Yuri Goegebeur

Armelle Guillou

Jing Qin

Bias-corrected estimation for conditional Pareto-type distributions with random right censoring

Keywords: Pareto-type model, random covariate, random right censoring, local estimation, bias-reduction

We consider bias-reduced estimation of the extreme value index in conditional Pareto-type models with random covariates when the response variable is subject to random right censoring. The bias-correction is obtained by fitting the extended Pareto distribution locally to the relative excesses over a high threshold using the maximum likelihood method. Consistency and asymptotic normality of the estimators are established under suitable assumptions. The finite sample behaviour is illustrated with a small simulation experiment and the method is applied to AIDS survival data.

Introduction

In many studies it is more interesting to focus on the tail of a distribution rather than its centre. For instance insurance companies will typically be interested in modelling the large claims generated by a portfolio, as these may pose a major threat to the solvency of the company. Accurate modelling of the upper tail of the claim size distribution is also crucial when such companies enter re-insurance contracts, in particular for the pricing of such contracts. As a second example, the design of secure coastal defense structures is naturally based on analysis of extreme storm events. Next to extremes one is in practical applications also often faced with data that are incomplete due to censoring, meaning that only a lower bound for the variable of interest is available. In this paper, we will consider the analysis of tails of heavy tailed distributions when there are random covariates and when the responses are subject to random right censoring.

Let Y be the random variable of main interest, and let C be the censoring random variable, where both have distributions that are of Pareto-type, i.e. the distribution functions are of the form 1 -F • (y) = y -1/γ• l • (y), y > 0, 1 with • denoting either Y or C, and where γ • , called the extreme value index, is a positive parameter, and l • is a slowly varying function at infinity meaning that l • (ty)/l • (t) → 1 as t → ∞ for all y > 0. The class of Pareto-type models corresponds with the max-domain of attraction of extreme value distributions with positive extreme value index and finds important applications in areas like geology, climatology, finance and actuarial science. We refer to Beirlant et al. The above described Pareto-type model will be extended to the situation where a random covariate X is observed together with the variable of main interest. In this case, we consider the conditional distributions of Y and C given X = x, and assume these are of Pareto-type, meaning that γ Y , γ C , l Y and l C will depend on x. The estimation of the positive extreme value index and extreme quantiles in presence of random covariates received a lot of attention in the extreme value literature, see e.g. [START_REF] Wang | Tail index regression[END_REF] (2016) estimators explicitly address the bias-issue that is typically present when estimating tail parameters. In the current paper, we will introduce a bias-corrected estimator for the conditional extreme value index when there is random right censoring and random covariates. The estimator is obtained by fitting the extended Pareto distribution locally with the maximum likelihood method to the response observations. In this, as is classical in the context of censoring (see e.g. [START_REF] Lawless | Statistical models and methods for lifetime data[END_REF], the likelihood contribution of non-censored observations consists of the density of the extended Pareto distribution, while censored observations contribute the survival function of the latter. A similar idea was applied by [START_REF] Beirlant | Bias reduced tail estimation for censored Pareto type distributions[END_REF] in the univariate extreme value context, and in a more restrictive Pareto-type model.

The remainder of the paper is organised as follows. In the next section, we introduce a second order Pareto-type model and the local likelihood estimation method. The asymptotic properties of the obtained estimators are studied in Section 3, where we prove consistency and asymptotic normality. The finite sample behaviour is examined by a simulation experiment in Section 4, while in Section 5 we illustrate the method to AIDS survival data. The Appendix contains the proofs of the results.

Construction of the estimator

Let Y denote the response variable of interest, and C be the random right censoring time, both having a distribution depending on a random covariate X, and conditionally on X we assume Y and C to be independent random variables. We observe the random covariate X, T := Y ∧ C, and a censoring indicator 1l {Y ≤C} . The covariate X has density function f X with support S X ⊂ R d , having non-empty interior. The conditional distributions of Y and C given X = x, denoted F Y (.|x) and F C (.|x), respectively, are assumed to be of the following form.

Assumption (D) For all x ∈ S X , the conditional survival functions of Y and C satisfy, with • denoting either Y or C,

F • (y|x) = A • (x)y -1/γ•(x) 1 + 1 γ • (x) δ • (y|x) ,
where

A • (x) > 0, γ • (x) > 0, and |δ • (.|x)| is normalised regularly varying with index -β • (x), β • (x) > 0, i.e. δ • (y|x) = B • (x) exp y 1 ε • (u|x) u du , with B • (x) ∈ R and ε • (y|x) → -β • (x) as y → ∞.
Moreover, we assume y → ε • (y|x) to be a continuous function.

Note that our model (D) is more general than [START_REF] Beirlant | Bias reduced tail estimation for censored Pareto type distributions[END_REF], where the Hall-class of Pareto-type models was considered [START_REF] Hall | On some simple estimates of an exponent of regular variation[END_REF]. Also, as mentioned before, Beirlant et al.

(2016) is in the univariate context and does not consider the situation of random covariates.

Under assumption (D), F Y (.|x) and F C (.|x) have density functions. Indeed, straightforward differentiation gives

f • (y|x) = A • (x) γ • (x) y -1/γ•(x)-1 1 + 1 γ • (x) -ε • (y|x) δ • (y|x) . (1)
For distribution functions satisfying (D), one can approximate the conditional distribution function of Z := Y /t, given that Y > t, or Z := C/t given that C > t, where t denotes a high threshold value, by the extended Pareto distribution [START_REF] Beirlant | Statistics of Extremes -Theory and Applications[END_REF][START_REF] Beirlant | Second-order refined peaks-over-threshold modelling for heavy-tailed distributions[END_REF], with distribution function given by

G(z; γ, δ, β) = 1 -[z(1 + δ -δz -β )] -1/γ , z > 1, 0, z ≤ 1, (2)
and density function

g(z; γ, δ, β) = 1 γ z -1/γ-1 [1 + δ(1 -z -β )] -1/γ-1 [1 + δ(1 -(1 -β)z -β )], z > 1, 0, z ≤ 1,
where γ > 0, β > 0, and δ > max{-1, -1/β}. Indeed, as shown in [START_REF] Beirlant | Second-order refined peaks-over-threshold modelling for heavy-tailed distributions[END_REF], one has that

sup z≥1 F • (tz|x) F • (t|x) -G(z; γ • (x), δ • (t|x), β • (x)) = o(|δ • (t|x)|), if t → ∞.
Clearly, based on this result, one can obtain an estimator for γ Y (x) by fitting the extended Pareto distribution to the relative excesses over a high threshold t n with the maximum likelihood method, adjusted for local estimation and censoring. Let (T i , X i , 1l {Y i ≤C i } ), i = 1, . . . , n, be independent copies of the random vector (T, X, 1l {Y ≤C} ), and take x 0 ∈ Int(S X ), the interior of S X . We estimate (γ Y (x 0 ), δ Y (t n |x 0 )) by maximising the following local log-likelihood function

(γ Y , δ Y ; β) := 1 n n i=1 K hn (x 0 -X i ) ln g T i t n ; γ Y , δ Y , β 1l {Y i ≤C i } + ln G T i t n ; γ Y , δ Y , β 1l {Y i >C i } 1l {T i >tn} , (3)
where

K hn (x) := K(x/h n )/h d n , K is a joint density function on R d , h n is a non-random sequence of bandwidths with h n → 0 if n → ∞, 1l
{A} is the indicator function on the event A, and t n is a local non-random threshold sequence satisfying t n → ∞ if n → ∞. In this paper, we only estimate γ Y (x 0 ) and δ Y (t n |x 0 ) with the local likelihood method, while the parameter β will be fixed at some value. In the framework of bias-corrected estimation it is quite common to fix the second order rate parameter at some value, see e.g. [START_REF] Feuerverger | Estimating a tail exponent by modelling departure from a Pareto distribution[END_REF], [START_REF] Gomes | Bias-reduction and explicit semi-parametric estimation of the tail index[END_REF], [START_REF] Dutang | Robust and bias-corrected estimation of the coefficient of tail dependence[END_REF][START_REF] Dutang | Robust and bias-corrected estimation of the probability of extreme failure sets[END_REF]. As we will see later, when using the correct value for this parameter one obtains theoretically asymptotically unbiased estimators, while one loses this unbiasedness when β is mis-specified, but in the latter case, the estimators perform typically still better than uncorrected estimators in practice.

Asymptotic properties

In this section, we derive the asymptotic properties of solutions ( γ Y,n (x 0 ), δ Y,n (t n |x 0 )) of the likelihood equations.

In order to deal with the regression context, the functions appearing in F Y (y|x) and F C (y|x) are assumed to follow the following Hölder conditions. Let . denote some norm on R d .

Assumption (H) There exist positive constants M f X , M A• , M γ• , M B• , M ε• , η f X , η A• , η γ• , η B• and η ε• , where • denotes either Y or C, such that for all x, z ∈ S X : |f X (x) -f X (z)| ≤ M f X x -z η f X , |A • (x) -A • (z)| ≤ M A• x -z η A• , |γ • (x) -γ • (z)| ≤ M γ• x -z ηγ • , |B • (x) -B • (z)| ≤ M B• x -z η B• , sup y≥1 |ε • (y|x) -ε • (y|z)| ≤ M ε• x -z ηε • .
For the kernel function K we assume the following:

Assumption (K) K is a bounded density function on R d , with support S K included in the unit hypersphere in R d .
For the sequel it is instructive to note that the random variable T has a conditional distribution function satisfying the same properties as those of Y and C. Indeed, by straightforward computations one obtains:

F T (y|x) = F Y (y|x)F C (y|x) = A T (x)y -1/γ T (x) 1 + 1 γ T (x) δ T (y|x) , where A T (x) := A Y (x)A C (x), γ T (x) := γ Y (x)γ C (x)/(γ Y (x) + γ C (x))
, and

δ T (y|x) :=    γ T (x)/γ Y (x)δ Y (y|x)(1 + o(1)) if δ Y (y|x)/δ C (y|x) → ±∞, γ T (x)/γ C (x)δ C (y|x)(1 + o(1)) if δ Y (y|x)/δ C (y|x) → 0, γ T (x)δ Y (y|x)/γ Y (x)(1 + γ Y (x)/(γ C (x)a))(1 + o(1)) if δ Y (y|x)/δ C (y|x) → a,
where 0 < |a| < +∞.

Set ln + x = ln max{1, x}, x > 0, and consider, with s ≤ 0 and s ≥ 0, T (1) n (K, s, s |x 0 ) :=

1 n n i=1 K hn (x 0 -X i ) T i t n s ln + T i t n s 1l {T i >tn} , T (2) n (K, s, s |x 0 ) := 1 n n i=1 K hn (x 0 -X i ) T i t n s ln + T i t n s 1l {Y i ≤C i ,T i >tn} .
Statistics of the type T (j)

n (K, s, s |x 0 ), j = 1, 2, are the basic building blocks for studying the asymptotic behaviour of the estimators for γ Y (x 0 ) and δ Y (t n |x 0 ). Indeed, the estimating equations resulting from (3) depend only on statistics of these types.

As a first step we establish the asymptotic expansions of E(T

(j) n (K, s, s |x 0 )), j = 1, 2. Let η := η γ Y ∧ η γ C ∧ η ε Y ∧ η ε C .
Theorem 1 Assume (D), (H), (K) and x 0 ∈ Int(S X ) with f X (x 0 ) > 0. Then if t n → ∞ and h n → 0 as n → ∞ in such a way that h η n ln t n → 0, we have the following asymptotic expansions

E(T (1) n (K, s, s |x 0 )) = F T (t n |x 0 )f X (x 0 )Γ(s + 1)γ s T (x 0 ) 1 (1 -sγ T (x 0 )) s +1 + δ T (t n |x 0 ) γ T (x 0 ) 1 + β T (x 0 )γ T (x 0 ) (1 + β T (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) + O(h η f X ∧η A Y ∧η A C n ) + O(h ηγ Y ∧ηγ C n ln t n ) and E(T (2) n (K, s, s |x 0 )) = F T (t n |x 0 )f X (x 0 )Γ(s + 1) γ s +1 T (x 0 ) γ Y (x 0 ) 1 (1 -sγ T (x 0 )) s +1 + δ C (t n |x 0 ) γ C (x 0 ) 1 (1 + β C (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) + δ Y (t n |x 0 ) γ Y (x 0 ) 1 (1 + β Y (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) - δ Y (t n |x 0 )ε Y (t n |x 0 ) (1 + β Y (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 (1 + o(1)) + O(h η f X ∧η A Y ∧η A C n ) + O(h ηγ Y ∧ηγ C n ln t n ) .
From Theorem 1 we obtain the convergence in probability of our main statistics.

Corollary 1 Under the assumptions of Theorem 1 and assuming additionally that nh d n F T (t n |x 0 ) → ∞ we have that

T (1) n (K, s, s |x 0 ) F T (t n |x 0 )f X (x 0 ) P → Γ(s + 1)γ s T (x 0 ) (1 -sγ T (x 0 )) s +1 , (4)
T (2) n (K, s, s |x 0 ) F T (t n |x 0 )f X (x 0 ) P → Γ(s + 1)γ s +1 T (x 0 ) γ Y (x 0 )(1 -sγ T (x 0 )) s +1 . (5)
Next we turn to establishing consistency and asymptotic normality of our local likelihood estimators. From now on we denote the true values of γ Y (x 0 ) and δ Y (t n |x 0 ) as γ (0)

Y (x 0 ) and δ (0)
Y (t n |x 0 ), respectively. Similarly, the superscript '(0)' will also be used to denote the true value of the other parameters/functions that appear in F Y (y|x), F C (y|x) and F T (y|x).

Theorem 2 Under the assumptions of Corollary 1, with probability tending to 1 there exists a sequence of solutions ( γ Y,n (x 0 ), δ Y,n (t n |x 0 )) to the likelihood equations such that

( γ Y,n (x 0 ), δ Y,n (t n |x 0 ) -δ (0) Y (t n |x 0 )) P → (γ (0) Y (x 0 ), 0).
In order to obtain the asymptotic normality of the consistent estimator sequence ( γ Y,n (x 0 ), δ Y,n (t n |x 0 )) we need a result concerning the joint convergence in distribution of several statistics of the form

T (1) n (K, s, s |x 0 ) and T (2) n (K, s, s |x 0 ). Let T n := 1 F T (t n |x 0 )f X (x 0 )            T (1) n (K, s 1 , s 1 |x 0 ) . . . T (1)
n (K, s J , s J |x 0 ) T (2) n (K, s J+1 , s J+1 |x 0 ) . . . T (2) n (K, s J+L , s J+L |x 0 )            , r n := nh d n F T (t n |x 0 )f X (x 0 )
, and ' ' denoting convergence in distribution. Theorem 3 Under the assumptions of Theorem 1 and assuming additionally that r n → ∞ we have

r n (T n -E(T n )) N (0, Σ),
where Σ is a symmetric ((J + L) × (J + L)) matrix with elements

σ jk := K 2 2 Γ(s j + s k + 1)γ (0)(s j +s k ) T (x 0 ) (1 -(s j + s k )γ (0) T (x 0 )) s j +s k +1
, j, k ∈ {1, . . . , J},

σ jk := K 2 2 Γ(s j + s k + 1)γ (0)(s j +s k +1) T (x 0 ) γ (0) Y (x 0 )(1 -(s j + s k )γ (0) T (x 0 )) s j +s k +1
, j, k ∈ {J + 1, . . . , J + L},

σ jk := K 2 2 Γ(s j + s k + 1)γ (0)(s j +s k +1) T (x 0 ) γ (0) Y (x 0 )(1 -(s j + s k )γ (0) T (x 0 )) s j +s k +1
, j ∈ {1, . . . , J}, k ∈ {J + 1, . . . , J + L}.

Finally, we obtain the asymptotic normality of any consistent solution sequence of the likelihood equations.

Theorem 4 Assume the conditions of Theorem 3 and additionally that r n h

η f X ∧η A Y ∧η A C n → 0, r n h ηγ Y ∧ηγ C n ln t n → 0 and r n δ (0) T (t n |x 0 ) → λ ∈ R. Then for any consistent sequence of solutions ( γ Y,n (x 0 ), δ Y,n (t n |x 0 )) to the likelihood equations one has that r n γ Y,n (x 0 ) -γ (0) Y (x 0 ) δ Y,n (t n |x 0 ) -δ (0) Y (t n |x 0 ) N (λζ, Λ),
where

Λ 11 := K 2 2 γ (0)3 Y (x 0 )(1 + βγ (0) T (x 0 )) 2 β 2 γ (0)3 T (x 0 ) , Λ 22 := K 2 2 γ (0) Y (x 0 )(1 + βγ (0) T (x 0 )) 2 (1 + 2βγ (0) T (x 0 )) β 4 γ (0)3 T (x 0 ) , Λ 12 := K 2 2 γ (0)2 Y (x 0 )(1 + βγ (0) T (x 0 ))(1 + 2βγ (0) T (x 0 )) β 3 γ (0)3 T (x 0 )
, and where ζ has elements

• Case 1: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → ±∞ ζ 1 := β (0) T (x 0 )γ (0)2 Y (x 0 )(1 + βγ (0) T (x 0 ))(β (0) T (x 0 ) -β) βγ (0) T (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) , ζ 2 := γ (0) Y (x 0 )(β + β (0) T (x 0 ) + β 2 γ (0) T (x 0 ) + 3ββ (0) T (x 0 )γ (0) T (x 0 ) + β 2 β (0) T (x 0 )γ (0)2 T (x 0 ))(β (0) T (x 0 ) -β) β 2 γ (0) T (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) , • Case 2: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → 0 ζ 1 = ζ 2 := 0, • Case 3: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → a, where 0 < |a| < ∞ ζ 1 := aβ (0) T (x 0 )γ (0)2 Y (x 0 )γ (0) C (x 0 )(1 + βγ (0) T (x 0 ))(β (0) T (x 0 ) -β) βγ (0) T (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 ))(γ (0) Y (x 0 ) + aγ (0) C (x 0 )) , ζ 2 := aγ (0) Y (x 0 )γ (0) C (x 0 )(β + β (0) T (x 0 ) + β 2 γ (0) T (x 0 ) + 3ββ (0) T (x 0 )γ (0) T (x 0 ) + β 2 β (0) T (x 0 )γ (0)2 T (x 0 ))(β (0)
T (x 0 ) -β) β 2 γ (0) T (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 ))(γ (0) Y (x 0 ) + aγ (0) C (x 0 )) .
Note that in case the correct value is used for the parameter β, so β = β (0) T (x 0 ), then the mean of the limiting distribution is zero, and we say the estimator is asymptotically unbiased. If β is mis-specified then obviously one loses this unbiasedness but the estimator still continues to perform well in practice and outperforms estimators that have no bias-correction, as is illustrated in the simulation section.

A simulation study

In this section we evaluate the finite sample performance of our bias-corrected estimator with a small simulation study. We compare the proposed estimator with an estimator which is not corrected for bias, obtained by constraining δ Y to zero in (3), leading to

γ B Y,n (x 0 ) := 1 n n i=1 K hn (x 0 -X i ) ln T i tn 1l {T i >tn} 1 n n i=1 K hn (x 0 -X i )1l {Y i ≤C i T i >tn} .
This can be seen as a local Hill-type estimator, as introduced by Goegebeur et al. (2014b), corrected for censoring by dividing it by the local proportion of uncensored observations among the excesses over t n . An estimator of this type was already studied by [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. For K, we use the bi-quadratic kernel function given by

K(x) = 15 16 (1 -x 2 ) 2 1l {x∈[-1,1]} .
For the practical implementation of the estimators we have to select the bandwidth parameter h n and the threshold t n , where, as is common in extreme value statistics, we use for the latter the (k + 1)-th largest response observation in the ball B(x 0 , h n ).

The parameter h n is selected in a data driven way by using the following cross-validation criterion

h cv := arg min h∈H n i=1 n j=1 1l {T i ≤T j } -F n,-i (T j |X i ) 2 ,
where H is a grid of values for h and

F n,-i (t|x) := n j=1,j =i K h (x -X j )1l {T j ≤t} n j=1,j =i K h (x -X j )
, We assume that the conditional distributions of Y and C given X = x are Burr distributions with

is
F • (y|x) = 1 - η • (x) η • (x) + y τ•(x) λ•(x) , y > 0 and η • (x), λ • (x), τ • (x) > 0. We have γ • (x) = 1/(λ • (x)τ • (x)) and β • (x) = τ • (x).
The covariate X is assumed to be uniformly distributed on [0, 1]. In our simulations we set η Y (x) = η C (x) = 1 and consider the following cases:

• Case 1: λ Y (x) = 2, τ C (x) = 4, λ C (x) = 1, and τ Y (x) = 1 10 + sin(πx) 11 10 - 1 2 exp -64 x - 1 2 2 -1 . • Case 2: λ Y (x) = 1, τ C (x) = 4, λ C (x) = 1, and τ Y (x) = 2 1 10 + sin(πx) 11 10 - 1 2 exp -64 x - 1 2 2 -1 . • Case 3: λ Y (x) = 2/3, τ C (x) = 4, λ C (x) = 1, and
τ Y (x) = 3 1 10 + sin(πx) 11 10 - 1 2 exp -64 x - 1 2 2 -1 . • Case 4: λ Y (x) = 0.5, τ C (x) = 4, λ C (x) = 1, and τ Y (x) = 4 1 10 + sin(πx) 11 10 - 1 2 exp -64 x - 1 2 2 -1
.

We simulate 100 datasets of size n = 1000. In Figure 1 we show the mean (left) and mean squared error, MSE, (right) of γ Y,n (x 0 ) for β = β (0)

T (x 0 ) (solid line), β = 2 (dashed line) and γ B Y,n (x 0 ) (dashed-dotted line) as a function of k, when x 0 = 0.4. In the left panels the true value of γ Y (x 0 ) is represented by a horizontal line. The four rows in the figure correspond with the four cases mentioned above. Figure 2 is constructed similarly but now with x 0 = 0.1. Concerning the selection of h we note the following. In a first step we compute the optimal h for each dataset using the above mentioned cross validation criterion with H = {0.05, 0.10, . . . , 0.30}. This implies that the range of k varies from one dataset to the next, so means and MSE's would be based on a different number of observations when plotted as a function of k. In order to avoid this we take the median of the h cv values obtained in the 100 simulations and use this for all estimations. In Figure 3 we plot the proportion of censoring (computed as γ Y (x)/(γ Y (x)+γ C (x))) as a function of x. In case of x 0 = 0.1 and x 0 = 0.4 we have a proportion of censoring of 0.47 and 0.64, respectively. These can be considered as very high values, and moreover in case x 0 = 0.1 we are near the boundary of the covariate support, implying that at both positions the estimation is difficult. From the simulation results reported in Figures 1 and2 we can make the following conclusions

• In all the cases and for both positions our estimator reduces the bias considerably compared to the estimator γ B Y,n (x 0 ). In terms of MSE we are also very competitive and in general we have also a longer stable range of the MSE.

• After an extensive simulation study we observed that taking β = 2 gives overall good estimation results, whatever the cases and covariate position.

• Using the value β = 2 is competitive instead of using the unknown true value β (0)

T (x 0 ). To get an idea about the mis-specification, note that the true beta values are β • In extreme value statistics the bias of estimators is typically determined by a second order parameter ρ Y (x 0 ), linked to

β Y (x 0 ) by ρ Y (x 0 ) = -β Y (x 0 )γ Y (x 0 ), which is in the particular case of the Burr distribution equal to ρ Y (x 0 ) = -1/λ Y (x 0 ). When ρ Y (x 0
) is close to zero then estimators for tail parameters typically suffer from bias. In the figures the four cases are ordered from the more difficult (ρ Y (x 0 ) = -0.5) to the more easy situation (ρ Y (x 0 ) = -2). We can indeed observe that for the uncorrected estimator γ B Y,n (x 0 ), the bias is largest when ρ Y (x 0 ) is close to zero.

• Comparing the two positions x 0 = 0.1 and x 0 = 0.4 we can see that the estimation is more difficult near the boundary of the covariate space, but this is a well-known problem in nonparametric estimation. We observe though that the bias-corrected estimator still performs well.

Application to a real dataset

We illustrate in this section the proposed methodology with a dataset of patients diagnosed with AIDS in Australia before 1 July 1991. These data were provided by Dr P.J. and six h-values evenly spaced. This grid is close to the one already used by Stupfler (2016), and with our procedure of selection it leads to the particular value h = 11.25.

In Figure 5 we estimate γ Y (x 0 ) at two positions x 0 = 25 and x 0 = 50 which correspond to positions close to the border of the x-values considered or close to the middle of the range of values of x, respectively. Our bias-corrected estimator γ Y,n (x 0 ) is plotted in Figure 5 in full line whereas γ B Y,n (x 0 ) is in dashed line. For both estimators, as suggested in the simulation study, K is the bi-quadratic kernel and t n is the (k + 1)-th largest response observation in the ball B(x 0 , h). The second order rate parameter β is set at the value β = 2 as recommended also in the simulation study. It is clear from this figure that estimating γ Y (x 0 ) at x 0 = 25 is much more difficult than at position x 0 = 50. This corroborates what we have already observed in the simulation study for the position x 0 = 0.1. Despite, the estimation of γ Y (x 0 ) is very stable as a function of k for a wide range of k-values, on the contrary of γ B Y,n (x 0 ) which increases drastically without being stable at some stage.

1 n n i=1 K hn (x 0 -X i ) - 1 γ Y 1l {Y i ≤C i } + 1 γ 2 Y ln T i t n + 1 γ 2 Y ln 1 + δ Y -δ Y T i t n -β 1l {T i >tn} , 2 (γ Y , δ Y ; β) = 1 n n i=1 K hn (x 0 -X i )   - 1 1 + δ Y -δ Y T i tn -β 1 - T i t n -β 1l {Y i ≤C i } + 1 1 + δ Y 1 -(1 -β) T i tn -β 1 -(1 -β) T i t n -β 1l {Y i ≤C i } - 1 γ Y 1 1 + δ Y -δ Y T i tn -β 1 - T i t n -β    1l {T i >tn} , 11 (γ Y , δ Y ; β) = 1 n n i=1 K hn (x 0 -X i ) 1 γ 2 Y 1l {Y i ≤C i } - 2 γ 3 Y ln T i t n - 2 γ 3 Y ln 1 + δ Y -δ Y T i t n -β 1l {T i >tn} , 22 (γ Y , δ Y ; β) = 1 n n i=1 K hn (x 0 -X i )      1 1 + δ Y -δ Y T i tn -β 2 1 - T i t n -β 2 1l {Y i ≤C i } - 1 1 + δ Y 1 -(1 -β) T i tn -β 2 1 -(1 -β) T i t n -β 2 1l {Y i ≤C i } + 1 γ Y 1 1 + δ Y -δ Y T i tn -β 2 1 - T i t n -β 2      1l {T i >tn} , 12 (γ Y , δ Y ; β) = 1 n n i=1 K hn (x 0 -X i ) 1 γ 2 Y 1 1 + δ Y -δ Y T i tn -β 1 - T i t n -β 1l {T i >tn} .

Proof of Theorem 1

We focus on deriving the asymptotic expansion for E(T

n (K, s, s |x 0 )). Then, E(T n (K, s, s |x 0 ), though it was analysed under their high level assumption called (M), which is avoided in the present paper, allowing us to obtain a more precise statement of the remainder terms using the Hölder exponents from Assumption (H).

We have

E(T (2) n (K, s, s |x 0 )) = E K hn (x 0 -X) T t n s ln + T t n s 1l {Y ≤C,T >tn} = E K hn (x 0 -X)E T t n s ln + T t n s 1l {Y ≤C,T >tn} X = E K hn (x 0 -X) ∞ tn y t n s ln y t n s f Y (y|X)F C (y|X)dy = R d K hn (x 0 -u) ∞ tn y t n s ln y t n s f Y (y|u)F C (y|u)dyf X (u)du = S K K(z) ∞ tn y t n s ln y t n s f Y (y|x 0 -h n z)F C (y|x 0 -h n z)dyf X (x 0 -h n z)dz.
In view of the various Hölder conditions, the latter is further decomposed as

E(T (2) n (K, s, s |x 0 )) = f X (x 0 ) ∞ tn y t n s ln y t n s f Y (y|x 0 )F C (y|x 0 )dy + ∞ tn y t n s ln y t n s f Y (y|x 0 )F C (y|x 0 )dy S K K(z)(f X (x 0 -h n z) -f X (x 0 ))dz +f X (x 0 ) S K K(z) ∞ tn y t n s ln y t n s f Y (y|x 0 )(F C (y|x 0 -h n z) -F C (y|x 0 ))dydz + S K K(z) ∞ tn y t n s ln y t n s f Y (y|x 0 )(F C (y|x 0 -h n z) -F C (y|x 0 ))dy(f X (x 0 -h n z) -f X (x 0 ))dz +f X (x 0 ) S K K(z) ∞ tn y t n s ln y t n s (f Y (y|x 0 -h n z) -f Y (y|x 0 ))F C (y|x 0 ))dydz + S K K(z) ∞ tn y t n s ln y t n s (f Y (y|x 0 -h n z) -f Y (y|x 0 ))F C (y|x 0 )dy(f X (x 0 -h n z) -f X (x 0 ))dz +f X (x 0 ) S K K(z) ∞ tn y t n s ln y t n s (f Y (y|x 0 -h n z) -f Y (y|x 0 ))(F C (y|x 0 -h n z) -F C (y|x 0 ))dydz + S K K(z) ∞ tn y t n s ln y t n s (f Y (y|x 0 -h n z) -f Y (y|x 0 ))(F C (y|x 0 -h n z) -F C (y|x 0 ))dy (f X (x 0 -h n z) -f X (x 0 ))dz =: T 1 + • • • + T 8 .
Concerning T 1 we have

T 1 = t n f Y (t n |x 0 )F C (t n |x 0 )f X (x 0 ) ∞ 1 z s (ln z) s f Y (t n z|x 0 ) f Y (t n |x 0 ) F C (t n z|x 0 ) F C (t n |x 0 ) dz.
A slight modification of Proposition 2.3 in [START_REF] Beirlant | Second-order refined peaks-over-threshold modelling for heavy-tailed distributions[END_REF] gives

sup z≥1 z 1/γ•(x) F • (t n z|x 0 ) F • (t n |x 0 ) -G(z; γ • (x 0 ), δ • (t n |x 0 ), β • (x 0 )) = o(|δ • (t n |x 0 )|), t n → ∞.
This leads to the decomposition

T 1 = t n f Y (t n |x 0 )F C (t n |x 0 )f X (x 0 ) ∞ 1 z s (ln z) s f Y (t n z|x 0 ) f Y (t n |x 0 ) G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz + ∞ 1 z s (ln z) s f Y (t n z|x 0 ) f Y (t n |x 0 ) F C (t n z|x 0 ) F C (t n |x 0 ) -G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 )) dz =: t n f Y (t n |x 0 )F C (t n |x 0 )f X (x 0 )(T 1,1 + T 1,2 ).
From (1) we can write

T 1,1 = ∞ 1 z s-1/γ Y (x 0 )-1 (ln z) s G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz + 1 1 + 1 γ Y (x 0 ) -ε Y (t n |x 0 ) δ Y (t n |x 0 ) δ Y (t n |x 0 ) γ Y (x 0 ) ∞ 1 z s-1/γ Y (x 0 )-1 (ln z) s (z -β Y (x 0 ) -1)G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz + δ Y (t n |x 0 ) γ Y (x 0 ) ∞ 1 z s-1/γ Y (x 0 )-1 (ln z) s δ Y (t n z|x 0 ) δ Y (t n |x 0 ) -z -β Y (x 0 ) G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz -ε Y (t n |x 0 )δ Y (t n |x 0 ) ∞ 1 z s-1/γ Y (x 0 )-1 (ln z) s ε Y (t n z|x 0 ) ε Y (t n |x 0 ) -1 δ Y (t n z|x 0 ) δ Y (t n |x 0 ) G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz -ε Y (t n |x 0 )δ Y (t n |x 0 ) ∞ 1 z s-1/γ Y (x 0 )-1 (ln z) s δ Y (t n z|x 0 ) δ Y (t n |x 0 ) -1 G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz =: T 1,1,1 + 1 1 + 1 γ Y (x 0 ) -ε Y (t n |x 0 ) δ Y (t n |x 0 ) (T 1,1,2 + • • • + T 1,1,5 ).
In order to deal with these integrals, the following expansion of the extended Pareto distribution is useful

G(z; γ • (x 0 ), δ • (t n |x 0 ), β • (x 0 )) = z -1/γ•(x 0 ) 1 - δ • (t n |x 0 ) γ • (x 0 ) (1 -z -β•(x 0 ) ) + O(δ 2 • (t n |x 0 )) , where O(δ 2 • (t n |x 0 )) is uniform in z ≥ 1.
A straightforward calculation gives then

T 1,1,1 = Γ(s + 1)γ s +1 T (x 0 ) (1 -sγ T (x 0 )) s +1 + δ C (t n |x 0 ) Γ(s + 1)γ s +1 T (x 0 ) γ C (x 0 ) × 1 (1 + β C (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 + O(δ 2 C (t n |x 0 )), T 1,1,2 = δ Y (t n |x 0 ) Γ(s + 1)γ s +1 T (x 0 ) γ Y (x 0 ) 1 (1 + β Y (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 +O(δ Y (t n |x 0 )δ C (t n |x 0 )).
For T 1,1,3 we use Proposition B.1.10 in de [START_REF] De Haan | Extreme Value Theory: An Introduction[END_REF], see also [START_REF] Drees | On smooth statistical tail functionals[END_REF]. Thus, for ε > 0 and 0 < δ < 1/γ T (x 0 ) + β Y (x 0 ) -s, arbitrary, and n sufficiently large, we have

|T 1,1,3 | ≤ ε |δ Y (t n |x 0 )| γ Y (x 0 ) ∞ 1 z -(1/γ Y (x 0 )+β Y (x 0 )-s-δ)-1 (ln z) s G(z; γ C (x 0 ), δ C (t n |x 0 ), β C (x 0 ))dz.
Since ε is arbitrary and by using calculations for the integral that are similar to those above, one finds that T 1,1,3 = o(δ Y (t n |x 0 )). In the same way T 1,1,4 = o(δ Y (t n |x 0 )), and

T 1,1,5 = -ε Y (t n |x 0 )δ Y (t n |x 0 )Γ(s + 1)γ s +1 T (x 0 ) 1 (1 + β Y (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 +o(δ Y (t n |x 0 )). Analogously one can show that T 1,2 = o(δ C (t n |x 0 )).
Collecting the terms gives then

T 1 = t n f Y (t n |x 0 )F C (t n |x 0 )f X (x 0 )Γ(s + 1)γ s +1 T (x 0 ) 1 (1 -sγ T (x 0 )) s +1 + δ C (t n |x 0 ) γ C (x 0 ) 1 (1 + β C (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) +δ Y (t n |x 0 ) 1 γ Y (x 0 ) -ε Y (t n |x 0 ) 1 (1 + β Y (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) .

Note that

t n f Y (t n |x 0 )F C (t n |x 0 ) = F T (t n |x 0 ) γ Y (x 0 )   1 - ε Y (t n |x 0 )δ Y (t n |x 0 ) 1 + δ Y (tn|x 0 ) γ Y (x 0 )   ,
whence

T 1 = F T (t n |x 0 )f X (x 0 )Γ(s + 1) γ s +1 T (x 0 ) γ Y (x 0 ) 1 (1 -sγ T (x 0 )) s +1 + δ C (t n |x 0 ) γ C (x 0 ) 1 (1 + β C (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) + δ Y (t n |x 0 ) 1 γ Y (x 0 ) -ε Y (t n |x 0 ) 1 (1 + β Y (x 0 )γ T (x 0 ) -sγ T (x 0 )) s +1 - 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) -δ Y (t n |x 0 )ε Y (t n |x 0 ) 1 (1 -sγ T (x 0 )) s +1 (1 + o(1)) .
For T 2 , we use the Hölder condition on f X and obtain

T 2 = O(h η f X F T (t n |x 0 )).
By rearranging terms we obtain the following bound for T 3

|T 3 | ≤ f Y (t n |x 0 )F C (t n |x 0 )f X (x 0 ) × S K K(z) ∞ tn y t n s ln y t n s f Y (y|x 0 f Y (t n |x 0 ) F C (y|x 0 ) F C (t n |x 0 ) F C (y|x 0 -h n z) F C (y|x 0 ) -1 dydz, (6)
and from Assumption (H), for n large, and some constants M 1 , M 2 and M 3 ,

F C (y|x 0 -h n z) F C (y|x 0 ) -1 ≤ M 1 h η A C n + y M 2 h ηγ C n h ηγ C n ln y + |δ C (y|x 0 )|h η B C n + |δ C (y|x 0 )|y M 3 h ηε C n h ηε C n ln y .
Plugging the above inequality into (6), and computing integrals similar to those encountered above yields

T 3 = O F T (t n |x 0 )(h η A C n + h ηγ C n ln t n + δ C (t n |x 0 )h η B C n + δ C (t n |x 0 )h ηε C n ln t n ) .
Using the Hölder condition on f X one easily verifies that T 4 is of smaller order than T 3 .

As for T 5 , we can write

|T 5 | ≤ f Y (t n |x 0 )F C (t n |x 0 )f X (x 0 ) × S K K(z) ∞ tn y t n s ln y t n s f Y (y|x 0 ) f Y (t n |x 0 ) F C (y|x 0 ) F C (t n |x 0 ) f Y (y|x 0 -h n z) f Y (y|x 0 ) -1 dydz,
which, combined with the inequality

f Y (y|x 0 -h n z) f Y (y|x 0 ) -1 ≤ M 1 h η A Y n + y M 2 h ηγ Y n h ηγ Y n ln y + |δ Y (y|x 0 )|h η B Y n + |δ Y (y|x 0 )|y M 3 h ηε Y n h ηε Y n ln y ,
valid for n large, where M 1 , M 2 and M 3 are some constants, leads to

T 5 = O F T (t n |x 0 )(h η A Y n + h ηγ Y n ln t n + δ Y (t n |x 0 )h η B Y n + δ Y (t n |x 0 )h ηε Y n ln t n ) .
After tedious calculations, but essentially involving integrals similar to the ones above, one can verify that T 6 , T 7 and T 8 , are of smaller order than terms that were already encountered before.

Collecting the terms then establishes Theorem 1.

Proof of Corollary 1

We only give the details for T

(1)

n (K, s, s |x 0 )/(F T (t n |x 0 )f X (x 0 )). The convergence in probability of T (2) n (K, s, s |x 0 )/(F T (t n |x 0 )f X (x 0 )) can be established analogously.
From Theorem 1 we have immediately

E T (1) n (K, s, s |x 0 ) F T (t n |x 0 )f X (x 0 ) → Γ(s + 1)γ s T (x 0 ) (1 -sγ T (x 0 )) s +1 .

By independence

Var

T (1) n (K, s, s |x 0 ) F T (t n |x 0 )f X (x 0 ) = Var K hn (x 0 -X) T tn s ln + T tn s 1l {T >tn} nF 2 T (t n |x 0 )f 2 X (x 0 )
.

Using (K) and Theorem 1 we obtain

Var K hn (x 0 -X) T t n s ln + T t n s 1l {T >tn} = K 2 2 h d n E 1 K 2 2 h d n K 2 x 0 -X h n T t n 2s ln + T t n 2s 1l {T >tn} + O(F 2
T (t n |x 0 )) = F T (t n |x 0 )f X (x 0 ) K 2 2 Γ(2s + 1)γ 2s T (x 0 ) h d n (1 -2sγ T (x 0 )) 2s +1 (1 + o(1)),

and thus

Var

T (1) n (K, s, s |x 0 ) F T (t n |x 0 )f X (x 0 ) = 1 nh d n F T (t n |x 0 )f X (x 0 ) K 2 2 Γ(2s + 1)γ 2s T (x 0 ) (1 -2sγ T (x 0 )) 2s +1 (1 + o(1)),
which tends to zero under the assumption nh d n F T (t n |x 0 ) → ∞. This establishes (4).

Proof of Theorem 2

To prove consistency we adapt the argument of the proof of Theorem 5.1 in Chapter 6 of [START_REF] Lehmann | Theory of Point Estimation[END_REF]

to our context. Let (γ Y , δ Y ; β) := (γ Y , δ Y ; β)/(F T (t n |x 0 )f X (x 0 )),
and similarly for j and jk . We denote by jkl , j, k, l = 1, 2, the third order derivatives of , and by jkl their rescaled version. As a first step we show that for any ν > 0 sufficiently small

P (γ (0) Y (x 0 ),δ (0)
Y (tn|x 0 )) ( (γ Y , δ Y ; β) < (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) for all (γ Y , δ Y ) on the surface of Q ν,n ) → 1
as n → ∞, where Q ν,n denotes the circle centered at (γ

(0) Y (x 0 ), δ (0)
Y (t n |x 0 )) with radius ν, and Q ν,n is assumed to belong to the parameter space. By Taylor's theorem

(γ Y , δ Y ; β) -(γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = 1 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β)(γ Y -γ (0) Y (x 0 )) + 2 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β)(δ Y -δ (0) Y (t n |x 0 )) + 1 2 [ 11 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β)(γ Y -γ (0) Y (x 0 )) 2 + 22 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β)(δ Y -δ (0) Y (t n |x 0 )) 2 +2 12 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β)(γ Y -γ (0) Y (x 0 ))(δ Y -δ (0) Y (t n |x 0 ))] + 1 6 [ 111 ( γ Y , δ Y ; β)(γ Y -γ (0) Y (x 0 )) 3 + 222 ( γ Y , δ Y ; β)(δ Y -δ (0) Y (t n |x 0 )) 3 +3 112 ( γ Y , δ Y ; β)(γ Y -γ (0) Y (x 0 )) 2 (δ Y -δ (0) Y (t n |x 0 )) +3 122 ( γ Y , δ Y ; β)(γ Y -γ (0) Y (x 0 ))(δ Y -δ (0) Y (t n |x 0 )) 2 ] =: S 1 + S 2 + S 3 ,
where ( γ Y , δ Y ) is a point on the line segment connecting (γ Y , δ Y ) and (γ

(0) Y (x 0 ), δ (0) Y (t n |x 0 )).
After linearisation, 1 and 2 can be written in terms of T n (K, s, s |x 0 ) as follows

1 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = - 1 γ (0) Y (x 0 ) T (2) n (K, 0, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) + 1 γ (0)2 Y (x 0 ) T (1) n (K, 0, 1|x 0 ) F T (t n |x 0 )f X (x 0 ) +δ (0) Y (t n |x 0 ) 1 γ (0)2 Y (x 0 ) T (1) n (K, 0, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) - T (1) n (K, -β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) +O P (δ (0)2 Y (t n |x 0 )), ( 7
) 2 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) = β T (2) n (K, -β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) - 1 γ (0) Y (x 0 ) T (1) n (K, 0, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) + 1 γ (0) Y (x 0 ) T (1) n (K, -β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) +δ (0) Y (t n |x 0 ) (2β -β 2 ) T (2) n (K, -2β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) -2β T (2) n (K, -β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) + 1 γ (0) Y (x 0 ) T (1) n (K, 0, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) + 1 γ (0) Y (x 0 ) T (1)
n (K, -2β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) - 2 γ (0) Y (x 0 ) T (1)
n (K, -β, 0|x 0 ) F T (t n |x 0 )f X (x 0 ) + O P (δ (0)2 Y (t n |x 0 )). (8)
Using Corollary 1 we have

1 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) P → 0 and 2 (γ (0)
Y (x 0 ), δ (0)
Y (t n |x 0 ); β) P → 0, so, for any ν > 0 we have that | j (γ (0)
Y (x 0 ), δ (0)
Y (t n |x 0 ); β)| < ν 2 , j = 1, 2
, with probability tending to 1, and hence on Q ν,n , |S 1 | < 2ν 3 with probability tending to 1.

Using again linearisation and Corollary 1 one has

11 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) P → ˙ 11 := - γ (0)
T (x 0 ) γ (0)3 Y (x 0 ) , 22 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) P → ˙ 22 := - β 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + 2βγ (0) T (x 0 )) , 12 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) P → ˙ 12 := βγ (0) T (x 0 ) γ (0)2 Y (x 0 )(1 + βγ (0) T (x 0 )) . Now write 2S 2 = ˙ 11 (γ Y -γ (0) Y (x 0 )) 2 + ˙ 22 (δ Y -δ (0) Y (t n |x 0 )) 2 + 2 ˙ 12 (γ Y -γ (0) Y (x 0 ))(δ Y -δ (0) Y (t n |x 0 )) +( 11 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) -˙ 11 )(γ Y -γ (0) Y (x 0 )) 2 +( 22 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) -˙ 22 )(δ Y -δ (0) Y (t n |x 0 )) 2 +2( 12 (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) -˙ 12 )(γ Y -γ (0) Y (x 0 ))(δ Y -δ (0) Y (t n |x 0 )).
It can easily be verified that the first three terms are a deterministic negative definite quadratic form in γ Y -γ

(0) Y (x 0 ) and δ Y -δ (0) Y (t n |x 0 )
. By the spectral decomposition this quadratic form can be rewritten as

λ 1 ξ 2 1 + λ 2 ξ 2 2
, where λ 2 ≤ λ 1 < 0 are the eigenvalues and ξ 1 and ξ 2 are the orthogonal transformations of γ Y -γ

(0) Y (x 0 ) and δ Y -δ (0) Y (t n |x 0 ). Note that in the new coordi- nate system Q ν,n becomes ξ 2 1 + ξ 2 2 = ν 2 . Thus λ 1 ξ 2 1 + λ 2 ξ 2 2 ≤ λ 1 ν 2 .
For the random terms of S 2 we use the above convergences in probability and conclude that the random part is in absolute value less than 4ν 3 with probability tending to 1. Combined, we have that there exists c > 0 and ν 0 > 0 such that for ν < ν 0 one has S 2 < -cν 2 with probability tending to 1.

For the third order derivatives one has that

| jkl (γ Y , δ Y ; β)| ≤ M jkl (V) for (γ Y , δ Y ) ∈ Q ν,n ,
where V := {(T i , X i , 1l {Y i ≤C i } ), i = 1, . . . , n}, and M jkl P → m jkl , bounded, j, k, l = 1, 2. These derivations are straightforward and for brevity omitted from the paper. We have then that, with probability tending to one,

| jkl ( γ Y , δ Y ; β)| < 2m jkl and thus |S 3 | < eν 3 on Q ν,n , where e = 1/3 2 j=1 2 k=1 2 l=1 m jkl .
Combining the above, with probability tending to 1, max Qν,n

(S 1 + S 2 + S 3 ) < -cν 2 + (2 + e)ν 3 , which is negative if ν < c/(2 + e).
To establish existence and consistency we adjust the line of argumentation in the proof of Theorem 3.7 in Chapter 6 of Lehmann and Casella (1998). For ν > 0, small enough such that Q ν,n is a subset of the parameter space, consider

S n (ν) := {v : (γ Y , δ Y ; β) < (γ (0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) for all (γ Y , δ Y ) on the surface of Q ν,n }.
From the above we have that P (γ (0)

Y (x 0 ),δ (0)
Y (tn|x 0 )) (S n (ν)) → 1 for any such ν, and hence there exists a sequence ν n ↓ 0 such that P (γ (0)

Y (x 0 ),δ (0)
Y (tn|x 0 )) (S n (ν n )) → 1 as n → ∞. By the differentiabil- ity of (γ Y , δ Y ; β) we have that v ∈ S n (ν n ) implies that there exists a point ( γ Y,n (ν n ), δ Y,n (ν n )) ∈ Q ν,n for which (γ Y , δ Y ; β) attains a local maximum, and thus j ( γ Y,n (ν n ), δ Y,n (ν n ); β) = 0, j = 1, 2. Now let ( γ * Y,n (x 0 ), δ * Y,n (t n |x 0 )) := ( γ Y,n (ν n ), δ Y,n (ν n )) for v ∈ S n (ν n
) and arbitrary otherwise. Clearly

P (γ (0) Y (x 0 ),δ (0)
Y (tn|x 0 )) ( 1 ( γ * Y,n (x 0 ), δ * Y,n (t n |x 0 )); β) = 0, 2 ( γ * Y,n (x 0 ), δ * Y,n (t n |x 0 )); β) = 0) ≥ P (γ (0) Y (x 0 ),δ (0)
Y (tn|x 0 )) (S n (ν n )) → 1,
as n → ∞. Thus with probability tending to 1 there exists a sequence of solutions to the likelihood equations. Also, for any fixed ν > 0 and n sufficiently large, and denoting by d the usual Euclidean distance,

P (γ (0) Y (x 0 ),δ (0) Y (tn|x 0 )) (d(( γ * Y,n (x 0 ), δ * Y,n (t n |x 0 )), (γ (0)
Y (x 0 ), δ (0)
Y (t n |x 0 ))) < ν) ≥ P (γ (0) Y (x 0 ),δ (0)
Y (tn|x 0 )) (d(( γ * Y,n (x 0 ), δ * Y,n (t n |x 0 )), (γ (0)
Y (x 0 ), δ (0)
Y (t n |x 0 ))) < ν n ) ≥ P (γ (0) Y (x 0 ),δ (0) Y (tn|x 0 )) (S n (ν n )) → 1,
which establishes the consistency of the sequence ( γ * Y,n (x 0 ), δ * Y,n (t n |x 0 )).

Proof of Theorem 3

The result of the theorem will be established by using the Cramér-Wold device (see e.g. van der Vaart, 1998, p.16). Take θ := (θ 1 , . . . , θ J+L ) T ∈ R J+L . Then

θ T r n (T n -E(T n )) = n i=1    J j=1 θ j h d n nF T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X i ) T i t n s j ln + T i t n s j 1l {T i >tn} + L j=1 θ J+j h d n nF T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X i ) T i t n s J+j ln + T i t n s J+j 1l {Y i ≤C i ,T i >tn} -E   J j=1 θ j h d n nF T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s j ln + T t n s j 1l {T >tn} + L j=1 θ J+j h d n nF T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s J+j ln + T t n s J+j 1l {Y ≤C,T >tn}      =: n i=1 W i .
Note that Var(W 1 ) = θ T Aθ/n, where A has elements

A jk := Cov h d n F T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s j ln + T t n s j 1l {T >tn} , h d n F T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s k ln + T t n s k
1l {T >tn} , j, k ∈ {1, . . . , J},

A jk := Cov h d n F T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s j ln + T t n s j 1l {Y ≤C,T >tn} , h d n F T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s k ln + T t n s k 1l {Y ≤C,T >tn} , j, k ∈ {J + 1, . . . , J + L}, A jk := Cov h d n F T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s j ln + T t n s j 1l {T >tn} , h d n F T (t n |x 0 )f X (x 0 ) 1/2 K hn (x 0 -X) T t n s k ln + T t n s k
1l {Y ≤C,T >tn} , j ∈ {1, . . . , J}, k ∈ {J + 1, . . . , J + L}.

Using Theorem 1 we obtain that A jk = Σ jk (1+o(1)), j, k ∈ {1, . . . , J+L}, and thus Var(θ T r n (T n -

E(T n ))) = θ T Σθ(1 + o(1)).
In order to establish the weak convergence to a Gaussian random variable we need to verify the Lyapounov condition (see e.g. Billingsley, 1995, p. 362), which simplifies in our setting to showing that lim n→∞ nE(|W 1 | 3 ) = 0. To this aim, note that W 1 is of the form V -E(V ), leading to the inequality

E(|W 1 | 3 ) ≤ E(|V | 3 ) + 3E(V 2 )E(|V |) + 4(E(|V |)) 3 .
Again using the result from Theorem 1, we obtain the following orders

E(|V | 3 ) = O   1 n 3/2 h d n F T (t n |x 0 )   , E(V 2 )E(|V |) = O   h d n F T (t n |x 0 ) n 3/2   , (E(|V |)) 3 = O h d n F T (t n |x 0 ) n 3/2
, so that nE(|W 1 | 3 ) → 0 under our assumption r n → ∞.

Proof of Theorem 4

We apply a Taylor series expansion of j ( γ Y,n (x 0 ), δ Y,n (t n |x 0 ); β), j =

Y (x 0 ), δ 1, 2, around (γ (0)
Y (t n |x 0 )) and obtain, after rearranging terms -r n 1 (γ

(0) Y (x 0 ), δ (0)
Y (t n |x 0 ); β) 2 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = ˇ 11 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) ˇ 12 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) ˇ 12 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) ˇ 22 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) r n ( γ Y,n (x 0 ) -γ (0) Y (x 0 )) r n ( δ Y,n (t n |x 0 ) -δ (0) Y (t n |x 0 )) (9)
where

ˇ 11 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = 11 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) + 1 2 [ 111 ( γ Y,n , δ Y,n ; β)( γ Y,n (x 0 ) -γ (0) Y (x 0 )) + 112 ( γ Y,n , δ Y,n ; β)( δ Y,n (t n |x 0 ) -δ (0) Y (t n |x 0 ))], ˇ 22 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = 22 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) + 1 2 [ 122 ( γ Y,n , δ Y,n ; β)( γ Y,n (x 0 ) -γ (0) Y (x 0 )) + 222 ( γ Y,n , δ Y,n ; β)( δ Y,n (t n |x 0 ) -δ (0) Y (t n |x 0 ))], ˇ 12 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = 12 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) + 1 2 [ 112 ( γ Y,n , δ Y,n ; β)( γ Y,n (x 0 ) -γ (0) Y (x 0 )) + 122 ( γ Y,n , δ Y,n ; β)( δ Y,n (t n |x 0 ) -δ (0) Y (t n |x 0 ))],
and ( γ Y,n , δ Y,n ) is a random value on the line segment between (γ

(0) Y (x 0 ), δ (0)
Y (t n |x 0 )) and ( γ Y,n (x 0 ), δ Y,n (t n |x 0 )).
We start by analysing the left hand side of [START_REF] Dierckx | Local robust and asymptotically unbiased estimation of conditional Pareto-type tails[END_REF]. As a first step we rewrite the expansion for

E(T (2)
n (K, s, s |x 0 )) in terms of δ (0)

T (t n |x 0 ).

• Case 1:

δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → ±∞ E(T (2) n (K, s, s |x 0 )) = F T (t n |x 0 )f X (x 0 )Γ(s + 1) γ (0)(s +1) T (x 0 ) γ (0) Y (x 0 ) 1 (1 -sγ (0) T (x 0 )) s +1 +δ (0) T (t n |x 0 ) γ (0) Y (x 0 ) γ (0) T (x 0 ) 1 γ (0) Y (x 0 ) 1 (1 + β (0) T (x 0 )γ (0) T (x 0 ) -sγ (0) T (x 0 )) s +1 - 1 (1 -sγ (0) T (x 0 )) s +1 + β (0) T (x 0 ) (1 + β (0) T (x 0 )γ (0) T (x 0 ) -sγ (0) T (x 0 )) s +1 (1 + o(1)) + O(h η f X ∧η A Y ∧η A C n ) + O(h ηγ Y ∧ηγ C n ln t n ) . • Case 2: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → 0 E(T (2) n (K, s, s |x 0 )) = F T (t n |x 0 )f X (x 0 )Γ(s + 1) γ (0)(s +1) T (x 0 ) γ (0) Y (x 0 ) 1 (1 -sγ (0) T (x 0 )) s +1 + δ (0) T (t n |x 0 ) γ (0) T (x 0 ) 1 (1 + β (0) T (x 0 )γ (0) T (x 0 ) -sγ (0) T (x 0 )) s +1 - 1 (1 -sγ (0) T (x 0 )) s +1 (1 + o(1)) + O(h η f X ∧η A Y ∧η A C n ) + O(h ηγ Y ∧ηγ C n ln t n ) . • Case 3: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → a, where 0 < |a| < ∞ E(T (2) n (K, s, s |x 0 )) = F T (t n |x 0 )f X (x 0 )Γ(s + 1) γ (0)(s +1) T (x 0 ) γ (0) Y (x 0 ) 1 (1 -sγ (0) T (x 0 )) s +1 +δ (0) T (t n |x 0 ) γ (0) Y (x 0 )γ (0) C (x 0 )a γ (0) T (x 0 )(γ (0) Y (x 0 ) + aγ (0) C (x 0 )) 1 γ (0) Y (x 0 ) + 1 γ (0) C (x 0 )a × 1 (1 + β (0) T (x 0 )γ (0) T (x 0 ) -sγ (0) T (x 0 )) s +1 - 1 (1 -sγ (0) T (x 0 )) s +1 + β (0) T (x 0 ) (1 + β (0) T (x 0 )γ (0) T (x 0 ) -sγ (0) T (x 0 )) s +1 (1 + o(1)) + O(h η f X ∧η A Y ∧η A C n ) + O(h ηγ Y ∧ηγ C n ln t n ) . Now let U n := 1 F T (t n |x 0 )f X (x 0 )         T (1) n (K, 0, 0|x 0 ) T (1) n (K, -β, 0|x 0 ) T (1) n (K, 0, 1|x 0 ) T (2) n (K, 0, 0|x 0 ) T (2) n (K, -β, 0|x 0 )         , µ :=            1 1 1+βγ (0) T (x 0 ) γ (0) T (x 0 ) γ (0) T (x 0 ) γ (0) Y (x 0 ) γ (0) T (x 0 ) γ (0) Y (x 0 )(1+βγ (0) T (x 0 ))            .
From Theorem 3 we have

r n (U n -µ) N (λ µ, Σ)
where Σ is a (5 × 5) symmetric matrix with elements

σ 11 := K 2 2 σ 12 := K 2 2 1 1 + βγ (0) T (x 0 ) σ 22 := K 2 2 1 1 + 2βγ (0)
T (x 0 )

σ 13 := K 2 2 γ (0) T (x 0 ) σ 33 := K 2 2 2 γ (0)2 T (x 0 ) σ 14 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 ) σ 44 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 ) σ 15 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 )) σ 55 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + 2βγ (0) T (x 0 )) σ 34 := K 2 2 γ (0)2 T (x 0 ) γ (0) Y (x 0 ) σ 23 := K 2 2 γ (0) T (x 0 ) (1 + βγ (0) T (x 0 )) 2 σ 35 := K 2 2 γ (0)2 T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 )) 2 σ 24 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 )) σ 45 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 )) σ 25 := K 2 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + 2βγ (0) T (x 0 ))
and µ is a vector with elements

µ 1 := 0, µ 2 := ββ (0) T (x 0 )γ (0) T (x 0 ) (1 + βγ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) , µ 3 := - β (0) T (x 0 )γ (0) T (x 0 ) 1 + β (0) T (x 0 )γ (0) T (x 0 ) , and • Case 1: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → ±∞ µ 4 := - β (0) T (x 0 )γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 )) + β (0) T (x 0 ) 1 + β (0) T (x 0 )γ (0) T (x 0 ) , µ 5 := - β (0) T (x 0 )γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) + β (0) T (x 0 ) 1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 ) , • Case 2: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → 0 µ 4 := - β (0) T (x 0 )γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 )) , µ 5 := - β (0) T (x 0 )γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) , • Case 3: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → a, where 0 < |a| < ∞ µ 4 := - β (0) T (x 0 )γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + β (0) T (x 0 )γ (0) T (x 0 )) + aβ (0) T (x 0 )γ (0) C (x 0 ) (γ (0) Y (x 0 ) + aγ (0) C (x 0 ))(1 + β (0) T (x 0 )γ (0) T (x 0 )) , µ 5 := - β (0) T (x 0 )γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + βγ (0) T (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) + aβ (0) T (x 0 )γ (0) C (x 0 ) (γ (0) Y (x 0 ) + aγ (0) C (x 0 ))(1 + βγ (0) T (x 0 ) + β (0) T (x 0 )γ (0) T (x 0 )) .
Based on [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF] and ( 8) we can write From the proof of the consistency, we know that C is a negative definite matrix, and thus invertible. Then, according to Lemma 5.2 in Chapter 6 of Lehmann and Casella (1998), for the solution of the system of equations ( 9), we have the following convergence

r n 1 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = ξ 1 r n (U n -µ) + r n δ (0) Y (t n |x 0 ) βγ (0) T (x 0 ) γ (0)2 Y (x 0 )(1 + βγ (0) T (x 0 )) (1 + o P (1)), r n 2 (γ (0) Y (x 0 ), δ (0) Y (t n |x 0 ); β) = ξ 2 r n (U n -µ) -r n δ (0) Y (t n |x 0 ) β 2 γ (0) T (x 0 ) γ (0) Y (x 0 )(1 + 2βγ
r n γ Y,n (x 0 ) -γ (0)
Y (x 0 ) δ Y,n (t n |x 0 ) -δ Working out the matrix products appearing in the mean and variance of the limiting distribution yields the result of the theorem.

  (2004) for some concrete practical examples where Pareto-type models have been successfully applied. In the present paper, interest is in estimating γ Y , but due to censoring one observes Y ∧ C rather than Y , along with a censoring indicator 1l {Y ≤C} . Estimation of the extreme value index based on censored data is considered by Beirlant et al. (2007), Einmahl et al. (2008), Gomes and Neves (2011), Worms and Worms (2014), Beirlant et al. (2016), to mention but a few.

  , Daouia et al. (2011), Dierckx et al. (2014), Goegebeur et al. (2014b). Also extensions to the estimation of the conditional extreme value index in the general max-domain of attraction have been considered, see Daouia et al. (2013), Stupfler (2013) and Goegebeur et al. (2014a). So far there are however very few contributions for the situation where one has random covariates and random right censoring of the response variable. Ndao et al. (2016) introduced a nonparametric estimator for γ Y (x) > 0 in presence of censoring, which is obtained by taking one of the local estimators of Goegebeur et al. (2014b), developed for the uncensored case, and dividing it by the local proportion of non-censored observations that exceed a high threshold. Similarly, in the general max-domain of attraction, Stupfler (2016) adjusted the local moment estimator, introduced in Stupfler (2013) to the censoring case, by dividing it by the local proportion of non-censored observations. Neither the Stupfler (2016) nor the Ndao et al.

  a kernel estimator for F T (t|x). This criterion was introduced in Yao (1999), and considered in an extreme value context by Daouia et al. (2011, 2013) and Escobar-Bach et al. (2018).

T

  (0.4) = 1.14, 2.28, 3.41, 4 and β (0) T (0.1) = 2.22, 4, 4, 4 in cases 1, 2, 3, 4, respectively.

( 1 )

 1 n (K, s, s |x 0 )) can be handled similarly, combined with some ideas from Dierckx et al. (2014). Note that Dierckx et al. (2014) also considered the statistic T (1)

( 1 )

 1 n (K, s, s |x 0 ) and T[START_REF] Beirlant | Statistics of Extremes -Theory and Applications[END_REF]

Y 2 γ 2 Y

 22 (t n |x 0 ); β) N (λκ, ∆),where ∆ is a symmetric (2 × 2) matrix with elements∆ 11 := K 2 (x 0 )(1 + βγ (0) T (x 0 )), and κ has elements• Case 1: δ

Y

  (t n |x 0 )/δ

C 1 γ

 1 (t n |x 0 ) → ±∞ κ 1 := -

C (t n |x 0 ) → 0 κ 1 =

 01 κ 2 = 0,• Case 3: δ (0) Y (t n |x 0 )/δ (0) C (t n |x 0 ) → a, where 0 < |a| < ∞ κ 1 := -aγ

.

  Now we focus on the right hand side of[START_REF] Dierckx | Local robust and asymptotically unbiased estimation of conditional Pareto-type tails[END_REF]. Concerning the terms ˇ jk (γ(0) Y (x 0 ), δ

Y

  (t n |x 0 ); β), j, k = 1, 2, we have by the consistency of ( γ Y,n (x 0 ), δ Y,n (t n |x 0 )) and because | jkl (γ Y , δ Y ; β)| ≤ M jkl (V ), in some open neighborhood of (γ (0) Y (x 0 ), δ

Y

  (t n |x 0 )), with M jkl (V ) = O P (1), j, k, l = 1, 2, that ˇ jk (γ (0) Y (x 0 ), δ

Y

  (t n |x 0 ); β) P → ˙ jk , j, k = 1, 2. Let C := ˙ 11 ˙ 12 ˙ 12 ˙ 22 .

Y

  (t n |x 0 ) N (-λC -1 κ, C -1 ∆C -1 ).

Figure 1 :

 1 Figure 1: Estimation of γ Y (0.4), case 1 (row 1), case 2 (row 2), case 3 (row 3), case 4 (row 4). Mean (left) and MSE (right) as a function of k for γ B Y,n (0.4) (dashed-dotted line), and γ Y,n (0.4) with β = β (0) T (0.4) (solid line) and β = 2 (dashed line).

Figure 2 :

 2 Figure 2: Estimation of γ Y (0.1), case 1 (row 1), case 2 (row 2), case 3 (row 3), case 4 (row 4). Mean (left) and MSE (right) as a function of k for γ B Y,n (0.1) (dashed-dotted line), and γ Y,n (0.1) with β = β (0) T (0.1) (solid line) and β = 2 (dashed line).

Figure 4 :Figure 5 :

 45 Figure 4: Scatterplot of the AIDS survival dataset.

  Solomon and the Australian National Centre in HIV Epidemiology and Clinical Research, see[START_REF] Venables | Modern Applied Statistics with S[END_REF]. This dataset was formerly analysed by[START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] also in an extreme value context but without covariates, and more recently considered in[START_REF] Ndao | Non parametric estimation of the conditional tail index and extreme quantiles under random censoring[END_REF] and[START_REF] Stupfler | Estimating the conditional extreme-value index under random rightcensoring[END_REF] but this time in a regression context. The information on each patient includes gender, date of diagnosis, date of death or end of observation and an indicator as to which of the two is the case. The dataset contains 2843 patients, among them 2754 males, of which 1708 died. Our methodology was applied in order to estimate the conditional extreme value index of the survival time Y of a patient conditionally on his age at the time of the diagnosis.A scatterplot of the dataset is shown in Figure4. From this figure, we can observe that most of the patients are between 20 and 65 years old at the time of diagnosis, so we only look at this interval of ages. Concerning the selection of h, we use the same procedure as in the simulation study with a grid of values in [h min , h max ] with h min = 0.05(x max -x min ) and h max = 0.30(x max -x min )

(γ Y , δ Y ; β) =

Acknowledgement

This work was supported by a research grant (VKR023480) from VILLUM FONDEN.

Appendix

Derivatives of the log-likelihood function