1

n n

i=1

K2h(χ (x)−X_i)

m⁽⁰⁾_n (χ (x)) 1_{Y_i>ω_{n,χ (x)}/2}

≤ 2

γ (logω_n,x)

n^−b ν_n(χ (x)) m⁽⁰⁾_n (χ (x))

1+

V_n(χ (x)) ν_n(χ (x)) −1

, (9)

where

νn(x):=E

K2h(x−X)1_{Y >ω_n,x/2}

and Vn(x):=1 n

n

i=1

K2h(x−Xi)1_{Y_i>ω_n,x/2}.

The family of sequences(ωn,x/2)clearly satisfies the hypotheses of Lemmas2and3: in particular sup

x∈Ω

ν_n(x) m⁽⁰⁾_n (x)

F (ω_n,x|x) F (ω_n,x/2|x)−1

→0 (10)

and there exists a positive constantκ such that fornlarge enough

∀x∈Ω, P V_n(x)

ν_n(x) −1

> ε ≤2 exp

−κ ε²nh^dF (ωn,x|x)

, (11)

where the inequalityF (ωn,x/2|x)≥F (ωn,x|x)was used. We conclude by noting that according to (4), lim sup

n→∞ sup

x∈Ω

log F (ω_n,x|x) F (ω_n,x/2|x)

≤log 2

γ <∞ ⇒ 0<lim sup

n→∞ sup

x∈Ω

F (ω_n,x|x) F (ω_n,x/2|x)<∞. This property together with (10) entails the convergences

δ_nsup

x∈Ω

ν_n(x)

m⁽⁰⁾_n (x)→0 and sup

x∈Ω

(logω_n,x)

n^−b ν_n(x)

m⁽⁰⁾_n (x)→0. (12) Reporting (10) along with (12) into (9), recalling condition(C)and using the triangular inequality together with (8) shows that fornlarge enough,

R_5,n≤

z∈Ωn

P

δ_nv_n(z) Vn(x)

ν_n(x)−1

> ε =O

n^cexp

−κ ε²logn δ²_n

,

where (11) was used in the last step. As a consequence,

nR5,nconverges in this case as well. This completes the

proof of Proposition1.

With Proposition1at hand, we can now prove Theorems1and2.

Proof of Theorem1. Notice that

γn(x)=μ⁽¹⁾_n (x) μ⁽⁰⁾n (x)

T_n⁽¹⁾(x) μ⁽¹⁾n (x)

μ⁽⁰⁾_n (x) Tn⁽⁰⁾(x)

. (13)

Applying Proposition1twice yields sup

x∈Ω

T_n⁽¹⁾(x) μ⁽¹⁾_n (x)

μ⁽⁰⁾_n (x) T_n⁽⁰⁾(x)−1

→0 almost surely asn→ ∞. (14)

Moreover, recalling that γ is continuous and therefore bounded on the compact setΩ, using Lemma2(i) and (iv) twice entails

sup

x∈Ω

μ⁽¹⁾_n (x)

μ⁽⁰⁾_n (x)−γ (x)

→0 asn→ ∞. (15)

The result follows by reporting (14) and (15) into (13).

Proof of Theorem2. Note that becausenh^d→ ∞, the hypothesis lim sup

n→∞ sup

x∈Ω

v_n(x) (logω_n,x)(h) <∞

entails condition(C). Besides, Proposition1yields sup

x∈Ω

v_n(x) μ⁽⁰⁾n (x)

T_n⁽⁰⁾(x)−1

=O(1) and sup

x∈Ω

v_n(x) Tn⁽¹⁾(x)

μ⁽¹⁾_n (x)−1

=O(1) (16) almost surely asn→ ∞. Moreover, Lemma2(iv) gives

sup

x∈Ω

1

α(ω_n,x|x)∨h^ηf ∨h^ηlogω_n,x

μ^(t)_n (x)

f (x)F (ωn,x|x)−γ^t(x) =O(1) fort∈ {0,1}, so that using condition (3),

sup

x∈Ω

v_n(x) μ⁽¹⁾_n (x)

μ⁽⁰⁾_n (x)−γ (x)

=O(1). (17)

The result follows by reporting (16) and (17) into (13).

Appendix: Auxiliary results and proofs

The first lemma of this section is a technical result that gives an upper bound for the oscillation of the log-conditional survival function.

Lemma 1. Assume that (SP),(A1)and(A2)hold.Let moreoverε:=εn,ε :=ε_n and ε :=ε_n be three positive sequences tending to0and assume that:

• infx∈Ωω_n,x→ ∞;

• ε ^ηsup_x∈Ωlogω_n,x→0;

• sup_x∈Ω (logω_n,x)(ε)→0;

• sup_x∈Ωα(y|x)→0asy→ ∞.

Then it holds that,fornlarge enough,

∀ x, x

∈Ω×Ω^ε,∀ z, z

∈B x, ε

×B x , ε

, logF (ωn,z|z)

F (ω_n,x|x)

≤M_Fε ^ηlogω_n,z+ 2

γ (logω_n,x) ε

.

In particular, sup

x∈Ω

sup

x∈Ω^ε

sup

z∈B(x,ε)

sup

z∈B(x,ε )

1

ε ^ηlogωn,x∨ (logωn,x)(ε)

F (ω_n,z|z) F (ω_n,x|x)−1

=O(1).

Proof. Pick(x, x)∈Ω×Ω^εand(z, z)∈B(x, ε)×B(x, ε ). Use (4) to get fornlarge enough logF (ω_n,z|z)

F (ωn,x|x)

≤M_Fx −z^ηlogω_n,z+ 1

γ (x)+α

ω_n,z∧ω_n,x|x |logω_n,x−logω_n,z|. For everyy≥1, inequality (7) entails

sup

x∈Ω^ε

α(y|x)≤sup

x∈Ω

α(y|x)+Mαε^ηα. (18)

Using then (2) withε instead ofh, we get infx∈Ωinfz∈B(x,ε)ωn,z∧ωn,x=infx∈Ωωn,x(1+o(1))→ ∞, so that sup

x∈Ω

sup

x∈Ω^ε

sup

z∈B(x,ε)

α

ω_n,z∧ω_n,x|x

→0.

Especially, since 0< γ ≤γ (x), we obtain fornlarge enough:

∀ x, x

∈Ω×Ω^ε,∀ z, z

∈B x, ε

×B x, ε

, log F (ω_n,z|z)

F (ω_n,x|x)

≤M_Fε ^ηlogω_n,z+2

γ (logω_n,x) ε

which is the first part of the result. To prove the second part, note that because sup_x∈Ω (logω_n,x)(ε)→0 it holds that fornlarge enough

∀ x, x

∈Ω×Ω^ε,∀ z, z

∈B x, ε

×B x, ε

, log F (ωn,z|z)

F (ω_n,x|x)

≤2M_Fε ^ηlogω_n,x+ 2

γ (logω_n,x) ε

.

Consequently sup

x∈Ω

sup

x∈Ω^ε

sup

z∈B(x,ε)

sup

z∈B(x,ε )

1

ε ^ηlogω_n,x∨ (logω_n,x)(ε)

log F (ωn,z|z) F (ω_n,x|x)

=O(1).

Using the equivalent e^u−1=u(1+o(1))therefore completes the proof of Lemma1.

The second lemma examines the behavior of the conditional moment m^(t)_n (x, z):=E

(logY−logω_n,x)^t₊1_{Y >ωn,x}|X=z

and that of its smoothed versionμ^(t)_n (x)=E(K_h(x−X)m^(t)_n (x, X)). Letbe Euler’s Gamma function:

∀t >0, (t ):=

_+∞

0

v^t−1e^−vdv.

Lemma 2. Assume that(SP),(A₁)and(A₂)hold.Pickt≥0and assume thatK is a bounded probability density function onR^dwith support included inB.If moreover:

• infx∈Ωωn,x→ ∞;

• h^ηsup_x∈Ωlogωn,x→0;

• sup_x∈Ωα(y|x)→0asy→ ∞

then,asn→ ∞,the following estimations hold:

(i) sup_x∈Ωsup_{z∈B(x,h)α(ω} ¹

n,x|x)∨h^ηα|_γt(z)(t^m(t)n+1)F (ω^(x,z) _n,x|z)−1| =O(1),