Julien Worms
email: julien.worms@uvsq.fr

Rym Worms
email: rym.worms@u-pec.fr

Estimation of extremes for heavy-tailed and light-tailed distributions in the presence of random censoring

Keywords: AMS Classification, Primary 62G32 ; Secondary 62N02 Keywords and phrases, Log-Weibull tail, Tail inference, Random censoring

) is considered for conducting tail inference of censored data. Both the censored and the censoring variables are supposed to belong to this family of distributions, and thus solutions for modeling the tail of censored data which are between Weibull-tail and Pareto-tail behavior are proposed. Estimators of the tail parameters and extreme quantiles are defined without prior knowledge of censoring strength and asymptotic normality results are proved. Various combinations of the tails of censored and censoring distributions are covered, ranging from rather mild censoring to severe censoring in the tail, i.e. when the ultimate probability of censoring in the tail is zero. Finite sample behavior is presented via some simulations and an illustration on real data is also provided.

Introduction

This paper proposes new contributions to the topic of extreme value statistics for data which are randomly censored from the right.

Consider the classical random censoring setup, where one observes a sample from a couple pZ, δq " pminpX, Cq, I XďC q with X denoting the variable of interest, and C a censoring variable (independent from X) which may prevent the user from observing the data X. The observed data is a sample pZ 1 , δ 1 q, . . . , pZ n , δ n q where pX 1 , . . . , X n q and pC 1 , . . . , C n q are independent samples of i.i.d. copies of X and C and δ i " I XiďCi . The topic of extreme value statistics for randomly censored data deals with the estimation of the tail of X (tail parameters, extreme quantiles, rare probabilities of exceeding a large value), while observing such an incomplete data sample.

A variety of topics can fit this formal random censoring framework. For instance, in the so-called survival analysis setting, the statistical units are patients suffering from a disease, and X 1 , . . . , X n denote the times elapsed between their inclusion in a study and some event of interest (recovery, recurrence of the disease, death, etc...). Due to loss of follow-up or end of study, one only observe durations pZ 1 , . . . , Z n q and noncensoring indicators pδ 1 , . . . , δ n q. An extreme quantile, in this survival analysis context, is a duration x p that the "lifetime" of a patient is expected to exceed only with a small probability p, typically smaller than 1{n. Due to scarcity of data in the tail, some sort of semi-parametric modeling is required to estimate such extreme quantile x p . This topic has benefited from a number of contributions in the recent years, which were stimulated by applications in a variety of domains, mainly reliability analysis, survival/lifetime analysis and insurance. [START_REF] Beirlant | Estimation of the extreme value index and extreme quantiles under random censoring[END_REF] and [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] presented a general method for adapting estimators of the extreme value index in this censorship framework. [START_REF] Worms | New estimators of the extreme value index under random right censoring, for heavy-tailed distributions[END_REF], [START_REF] Beirlant | Asymptotic distribution for an extreme value index estimator in a censorship framework[END_REF] and [START_REF] Worms | Moment estimators of the extreme value index for randomly censored data in the Weibull domain of attraction[END_REF] proposed a more survival analysis-oriented approach, the first two being restricted to the heavy tail case. [START_REF] Worms | Extreme value statistics for censored data with heavy tails under competing risks[END_REF] extended this survival analysis approach to competing risks. The Weibull-tail class of distributions is studied in [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF]. [START_REF] Ndao | Nonparametric estimation of the conditional tail index and extreme quantiles under random censoring[END_REF], [START_REF] Ndao | Nonparametric estimation of the conditional extreme-value index with random covariates and censoring[END_REF] and [START_REF] Stupfler | Estimating the conditional extreme-value index in presence of random right-censoring[END_REF] extended the framework to data with covariate information. [START_REF] Beirlant | Bias reduced tail estimation for censored Pareto type distributions[END_REF], [START_REF] Beirlant | Penalized bias reduction in extreme value estimation for censored Pareto-type data, and long-tailed insurance applications[END_REF] and Goegebeur et al. (2019a) considered the bias-reduction problem. The multivariate case is studied in Goegebeur et al. (2019b) and [START_REF] Hashorva | Modeling of censored bivariate extremal events[END_REF]. See also [START_REF] Beirlant | Peaks-Over-Threshold modeling under random censoring[END_REF] , [START_REF] Gomes | Estimation of the extreme value index for randomly censored data[END_REF], [START_REF] Sayah | On robust tail index estimation under random censorship[END_REF], [START_REF] Brahimi | Necir Approximations to the tail index estimator of a heavy-tailed distribution under random censoring and application[END_REF], [START_REF] Brahimi | Nelson-Aalen tail product-limit process and extreme value index estimation under random censorship[END_REF], [START_REF] Brahimi | Tail empirical process and a weighted extreme value index estimator for randomly right-censored data[END_REF], [START_REF] Stupfler | On the study of extremes with dependent random right censoring[END_REF] and Bladet al. (2020) for other papers on the subject.

A characteristic of most of these papers is that X and C are always supposed to share the same type of tail, i.e. a heavy tail censored by a heavy tail, a light (Gumbel) tail censored by another light tail, or a finite tail censored by a finite tail. This is for instance very well described by the 3 cases exhibited in formula (7) of the insightful paper [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF].

The main and initial objective of this paper is to broaden the type of tails in the Gumbel domain that the user will be able to deal with, for estimating tail parameters and extreme quantiles based on censored data. As a matter of fact, the lighter-than-Pareto-tails situation was slightly overlooked in censored extremes works, and this may be considered unfortunate since several applications of the censored extremes question do not necessarily exhibit a heavy tail behavior (particularly in survival/lifetime analysis). Essentially only two research papers proposed so far solutions for dealing with light tails. The first one is [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] which proposed estimators of the extreme value index and of extreme quantiles in various cases and in particular in the double Gumbel case (a distribution in the Gumbel domain of attraction censored by another distribution in the same domain). However, the results on the extreme value index, in this case, are stated with a restrictive assumption on the ultimate probability of non-censoring in the tail and there is no formal convergence statement for the proposed extreme quantiles estimator. The second one is [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF] which considered the general two Weibull-tails framework (a distribution in the Weibull-tail class censored by another distribution in the same class) : this is a strict subset of the double Gumbel case, allowing however interesting configurations where the ultimate probability of non-censoring in the tail can be zero (see its definition in next Section).

The basement of the present work is the flexible semi-parametric model proposed in [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF] (model A 1 pτ, θq described in the next section), which encompasses a large part of the Gumbel maximum domain of attraction and the whole Fréchet one, and therefore provides a more flexible option for modeling various phenomena. In this paper, estimation of the parameters of this model will be made possible in the presence of censoring, with very simple expressions for the estimators. In addition, this setup will allow for a more diverse combination of tails (without prior knowledge of that combination) than the Fréchet versus Fréchet or the Weibull-tail versus Weibull-tail cases (see next section).

The paper is organized as follows. Section 2 formally settles the framework and describes how the parameters of the observed Z can be deduced from those of X and C, thus explaining what is statistically at stake. Section 3 explains how the parameters and extreme quantiles of X can be estimated from the observed censored data, while Section 4 states the main results of this paper, along with the required assumptions on the number k n of order statistics retained for the estimation. Section 5 contains simulations to illustrate the performance of our estimators and Section 6 an illustration on real-data. Part A to D of the Appendix are devoted to the proofs of our asymptotic results, while part E contains important technical results. Technical aspects of the proofs can be found in a Supplementary Material document provided by the authors [START_REF] Worms | Supplementary Material for : Estimation of extremes for heavy-tailed and light-tailed distributions in the presence of random censoring[END_REF]).

Description of the framework and assumptions

Model and main assumptions

In the sequel, F ´denotes the general inverse of a function F , F ´ptq " inftx P R; F pxq ě tu for any t P r0, 1s.

The formal framework of this paper is the following. Defining for τ P r0, 1s the Box-Cox function

K τ pxq " ż x 1 u τ ´1du " " px τ ´1q{τ if τ Ps0, 1s, logpxq if τ " 0,
we consider, for parameters τ P r0, 1s and θ ą 0, that a distribution function F belongs to the semi-parametric family A 1 pτ, θq if the following holds (see [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF] where this model was initially introduced in a complete data setting, and El [START_REF] Methni | Estimation of extreme quantiles from heavy and light tailed distributions[END_REF] for additional developments):

A 1 pτ, θq : for some x ˚ą 0 and every x ě x ˚, we have

1 ´F pxq " expp´K τ plogpHpxqqqq,
where H is an increasing positive function such that H ´is regularly varying at infinity with index θ (which will be denoted by H ´P RV θ ).

Let us highlight that the tail heaviness of a distribution belonging to A 1 pτ, θq is mainly driven by τ , although in practice both shape parameters τ and θ play an important role in the properties and shape of the upper tail. It is easy to see that (for more details see Proposition 2 in [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF]) :

• A 1 p1, θq corresponds to distributions in the Fréchet domain of attraction with extreme value index θ (i.e. when 1 ´F pxq " x ´1{θ lpxq with l slowly varying).

• A 1 p0, θq corresponds to Weibull-tail distributions with Weibull-tail coefficient θ (i.e. when 1 ´F pxq " expp´x 1{θ lpxqq with l slowly varying).

• The case τ Ps0, 1r corresponds to distributions in the Gumbel domain having tails heavier than Weibulltype ones : such distributions can be conveniently qualified as having log-Weibull-type tails, and lognormal distributions belong to this category with τ " 1{2 (see [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF] for more examples). The larger τ is, the heavier the tail can be considered.

In this work, the main assumption is that both the censored and the censoring variables have their distribution belonging to the A 1 pτ, θq family. This assumption covers a quite flexible setting. Indeed, the users will not need to decide in advance whether a Pareto, Weibull or Log-Weibull tail is convenient for their data, either for the target X or the censoring C. In particular, it is important to note that X and C do not necessarily share the same type of tail. For instance, the cases of Weibull-type data censored by a log-Weibull-type distribution, or of Pareto-type data censored by a log-Weibull-type distribution, are covered. We thus assume the following : Assumption (A1) : there exist τ X P r0, 1s, τ C P r0, 1s, θ X ą 0, θ C ą 0 such that

F X P A 1 pτ X , θ X q and F C P A 1 pτ C , θ C q.
This means that there exists positive functions H X and H C such that s F X pxq " 1 ´FX pxq " expp´K τX plogpH X pxqqqq and s F C pxq " 1 ´FC pxq " expp´K τC plogpH C pxqqqq and, for some slowly varying functions lX and lC at infinity, H X pxq " x θ X lX pxq and H Ć pxq " x θ C lC pxq.

It is clear that under this condition we also have H X pxq " x 1{θ X l X pxq and H C pxq " x 1{θ C l C pxq where both l X and l C are slowly varying functions at infinity.

The estimation of the parameters τ X and θ X is the main objective of this work (with the estimation of extreme quantiles of F X being its main application). A first step to do so is to find a relation between the parameters of X and C and those of the observed variable Z " mintX, Cu.

Under assumption (A1), the following proposition states that the distribution of Z also belongs to the same family of distributions as those of X and C, for some parameters τ Z and θ Z specified below :

Proposition 1. Under Assumption (A1), the distribution function of Z " minpX, Cq satisfies condition A 1 pτ Z , θ Z q, where τ Z " minpτ X , τ C q and θ Z " $ ' ' & ' ' % θ X if 0 ď τ X ă τ C ď 1 θ C if 0 ď τ C ă τ X ď 1 pθ ´1{τ Z X `θ´1{τ Z C q ´τZ if 0 ă τ X " τ C ď 1 minpθ X , θ C q if τ X " τ C " 0
Therefore, there exists x ˚ą 0 such that for any x ě x ˚, we have

PpZ ą xq " expp´K τZ plogpH Z pxqqqq,
where H Ź P RV θ Z . Consequently, if E denotes a standard exponential variable, we have

Z d " H Ź pexp K τ Z pEqq.
Remark 1. It is interesting to note that :

´in the two-heavy-tails case τ X " τ C " 1, we recover the well-known fact that θ

Z " γ Z " pγ ´1 X `γ´1 C q ´1
where γ X and γ C are the extreme value indices of X and C (see [START_REF] Beirlant | Estimation of the extreme value index and extreme quantiles under random censoring[END_REF]).

´in the two-Weibull-tails case τ X " τ C " 0, we recover the fact that the Weibull-tail parameter of Z is equal to the minimum of those of X and C (see [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF]).

´when τ X " τ C , we have θ Z ď minpθ X , θ C q, but otherwise this is not necessarily the case.

´the expression of θ Z in the fourth case is coherent with the third one in the sense that minpθ X , θ C q is indeed the limit of pθ ´1{τ X

`θ´1{τ

C q ´τ as τ Ñ 0.

In this paper, we will exclude the first two situations evoked in Remark 1 above, which have already been explored in anterior works, and therefore suppose that pτ X , τ C q P r0, 1s 2 zt p0, 0q , p1, 1q u.

Let us close this subsection by now describing the more technical assumptions required for our results to hold. This part of the section may be skipped on first reading.

In order to achieve asymptotic normality of the estimators defined in this paper, the slowly varying functions l X and l C associated to H X and H C are supposed to satisfy a classical second order condition (usually called the SR2 condition) :

Assumption (A2) : there exist some negative constants ρ X and ρ C , and some rate functions b X and b C having constant sign at `8 and satisfying |b X | P RV ρ X and |b C | P RV ρ C , such that, as t Ñ `8,

l X ptxq{l X ptq ´1 b X ptq ÝÑ K ρ X pxq, and l C ptxq{l C ptq ´1 b C ptq ÝÑ K ρ C pxq, @x ą 0.
(1) According to the last statement of Proposition 1 and to the expression of our estimators (see next Section), it will be important in the sequel to consider the functions H Ź pxq " x θ Z lpxq and H X ˝HŹ pxq " x a lpxq with a :"

θ Z θ X , (2)
where both l and l are slowly varying. The crucial parameter a " θ Z {θ X is equal to 1 in "milde censoring" situations (in particular when τ X ă τ C ).

In addition, our important technical Lemma 1, stated in Appendix E.1, ensures that functions H Ź and H X ˝HŹ also satisfy a second order condition SR2. For technical reasons though, we need to consider the following stronger conditions on l and l, respectively noted R lp b, ρq and R l pb, ρq, and defined by : Assumption R pB, ρq : for some constant ρ ď 0 and a rate function B satisfying lim xÑ`8 Bpxq " 0, such that for all ą 0, we have sup λě1

ˇˇˇ pλxq{ pxq ´1 BpxqK ρ pλq ´1ˇˇˇˇď , for x sufficiently large .

Note that, according to Lemma 1 (see Appendix E.1), we have ρ " ρ, and that this parameter is negative when either τ X " 0 or τ C " 0, but otherwise (i.e. in most cases) it is zero, an unpleasant fact which often implies some challenge in the proofs, and affects the rates of convergence of our estimators (with respect to the non-censored framework).

Proportion of censoring in the tail

It is well known that the strength of censoring affects the statistical performance of estimators in survival analysis. This is naturally also the case for tail estimation under random censoring. Indeed, the ultimate proportion of non-censoring in the tail, denoted p below, explicitly appears in asymptotic variances in the context of extremes of censored data. For instance, the adaptation of the Hill estimator introduced in [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] multiplies the asymptotic variance of the non-censored case by 1{p (see Corollary 1 therein), provided that this ultimate probability p is positive. Moreover, the pre-asymptotic probability ppxq " Ppδ " 1|Z " xq of being non-censored at level x (for large x) often plays a crucial role for proving asymptotic results.

In the context of this paper, Lemma 2 (stated in Appendix E.1) provides precise expansions for ppxq, for large x, which turn out to be useful in the proofs of our asymptotic results. In particular, its statement (i) yields the following :

lim xÑ`8 ppxq " p :" $ ' & ' % 1 if 0 ď τ X ă τ C ď 1, 0 if 0 ď τ C ă τ X ď 1, θ 1{τ X X { pθ 1{τ X X `θ1{τ X C q if 0 ă τ X " τ C ă 1.
Note that when X and C are both in the Fréchet or both in the Weibull (i.e. finite tail, not Weibull-type tail) maximum domain of attraction, p necessarily belongs to s0, 1r (see [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] page 214, for instance). When X and C are both in the Gumbel maximum domain of attraction, things are more complicated, and in this case, [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] considered the assumption p Ps0, 1r, which is difficult to check in practice and somehow restrictive.

In the model considered in this paper, p can thus span the whole r0, 1s interval. In the first situation above (the light censoring one), the fact that the ultimate probability p of non-censoring in the tail is 1 and that the parameters of X are the same as those of Z (see Proposition 1) would suggest that taking into account the censoring is useless. However, as [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF] already put forward, this is not advisable because those settings produce finite size data where censoring is still present and needs to be taken into account. Similarly, the second situation (strong censoring) where the ultimate probability p is 0 produces, in practice, data which are not completely censored in the tail, and thus the statistical problem of estimating the tail parameters and extreme quantiles of X should and can be addressed. Finally, one can note that the particular situation where tails of X and C have the same heaviness (τ X " τ C ) is interesting on its own.

Construction of the estimators

Let us denote by Λ X and Λ C the cumulative hazard functions associated to F X and F C , respectively

Λ X pxq " ´log s F X pxq and Λ C pxq " ´log s F C pxq,
and let ΛnX denote the Nelson-Aalen estimator of Λ X defined as

ΛnX pxq " ÿ

Zi,nďx

δ i,n n ´i `1 , (3)
where Z 1,n ď . . . ď Z n,n are the order statistics of the sample pZ i q and δ 1,n , . . . , δ n,n are the corresponding indicators associated to these reordered Z values. Let k n " opnq be an intermediate sequence of integers (which will often be simply denoted by k), representing the number of upper data values retained for tail estimation.

In the following lines, we derive the approximations that inspired our estimators defined below in relation ( 6). Under assumption (A1), H X is regularly varying with index 1{θ X and K τ X pΛ X pxqq " logpH X pxqq, hence, for u large, we have

K τ X puq « 1 θ X logpΛ X puqq.
Moreover, for s large and any u ą

1 log ˆKτ X psuq K τ X psq ˙" log ˆpsuq τ X ´1 psq τ X ´1 ˙» τ X log u.
Combining these two results, we obtain a first approximation, for u and s large, relating τ X to Λ X :

τ X log u « log logpΛ X psuqq ´log logpΛ X psqq. (4)
The second approximation comes from the fact that, for t large and any given x ą 1, we have

H X ptxq H X ptq " exppK τ X pΛ X ptxqq ´Kτ X pΛ X ptqqq » x 1{θ X , hence θ X is related to τ X and Λ X via the formula : 1 θ X log x « K τ X pΛ X ptxqq ´Kτ X pΛ X ptqq. (5)
Therefore, applying approximation (4) to s " Λ X pZ n´kn,n q and u " Λ X pZ n´j`1,n q{Λ X pZ n´kn,n q on one hand, and approximation (5) to t " Z n´kn,n and x " Z n´j`1,n {Z n´kn,n on the other hand, and then plugging in the Nelson-Aalen estimator of Λ X and summing for 1 ď j ď k lead to our proposed estimators of τ X and θ X :

τX :" HH k,n D k,0 and θX,τ X :" H k,n D k,τ X (6)
with

H k,n :" 1 k n kn ÿ j"1 logpZ n´j`1,n q ´logpZ n´kn,n q, HH k,n :" 1 k n kn ÿ j"1 log logpZ n´j`1,n q ´log logpZ n´kn,n q, D k,τ X :" 1 k n kn ÿ j"1 K τ X p ΛnX pZ n´j`1,n qq ´Kτ X p ΛnX pZ n´kn,n qq.
The two estimators above are thus ratios involving on one hand the mean of either the log-spacings (i.e. the Hill statistic) or the log-log-spacings, and on the other hand a denominator involving the Nelson-Aalen estimator at the k upper values of the observed Z sequence.

Note that the expressions of the estimators defined in (6) do not depend on the relative positions of τ X and τ C (or of θ X and θ C ). They can be calculated whatever the combinations of the tails of X and C are, with the same formulas. However, we will see in the next Section that the rates of convergence, performances, and assumptions of these estimators can differ depending on the strength of censoring.

Remark 2. In the case τ X " τ C " 0, corresponding to the purely Weibull-tail framework, the estimator θX,0 corresponds to the one studied in [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF], because K τ Z pxq " logpxq in that case. In the case τ X " τ C " 1, corresponding to the purely heavy-tail framework, the estimator θX,1 corresponds to the adapted Hill estimator studied in [START_REF] Beirlant | Estimation of the extreme value index and extreme quantiles under random censoring[END_REF], because in that case K τ Z pxq " x ´1 and thus we have exactly D k,1 " pk (see formula (7) below). As said earlier, these two particular cases are excluded from the scope of the statements of this paper because properties of θX,0 and θX,1 are already known.

The main issue in the proofs lies in the treatment of the denominators D k,τ X and D k,0 . In fact, the statistic D k,τ X defined below p6q turns out to be related to the proportion pk of uncensored data in the tail via the relation (see Lemma 3 in the Appendix for the details)

D k,τ X « ´Λ nX pZ n´k,n q ¯τX ´1 pk where pk :" 1 k k ÿ j"1 δ n´j`1,n
because of the nature of the Box-Cox transformation K τ X , Taylor's formula, and of the fact that

1 k k ÿ j"1 ´Λ nX pZ n´j`1,n q ´Λ nX pZ n´k,n q ¯" 1 k k ÿ j"1 k ÿ l"j δ n´l`1,n l " 1 k k ÿ j"1 δ n´j`1,n . (7)
Therefore, the properties of our estimators will rely on a careful study of two sequences. The first one is ΛnX pZ n´k,n q (in particular, how it can be approximated by Λ X pZ n´k,n q and written as an increasing function of log n{k ; see Lemma 4 in the Appendix). The second one is the sequence pk , which converges to 0, 1 or a value p Ps0, 1r depending on the position of τ X with respect to τ C (Proposition 2 in section Appendix A provides the full details about this, and relies on sharp second order expansions of the different regularly varying functions that appear in this framework, cf the important technical Lemmas 1 and 2 in the Appendix).

Finally, let us deal with the estimation of an extreme quantile x pn :" s F X pp n q of the distribution of X, with p n Ñ 0, as n Ñ `8. Applying the approximation (5) now to t " Z n´k,n and x " x pn {Z n´k,n , we can propose the following estimator of x pn (with both θ X and τ X being unknown) :

xpn :" Z n´k,n exp ! θX,τ X ´Kτ X p´logpp n qq ´Kτ X p ΛnX pZ n´k,n qq ¯) . (8)
Note that if we know that τ X " 0 and we then set τX " 0, then this estimator is the same as the one proposed in [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF].

Asymptotic results

The main assumptions on the model and the different notations have been stated in the previous sections. In order to obtain the asymptotic normality of our estimators, we naturally need the sequence pk n q (number of top order statistics to use) to satisfy some conditions (we will note k " k n from now on). The first one is standard in the literature on Weibull-tail or log-Weibull-tail models :

H 1 : k Ñ `8, k n Ñ 0, log k log n Ñ 0, as n Ñ `8.

Moreover, introducing the important notation

L nk " logpn{kq, let v n be a factor which will contribute to the rates of convergence of our estimators, and which depends on the censoring strength in the tail :

v n :" $ ' ' & ' ' % 1 if 0 ă τ X ă τ C ď 1 or 0 ă τ X " τ C ă 1 or 0 " τ X ă τ C ď 1, L 1 2 p τ C τ X ´1q nk if 0 ă τ C ă τ X ď 1, L ´1{2 nk plog L nk q 1 2 p 1 τ X ´1q if 0 " τ C ă τ X ď 1.
Note that v n " 1 in the mild or moderate censoring cases (when p " 1 or at least p ą 0, see Section 2.2), and v n Ñ 0 in the strong censoring cases (when p " 0). We also consider the following conditions

H 2 : 0 ă τ X ă τ C ď 1 and # piq ? kL τ X {τ C ´1 nk Ñ 0 if 1 τ C ´1 τ X ě ´1 piiq ? kL ´τX nk Ñ 0 if 1 τ C ´1 τ X ă ´1 H 3 : 0 ă τ C ă τ X ď 1 and $ ' & ' % piq ? kv n Ñ `8 piiq ? kv n L τ C {τ X ´1 nk Ñ 0 if 1 τ X ´1 τ C ě ´1 piiiq ? kv n L ´τC nk Ñ 0 if 1 τ X ´1 τ C ă ´1 H 4 : 0 ă τ X " τ C ă 1 and ? kL ´τX nk Ñ 0 H 5 : 0 " τ X ă τ C ď 1 and Dδ ą 0, ? kL ρ`δ nk Ñ 0 H 6 : 0 " τ C ă τ X ď 1 and " piq ? kv n Ñ `8 piiq ?
kv n plog L nk q ´1 Ñ 0 (in assumption H 5 above, ρ denotes the second order parameter associated to the slowly varying function l, which is negative in this case ; see formula (2) in Section 2 as well as Lemma 1 in Appendix E.1) Remark 3. A possible choice of the sequence pk n q satisfying the conditions above, as well as the additional conditions in the theorems stated below, is :

k n " # plog nq a if τ C ‰ 0, log n plog log nq b if τ C " 0.
The choice and scope of exponents a and b depend on which condition H 2 , . . . , or H 6 is considered. Note that these sequences tend to infinity a bit more slowly than the sequences pk n q considered in El [START_REF] Methni | Estimation of extreme quantiles from heavy and light tailed distributions[END_REF], in the non-censoring situation (see the paragraph following the statement of their Theorem 2).

The following four theorems respectively state the convergence in distribution of the estimators θX,τ X (with τ X known), τX , θX,τ X , and xpn , all of them being defined in the previous section.

Theorem 1. Let assumptions (A1) and (A2) hold, with pτ X , τ C q P r0, 1s 2 zt p0, 0q , p1, 1q u, as well as R l pb, ρq and R lp b, ρq. If pk n q satisfies H 1 and one of the conditions H 2 , . . . , H 6 , then we have, as n Ñ 8, ?

kv n p θX,τ X ´θX q d ÝÑ N `0, θ 2 X σ 2 ˘,
where a " θ Z {θ X and

σ 2 " $ ' ' ' ' & ' ' ' ' % 1 if 0 ď τ X ă τ C ď 1, a ´1{τ X ´τX τ C ¯1´1{τ X if 0 ă τ C ă τ X ď 1, a ´1{τ X if 0 ă τ X " τ C ă 1, a ´1{τ X τ 1´1{τ X X if 0 " τ C ă τ X ď 1.
Remark 4. When τ X ă τ C , the ultimate probability p of non-censoring is 1, this is the mild censoring situation. When τ X " τ C , it is easy to see that θ Z ă θ X and thus the asymptotic variance is larger than in the case τ X ă τ C (i.e. we have σ 2 ą 1). When 0 ă τ C ă τ X (strong censoring setting), the ultimate probability of non-censoring p is zero, and the factor σ 2 is ă 1 when θ C ą θ X , but otherwise this is not necessarily the case.

Theorem 2. Under the same assumptions as Theorem 1, we have, as n Ñ 8,

if τ X ‰ 0, ? kv n pτ X ´τX q d ÝÑ N `0, τ 2 X σ 2 ˘, if τ X " 0, τX " O P p1{ logpL nk qq P ÝÑ 0,
Theorem 3. Under the same assumptions as Theorem 1, if τ X ą 0 and if we further assume that ?

kv n log L nk Ñ `8 pif τ C ‰ 0q or ? kv n log log L nk Ñ `8 pif τ C " 0q, (9)
we then have, as n Ñ 8,

if τ C ‰ 0 ? kv n log L nk p θX,τ X ´θX q d ÝÑ N `0, θ 2 X σ 2 τ 2 Z ˘, if τ C " 0 ? kv n log log L nk p θX,τ X ´θX q d ÝÑ N `0, θ 2 X σ 2 ˘.
Remark 5. Note that the rate of convergence and asymptotic variance of θX,τ X are altered and different from that of θX,τ X due to the plug-in of τX .

Theorem 4. Under the same assumptions as Theorem 3, if moreover ? kv n log logp1{p n qp´logpp n qq τ X Ñ `8

(10)

and log L nk log logp1{p n q Ñ 0 pif τ C ‰ 0q or log log L nk log logp1{p n q Ñ 0 pif τ C " 0q, (11)
we then have, as n Ñ 8, ? kv n log logp1{p n qp´logpp n qq τ X ˆx pn

x pn ´1˙d ÝÑ N `0, θ 2 X σ 2 ˘.

Remark 6. Note that condition p11q allows for an order p n of the high quantile which is lower than 1 n , making it really extreme, while condition p10q is a restriction on this order.

τ X τ C p=1, k rate p=1, k rate, H 5 H 2 (ii) H 2 (i) 0 < p < 1 , k r a t e , H 4 p=0, k Lnk - 1 2 τC (τ C -1 -τ X -1 ) rate p=0, k Lnk - 1 2 log(L nk ) 1 2 (τ X -1 -1) rate, H 6 H 3 (ii) H 3 (iii)
strong censoring mild censoring m o d e r a t e c e n s o r i n g , Figure 1: Illustration of the possible combinations of τ -parameters and the impact on rates and assumptions (τ X " τ C " 0 and τ X " τ C " 1 are excluded).

Remark 7. There is some sort of phase transition phenomenon in the above results. As a matter of fact, not only the rate of convergence of our estimators vary whether τ X is ď τ C or not, but the closeness of the parameters τ X and τ C also play a role (see assumptions H 2 and H 3 ) : the assumptions vary whether τ X is lower than τ C but not too close to it (i.e. 1 ă 1 τ X ´1 τ C ), lower than τ C but close to it (i.e. 0 ă 1 τ X ´1 τ C ď 1), equal to τ C , larger than and close to τ C (i.e. 0 ă 1 τ C ´1 τ X ď 1), or sufficiently larger than τ C (i.e. 1 ă 1 τ C ´1 τ X ). Figure 1 helps to understand these facts. However, in practice, for finite and moderate values of n, visualizing these findings on simulations is not easy, because other factors (than just the tail parameters) play a non-negligible role in the estimation quality.

Let us close this section by providing a hint of the proof of the consistency of our estimators (consistency alone is not considered in the full proofs, only asymptotic normality is detailed

ˆµ1,τ Z pL nk q D k,τ X and τX " τ pcq Z ˆµ1,0 pL nk q lµ 1,τ Z pL nk q ˆlµ 1,τ Z pL nk q D k,0 (13)
where lµ 1,τ ptq :" ş 8 0 plogpK τ px `tqq ´logpK τ ptqqq e ´x dx. The consistency of θX,τ X will thus come from the convergence of the ratio µ 1,τ Z pL nk q{D k,τ X to 1{a " θ X {θ Z , which is deduced from Corollary 1 (stated in Appendix A) of the present paper. The consistency of τX comes from the convergence of lµ 1,τ Z pL nk q{D k,0 to τ X , which is deduced from Corollary 2 (stated in Appendix B), and from the fact that µ 1,0 ptq{lµ 1,τ Z ptq converges to 1{τ Z as t Ñ 8 (which is deduced from relations (A.3) and (B.3) in the Appendix).

It is noteworthy that equation ( 13 In this section, we illustrate, using few simulations, the finite sample performances of our estimators of τ X , θ X and x pn (for small p n ), in terms of observed bias and mean squared error (MSE). Note that numerous different situations could be considered with our flexible framework : a thorough and extensive simulation study is however not possible within the limits of the present paper. We generate N " 1000 samples of size n " 500.

We consider three classes of distributions of Log-Weibull-tail type, for the target variable X and the censoring variable C (for the first two classes, see Proposition 3 in [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF] for the justification that they fit our framework) :

• Log-Weibullpθq distribution such that its logarithm has c.d.f. 1 ´expp´x 1{θ q (x ą 0). It satisfies assumption A 1 pθ, θq.

• Log-Normal distribution LN pµ, σ2 q, which satisfies assumption A 1 p 1 2 , σ ? 2 2 q.

• Model F with c.d.f. F τ satisfying A 1 pτ, 1{5q, with H ´pxq " x 1{5 p1 `x´1{2 q p@xq.

We then consider three cases : a Log-Weibullpθ X q distribution censored by the Log-Normalp1, 1{2q distribution (Figure 2), the Log-Normalp1, 1{2q distribution censored by a Log-Weibullpθ C q distribution (Figure 3), and then a distribution in the F model censored by another distribution in the F model (Figure 4). In each case, we consider three situations with τ X ă τ C , τ X " τ C or τ X ą τ C , corresponding to different (ultimate) intensities of censoring in the tail.

In parts paq,pcq,peq of Figures 2, 3 and 4, we present the bias and the MSE of our estimators τX and θX,τ X as a function of k. In parts pbq,pdq,pf q of Figures 2, 3 and 4, we present the relative bias and the relative MSE of our estimator xpn for the value p n " 0.001, compared with those of the existing estimator defined, in a more general censored setting, by equation p8q in [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] :

xEF G pn " Z n´k,n `â k pp1 ´F n pZ n´k qq{p n q γc,Mom ´1 γc,Mom , (14)
where γc,Mom is the moment estimator of the extreme value index γ X of F adapted to censoring and Fn stands for the Kaplan-Meier estimator of the c.d.f. F . We refer to [START_REF] Einmahl | Statistics of extremes under random censoring[END_REF] for the expression of âk . Note that no formal asymptotic result is currently available for xEF G pn .

Concerning the performance of the estimators θX,τ X and τX , we observe that when X has a Log-Weibull tail, the bias and the MSE for both estimators are very small. When one deviates from this situation, though, they are not very satisfactory on the situations presented here. Note however that these estimators are the first to be proposed in this context, which is why no comparison to competitors is presented . Another remark is that the quality of the estimators do not systematically deteriorate when censoring gets stronger.

Concerning the performance of the high quantile estimator, the figures show very good performances when X has a Log-Weibull tail. When one deviates from this situation, things may become worse. It is particularly true here in the Log-Normal versus Log-Weibull case. However, our estimator remains competitive in terms of bias and MSE in a number of situations, for instance in Figure 4.

Real data analysis

In this section, we apply our methodology to one of the datasets included in the Cancer Genome Atlas (TCGA, more information at cancergenome.nih.gov), namely the dataset concerning women suffering from an ovarian cancer. The ov.clinical dataset, accessible using the R package RTCGA (or manually), contains informations about 574 women 2 . In this section we will only be interested in the vital status of the patient (δ " 1 if the patient died during the study, or δ " 0 if the patient was still alive at the end of the study or was lost in the follow-up), and the observed duration Z " minpX, Cq (Z " X if survival time is actually observed, and Z " C if censoring occurred). The tail of the survival time distribution F X will be estimated using the model (A1), and in this context an extreme quantile x p associated to a small probability p, is a survival time that a patient is expected to exceed only with probability p.

We plot, in figure 5, the proportion pk of non-censoring as a function of k n . We observe that the censoring is rather strong, overall censoring rate is around 40% and more around 55% in the tail. If we consider the quite stable zone where k n is between 55 and 80, we can estimate the probability of non-censoring in the tail by 0.46.

We plot, on the left part of Figure 6, the values of our estimators τX (thick blue) and θX,τ X (thin red) against k n . The estimate curve, as a function of k n , is particularly stable for τ X (an estimate of 0.12, possibly suggesting a Weibull-tail underlying distribution), but not very stable for θ X (an estimation between 0.85 and 0.9 if we consider the range of k n cited above).

On the right part of Figure 6, we plot the values of our estimator xpn (thick blue) of the extreme quantile xpn for the value p n " 0.001, as well as the estimator xEF G pn (thin green) against k n . We observe that it is quite hard to propose as estimation of x pn relying on the estimator xEF G pn , as it is particularly unstable in the tail. Concerning xpn , the choice of the sample fraction is delicate. However, if we consider the quite stable area where k n is between 55 and 80, as for the estimation of the parameters, then a possible estimation the extreme quantile is around 28 years. A concluding remark could be that estimating extreme quantiles under strong censoring is still a research subject in progress and every new contribution is welcome.

Conclusion

In this paper we propose a solution for dealing with tail and extreme quantile estimation of data which are randomly right censored, within a rather large family of distributions encompassing power tail distributions, Weibull-tail distributions, and intermediary situations such as (for instance) log-normal distributions. This 14 family was first introduced in a complete data context in [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF]. Our asymptotic normality results support all possible amounts of censoring in the tail, even very strong ones where the ultimate probability of being censored in the tail is equal to one.

The main two contributions of this work are that very diverse combinations of tails of the censored and censoring distributions are dealt with (not just a combination of tails from the same category), and that tail estimation of log-Weibull-type distributions (not heavier than Pareto tails though) are dealt with as well. The fact that one can estimate the tail parameters of this flexible model, and not just the extreme quantiles, means that the user may consider estimating more elaborated parameters than the extreme quantiles (for instance, expected tail losses EpX|X ą F X p1 ´pqq for small p, of course with additional efforts in order to formally prove convergence results).

Concerning the performances, the bias of our estimators of θ and τ remains a problem, as soon as one moves away from the pure log-Weibull situation. However our opinion is that this bias problem was already present for the original estimators of τ and θ (which inspired ours) in the non-censoring context. This topic of bias reduction still needs to be explored for this family of distributions, even in the non-censored situation. In this paper, we did not try to detail the asymptotic bias, mainly because of the great diversity of situations that our model handled, which already made the exposition a bit complicated. This would require further work.

Finally, a continuation of this work could be to look for estimators of τ and θ which are weighted modifications of their non-censored versions (the estimators in equation ( 12)), but with varying weights, not the constant weights D k,0 and D k,τ X , with in mind a possible improvement in terms of bias and mean-squared error.

intermediate relation :

θX,τ X ´θX d " θ Z M n ´θX Λ τ X ´1 k pk D k,τ X `3 ÿ i"1 T i,n ,
where

T 1,n :" R n, l D k,τ X T 2,n :" ´θX R 1,n D k,τ X T 3,n :" ´θX Λτ X ´1 k ´Λτ X ´1 k D k,τ X pk .
Concerning now pk , recalling that a :" θ Z {θ X , we prove in Lemma 5 (stated in Appendix E.2) that, when τ X ě 0 and τ C ą 0,

Λ τ X ´1 k pk " ˆaτ X τ Z ˙1´1{τ X E τ Z p1´1{τ X q n´k pk `R2,n
(note that the first term is equal to E ´1 n´k pk when 0 " τ X ă τ C ď 1, since then τ Z " τ X and a " 1), and when τ X ą 0 and τ C " 0,

Λ τ X ´1 k pk " paτ X q 1´1{τ X plog E n´k q 1´1{τ X pk `R2,n ,
where the remainder term R 2,n is detailed for each case in the statement of Lemma 5.

Consequently, defining T 4,n :" ´θX

R2,n D k,τ X
, we obtain the following decomposition : when τ X ě 0 and

τ C ą 0 θX,τ X ´θX d " σ 1,τ Z pE n´k q D k,τ X ˜θZ A 1,n ´θX µ 1,τ Z pE n´k q σ 1,τ Z pE n´k q ˜ˆaτ X τ Z ˙1´1{τ X E τ Z p1´1{τ X q n´k µ 1,τ Z pE n´k q pk ´θZ θ X ¸¸`4 ÿ i"1 T i,n ,
and, when τ X ą 0 and

τ C " 0, θX,τ X ´θX d " σ 1,τ Z pE n´k q D k,τ X ˜θZ A 1,n ´θX µ 1,τ Z pE n´k q σ 1,τ Z pE n´k q ˜paτ X q 1´1{τ X plog E n´k q 1´1{τ X µ 1,τ Z pE n´k q pk ´θZ θ X ¸¸`4 ÿ i"1 T i,n .
Then, recalling that µ 1,τ Z ptq " t τ Z ´1 as t Ñ 8, we define the following remainder term as (note again that aτ X {τ Z " 1 and τ Z p1 ´1{τ X q " ´1 when τ X " 0 ă τ C )

R 3,n :" $ & % ´aτ X τ Z ¯1´1{τ X pk ´pE n´k q τ Z p1´1{τ X q µ1,τ Z pE n´k q ´L1´τ Z {τ X nk ¯when τ X ě 0, τ C ą 0, paτ X q 1´1{τ X pk ´plog E n´k q 1´1{τ X µ1,0pE n´k q ´Lnk plog L nk q 1´τ Z {τ X ¯when 0 " τ C ă τ X .
Finally, using the additional fact that, thanks to pA.3q, µ1,τ Z pE n´k q σ1,τ Z pE n´k q P ÝÑ 1, we can state the main relation of the proof of Theorem 1 :

θX,τ X ´θX d " µ 1,τ Z pE n´k q D k,τ X pθ Z A 1,n ´θX A 2,n p1 `oP p1qqq `5 ÿ i"1 T i,n , (A.6)
where the second important term A 2,n is defined as

A 2,n :" $ ' & ' % ˆaτ X τ Z ˙1´1{τ X L 1´τ Z {τ X nk pk ´a if τ X ě 0 and τ C ą 0, paτ X q 1´1{τ X L nk plog L nk q 1´τ Z {τ X pk ´a if 0 " τ C ă τ X ,
and the last remainder term to be introduced is T 5,n :" θ Z R 3,n p1 `oP p1qq.

We deal with the asymptotic normality of A 2,n and the reminder terms T i,n in the following two propositions. Recall that the rate v n is defined as

v n :" $ ' ' & ' ' % 1 if 0 ă τ X ă τ C ď 1 or 0 ă τ X " τ C ă 1 or 0 " τ X ă τ C ă 1 L 1 2 p τ C τ X ´1q nk if 0 ă τ C ă τ X ď 1 L ´1{2 nk plog L nk q 1 2 p 1 τ X ´1q if 0 " τ C ă τ X ă 1 Proposition 2. Under the conditions of Theorem 1, if 0 ď τ X ă τ C ď 1, ? kv n A 2,n " ? kv n pp k ´aq " ? kpp k ´1q P ÝÑ 0, if 0 ă τ C ă τ X ď 1, ? kv n A 2,n " ? kv n ˆ´aτ X τ Z ¯1´1 τ X L 1´τ Z {τ X nk pk ´a˙d ÝÑ N ˆ0, a 2´1{τ X ´τX τ C ¯1´1{τ X ˙, if 0 ă τ X " τ C ă 1, ? kv n A 2,n " ? kpa 1´1{τ X pk ´aq d ÝÑ N `0, a 2´1{τ X p1 ´a1{τ X q ˘, if 0 " τ C ă τ X ă 1, ? kv n A 2,n " ? kv n ´paτ X q 1´1 τ X L nk plog L nk q 1´1 τ X pk ´a¯d ÝÑ N ´0, a 2´1{τ X τ 1´1{τ X X ¯.
Proposition 3. Under the conditions of Theorem 1, for all 1 ď i ď 5 , ? kv n T i,n P ÝÑ 0, as n tends to infinity.

The following result is a corollary of Proposition 2 and part of Proposition 3 (concerning the term T 2,n ). As explained at the end of Section 4, the statement of this corollary is helpful for understanding how consistency of an estimator of θ Z transfers to consistency of our estimator of θ X .

Corollary 1. Under the conditions of Theorem 1, we have D k,τ X µ 1,τ Z pE n´k q P ÝÑ a, as n tends to infinity.

Indeed, according to pA.5q, and since µ 1,τ Z ptq " t τ Z ´1 as t Ñ 8 (see relation

(A.3)), D k,τ X µ 1,τ Z pE n´k q " L 1´τ Z nk Λ τ X ´1 k pk p1 `op1qq d " pA 2,n `aqp1 `op1qq P ÝÑ a.
Of course, Corollary 1 certainly holds with weaker conditions than those of Theorem 1.

Let us end this proof by explaining how the combination of relations (A.6) and (A.4), Propositions 2 and 3, as well as Corollary 1 imply that ? kv n p θX,τ X ´θX q d ÝÑ N p0, vq where v " θ 2 X σ 2 .

´When 0 ď τ X ă τ C ď 1, Proposition 2 states that ? kA 2,n converges to 0. Hence, the leading term in pA.6q is ?

kA 1,n which converges in distribution to N p0, 1q (see pA.4q), and we thus obtain the desired value of v " p 1 a q 2 θ 2 Z " θ 2 X .

´When 0 ă τ X " τ C ă 1, Proposition 2 states that ? kA 2,n d ÝÑ N `0, a 2´1{τ X p1 ´a1{τ X q ˘. Moreover ? kA 1,n converges in distribution to N p0, 1q. Since A 1,n and A 2,n are independent (under our independent censoring setting), we obtain as desired

v " θ 2 Z a 2 `θ2 X a 2 a 2´1{τ X p1 ´a1{τ X q " θ 2 X `θ2 X pa ´1{τ X ´1q " θ 2 X a ´1{τ X .
´In the other two cases, since v n Ñ 0, ? kv n A 1,n converges in probability to 0, and on the other hand Proposition 2 states that ? kv n A 2,n converges in distribution to N p0, Dq with a variance described above, and it is not difficult to check that p 1 a q 2 θ 2 X D equals to θ 2 X σ 2 as stated.

Appendix B. Proof of Theorem 2

The proof is very similar to the previous one. First, recall that τX " where

HH k,n D k,0 .
R 1,n " 1 k k ÿ j"1 ˜log ˜1 `∆ j,k Λk ¸´∆ j,k
Λk where the ∆j,k are defined in Lemma 3 and pk denotes the proportion of uncensored data in the tail. From now on we consider that τ X ‰ 0 (see Remark 8 below for the τ X " 0 case). Formulas pB.1q and pB.5q easily entail the following important intermediary relation :

τX ´τX d " LM n ´τX Λ ´1 k pk D k,0 `3 ÿ i"1 T T i,n ,
where

T T 1,n :" RR n, l D k,0 T T 2,n :" ´τX R 1,n D k,0
T T 3,n :" ´τX p Λ´1 k ´Λ´1 k qpD k,0 q ´1 pk . Moreover, we prove in Lemma 6 (stated in Appendix E.2) that, when τ X ą 0 and τ C ą 0 (the case τ X ą 0 and τ C " 0 is omitted for brevity),

Λ ´1 k pk " ˆaτ X τ Z ˙´1{τ X E ´τZ {τ X n´k pk `RR 2,n ,
the expression for the remainder term RR 2,n being detailed for each case in the statement of Lemma 6.

Consequently, defining T T 4,n :" ´τX RR2,n D k,0 , we obtain the following decomposition : when τ X ą 0 and

τ C ą 0 τX ´τX d " lσ 1,τ Z pE n´k q D k,0 ˜LA 1,n ´τX lµ 1,τ Z pE n´k q lσ 1,τ Z pE n´k q ˜ˆaτ X τ Z ˙´1{τ X E ´τZ {τ X n´k lµ 1,τ Z pE n´k q pk ´1 τ X ¸¸`4 ÿ i"1 T i,n .
But lµ 1,τ Z ptq " τ Z t ´1, so we define the following remainder term as

RR 3,n :" ˆaτ X τ Z ˙´1{τ X pk ˜pE n´k q ´τZ {τ X lµ 1,τ Z pE n´k q ´1 τ Z L 1´τ Z {τ X nk ¸.
Finally, using the additional fact that lµ1,τ Z pE n´k q lσ1,τ Z pE n´k q P ÝÑ 1, we can state the main relation of the proof of Theorem 2 :

τX ´τX d " lµ 1,τ Z pE n´k q D k,0 `LA 1,n ´a´1 A 2,n p1 `oP p1qq ˘`5 ÿ i"1 T T i,n , (B.6)
where LA 1,n is defined in (B.4), the second main term A 2,n is defined in section Appendix A and the last remainder term to be introduced is T T 5,n :" ´τX RR 3,n p1 `oP p1qq. The asymptotic normality of A 2,n is dealt with in Proposition 2. Concerning the remainder terms T T i,n , we prove the following proposition :

Proposition 4. Under the conditions of Theorem 1, for all 1 ď i ď 5, ? kv n T T i,n P ÝÑ 0, as n tends to infinity.

The proof of Proposition 4 is very similar to the proof of Proposition 3.

The following statement is a Corollary of Propositions 2 and 4, in the same way that Corollary 1 was deduced from Propositions 2 and 3.

Corollary 2. Under the conditions of Theorem 1, when τ X ‰ 0 we have

D k,0 lµ 1,τ Z pE n´k q P ÝÑ 1 τ X
and, when 0 " τ X ă τ C , we have as n Ñ 8 D k,0 lµ 1,0 pE n´k q " plog L nk qp1 `oP p1qq.

The proof of Theorem 2 can be concluded in the same way as was that of Theorem 1. Details are omitted. l Remark 8. In the case 0 " τ X ă τ C , we have D k,0 {lµ 1,0 pE n´k,n q P " log L nk , and thus the estimator τX d " lθ n,1 pE n´k q { D k,0 `T T 1,n is contiguous to lµ 1,0 pE n´k q { D k,0 `T T 1,n , which is itself equivalent in probability to 1{ log L nk . Thus only the consistency and rate of convergence of τX is obtained in this case.

Appendix C. Proof of Theorem 3

Recall that θX,τ X " H k,n {D k,τ X where

H k,n " 1 k k ÿ j"1 logpZ n´j`1,n q ´logpZ n´k,n q and D k,τ X " 1 k k ÿ j"1 K τX p ΛnX pZ n´j`1,n qq ´Kτ X p ΛnX pZ n´k,n qq. Moreover log ˜θ X,τ X θ X ¸" log ˜θ X,τ X θX,τ X ¸`log ˜θ X,τ X θ X ¸. (C.1)
Theorem 1 and the delta-method yields that the second term of the right-hand side in pC.1q satisfies

? kv n log ˜θ X,τ X θ X ¸d ÝÑ N `0, σ 2 ˘. (C.2)
Now let us treat the first term. Since D k,τ X " p Λk q τ X ´1 pk `R1,n (see Lemma 3) and, similarly, D k,τ X " p Λk q τX ´1 pk `R 1,n , where R1,n is obtained by replacing τ X by τX in the expression for R 1,n , we obtain log

˜θ X,τ X θX,τ X ¸" pτ X ´τX q logp Λk q ´log ˜1 `R1,n Λτ X ´1 k pk ¸`log ˜1 `R 1,n Λτ X ´1 k pk ¸.
Let us study separately the first two terms in the expression above (the third one being similar to the second one). The starting point is pτ X ´τX q logp Λk q " pτ X ´τX q logpΛ k q `pτ X ´τX q log ˜Λ k Λ k

¸.

Let us continue with the case τ X ‰ 0 and τ C ‰ 0 (the case 0 " τ C ă τ X being similar and the case 0 " τ X ă τ C being excluded, see Remark 9 below).

Since ? kv n pτ X ´τX q d ÝÑ N `0, σ 2 τ 2 X ˘(Theorem 2), and, according to Lemma 7, logpΛ k q " τ Z τ X plog L nk qp1 òP p1qq, we obtain that ?

kv n log L nk pτ X ´τX q logpΛ k q d ÝÑ N `0, σ 2 τ 2 Z ȃnd ? kvn log L nk pτ X ´τX q log ´Λ k Λ k ¯" o P p1q (because Λk Λ k " O P p1q). Now, log ´1 `R1,n p Λk q τ X ´1 pk ¯"
R1,n p Λk q τ X ´1 pk p1 `oP p1qq, and we prove in Proposition 3 that

? kv n R1,n p Λk q τ X ´1 pk " o P p1q. Hence ? kvn log L nk log ´1 `R1,n p Λk q τ X ´1 pk ¯" o P p1q. This ensures that ? kv n log L nk log ˜θ X,τ X θX,τ X ¸d ÝÑ N `0, σ 2 τ 2 Z ˘.
Finally, pC.1q and pC.2q yield

? kv n log L nk log ˜θ X,τ X θ X ¸d ÝÑ N `0, σ 2 τ 2 Z ˘.
This entails the announced asymptotic normality, via the delta-method. l

Remark 9. In the case τ X " 0, logpΛ k q " aplog L nk qp1 `oP p1qq, according to Lemma 7. Hence, τX logpΛ k q does not converge to 0, in this case. This is why τ X " 0 is excluded from the asymptotic result of θX,τ X .

Appendix D. Proof of Theorem 4

Recall that x pn " s F X pp n q " H X pexppK τ X p´log p n qqq and xpn " Z n´k,n exp ´θ X,τ X ´Kτ X p´logpp n qq ´Kτ X p Λk q ¯where H X pxq " x θ X lX pxq, and lX is slowly varying at infinity. Moreover, since Z n´k,n " s F X pexpp´Λ k qq, it is easy to prove that log ´xp n xp n ¯" θX,τ X tpK τX p´logpp n qq ´Kτ X pΛ k qq ´pK τ X p´logpp n qq ´Kτ X pΛ k qqu `pθ X,τ X ´θX qK τ X p´logpp n qq `θ X,τ X ´Kτ X pΛ k q ´Kτ X p Λk q pθ X,τ X ´θX qK τ X pΛ k q `log ´l X pexppKτ X pΛ k qqq lX pexppKτ X p´logppnqqq ":

Q 1 `Q2 `Q3 `Q4 `Q5 .
Let us treat separately these five terms, in the case τ X ‰ 0 and τ C ‰ 0, the case 0 " τ X ă τ C being similar. Note that Q 1 will turn out to be the main term. Recall that

L k :" # paτ X {τ Z q 1{τ X pL nk q τ Z {τ X if τ X ‰ 0 and τ C ‰ 0, paτ X q 1{τ X plog L nk q 1{τ X if τ X ‰ 0 and τ C " 0.
Consider the temporary notations σ n :" ´?kv n ¯´1 and w n :"

ż ´logppnq L k u τx´1 log u du.
By integration by parts, and under assumption p11q (which implies that L k " op´logpp n qq), we can prove that w n " 1 τ X logplogp1{p n qq p´logpp n qq τ X p1 `op1qq, (D.1) and similarly wn :"

ş ´logppnq L k u τx´1 log 2 u du " 1 τ X plogplogp1{p n qqq 2 p´logpp n qq τ X p1 `op1qq. • Let us prove that σ ´1 n w ´1 n Q 1 converges in distribution to N p0, θ 2 X τ 2 X σ 2 q, which (via (D.1)) will imply that ? kv n log logp1{p n qp´log p n q τ X Q 1 d ÝÑ N p0, θ 2 X σ 2 q. (D.2)
According to Theorem 2, τX " τ X `σn ξ n , where ξ n converges in distribution to N p0, τ 2 X σ 2 q. Hence,

Q 1 " θX,τ X ´ş´log pn Λ k u τ X `σnξn´1 du ´ş´log pn Λ k u τ X ´1du " θX,τ X ´ş´log pn L k u τ X ´1pu σnξn ´1qdu ´şΛ k L k u τ X ´1pu σnξn ´1qdu ¯.
Let us introduce φpxq " e x ´1 ´x. Consequently,

Q 1 " 4 ÿ i"1 Q piq 1 , where Q p1q 1 " θX,τ X ş ´log pn L k u τ X ´1φpσ n ξ n log uqdu Q p2q 1 " θX,τ X σ n ξ n ş ´log pn L k u τ X ´1 log u du Q p3q 1 " ´θ X,τ X ş Λ k L k u τ X ´1φpσ n ξ n log uqdu Q p4q 1 " ´θ X,τ X σ n ξ n ş Λ k L k u τ X ´1 log u du
Now, there exists η ą 0, such that x ă log η implies that |φpxq| ă pη{2qx 2 . As a consequence, since σ n log logp1{p n q Ñ 0 and σ n log L k Ñ 0 (according to p10q and p11q),

|Q p1q 1 | ă θX,τ X η 2 σ 2 n ξ 2 n ż ´log pn L k
u τ X ´1plog uq 2 du " η O P p1q σ 2 n wn .

Hence, via (10) and the previous approximations of w n and wn , Let us now consider Q p3q 1 . We proceed as for Q Since σ n log Λ k P ÝÑ 0 (this is an easy consequence of assumption p11q and Lemma 7), the right hand-side tends to 0, according to Lemma 8 and assumption p11q.

Concerning Q p4q 1 , Lemma 8 and assumption p11q entails that σ ´1 n w ´1 n Q p4q 1 tends to 0. This completes the proof of (D.2).

• Let us prove that σ ´1 n w ´1 n Q 2 " o P p1q : according to Theorem 3, Q 2 " σ n plog L nk qK τ X p´logpp n qqδ n , where δ n converges to a gaussian distribution. Hence,

σ ´1 n w ´1 n Q 2 " plog L nk q K τ X p´logpp n qq ş ´logppnq L k u τx´1 log u du δ n ,
and assumption p11q yields the result. Kτ X p´logppnqq tends to 0 under assumption p11q), we obtain that K θ X ρ X pλ n q tends to ´1{pθ X ρ X q. Moreover, ? kv n bX pexppK τ X pL k qqq tends to 0 under the appropriate assumption among H 2 , . . . , H 5 . Hence,

?

kv n Q p1q 5

K τ X p´logpp n qq " ? kv n bX pexppK τ X pL k qqq K ρ X pλ n q K τ X p´logpp n qq , tends to 0. Then,

σ ´1 n w ´1 n Q p1q 5 " ? kv n Q p1q 5 K τ X p´logpp n qq K τ X p´log p n q ş ´logppnq L k u τx´1 log u du ,
which tends to 0 thanks to pD.1q.

Similarly, we have Q p2q 5

" log ´l X pλnxnq lX pxnq " bX px n qK θ X ρ X pλ n qp1 `oP p1qq, where x n " exppK τ X pL k qq and

λ n " exppK τ X pΛ k qq exppK τ X pL k qq " exppτ ´1 X pΛ τ X k ´Lτ X k qq " exppcst.L τ Z ´α nk p1 `op1qqq,
where, according to Lemma 4, the constant above is negative and α " " τ Z when either τ X " τ C , or τ X ‰ τ C and r ď 0, τ Z p1 ´rq when τ X ‰ τ C and r Ps0, 1r.

In the case where α " τ Z , K θ X ρ X pλ n q converges to a constant. Hence we obtain, for the term Q Recall now that the function pp¨q is defined by ppxq " Ppδ " 1|Z " xq.

The following lemma provides useful developments of functions pp¨q and rp¨q rptq " p ˝HŹ pexppK τ Z p´log tqqq, which are crucial to derive the properties of the random proportion pk (and therefore the statements of Proposition 2). Its proof is essantially based on the fact that ppxq " s F C pxqf X pxq s F C pxqf X pxq `s F X pxqgpxq " ˆ1 `pK τC q 1 plog H C pxqq pK τX q 1 plog H X pxqq

H 1 C pxq{H C pxq H 1 X pxq{H X pxq

˙´1

(where f X and f C are the respective probability density functions of X and C), as well as on the results of Lemma 1.

Lemma 2. Let us define the constants

A X " θ X pτ ´1 X ´1qpτ ´1 X `log c X q , A C " θ C pτ ´1 C ´1qpτ ´1 C `log c C q and A " A C ´AX and B " θ X θ C ˆτX θ X ˙1´1{τ X ˆτC θ C ˙1{τ C ´1 .

Figure 2 :

 2 Figure 2: Simulation with X log-Weibull censored by C log-Normal, where τ X " 0.4 ă τ C " 0.5 in first line (figures (a)-(b), mild censoring p " 1), τ X " 0.5 " τ C in second line (figures (c)-(d), moderate censoring p Ps0, 1r), and τ X " 0.6 ą τ C " 0.5 in third line (figures (e)-(f), strong censoring p " 0). The graphs represent observed bias and MSE of estimators τX (blue) and θX,τ X (dashed red) in figures (a)-(c)-(e), and relative bias and MSE of estimators xpn (black) and xEF G pn (dashed green) in figures (b)-(d)-(f).

Figure 3 :

 3 Figure 3: Simulation with X log-Normal censored by C log-Weibull, where τ X " 0.5 ă τ C " 0.6 in first line (figures (a)-(b), mild censoring p " 1), τ X " 0.5 " τ C in second line (figures (c)-(d), moderate censoring p Ps0, 1r), and τ X " 0.5 ą τ C " 0.4 in third line (figures (e)-(f), strong censoring p " 0). The graphs represent observed bias and MSE of estimators τX (blue) and θX,τ X (dashed red) in figures (a)-(c)-(e), and relative bias and MSE of estimators xpn (black) and xEF G pn (dashed green) in figures (b)-(d)-(f).

Figure 4 :

 4 Figure 4: Simulation with X and C in the F model, where τ X " 0.4 ă τ C " 0.6 in first line (figures (a)-(b), mild censoring p " 1), τ X " 0.5 " τ C in second line (figures (c)-(d), moderate censoring p Ps0, 1r), and τ X " 0.6 ą τ C " 0.4 in third line (figures (e)-(f), strong censoring p " 0). The graphs represent observed bias and MSE of estimators τX (blue) and θX,τ X (dashed red) in figures (a)-(c)-(e), and relative bias and MSE of estimators xpn (black) and xEF G pn (dashed green) in figures (b)-(d)-(f).

Figure 5 :Figure 6 :

 56 Figure 5: Plot of the proportion of non-censuring pk as a function of kn.

P

  η O P p1q σ n wn {w n " η O P p1q σ n log logp1{p n q

  ´1 log u du ă θX,τ X η 2 σ n maxplog Λ k , log L k qξ 2 n ş Λ k L k u τ X ´1 log u du ş ´log pn L k u τ X ´1 log u du .

P

  ÝÑ 0, thanks to pD.1q.When τ X " τ C or r ď 0, both c and c are positive. When τ X ‰ τ C and r Ps0, 1r, both c and c are zero and the following relation holds for some ν ą 0, as x Ñ 8loglpexp xq x " ´ν.x r´1 p1 `op1qq ÝÑ 0 and log lpexp xq x " ´θX ν.x r´1 p1 `op1qq ÝÑ 0 (E.1) Remark 10. A consequence of this Lemma is that lX and lC also satisfy the SR2 condition with rate functions | bX | P RV θ X ρ X and | bC | P RV θ C ρ C respectively.

  Concerning the numerator, we have by Proposition 1 that Z i " H Ź pexppK τ Z pE i qqq, where E 1 , . . . , E n are standard exponential, and thus logpK τ Z pE n´i`1,n qq´logpK τ Z pE n´k,n qq and RR n, l :" 1 k `logp lpexppKτ Z pE n´k,n qqqq θ Z Kτ Z pE n´k,n qBy the Renyi representation, for some independent standard exponential random variables F 1 , . . . , F k we have logpK τ Z pF i `tqq ´logpK τ Z ptqq. Z px `tqq ´logpK τ Z ptqqq q e ´x dx

			LM n	d " lθ n,1 pE n´k q where lθ n,1 ptq :"	1 k	k ÿ j"1	(B.2)
	Introducing, for q P N ˚,
									ż 8
	lµ q,τ Z ptq :" Eplθ n,q ptqq " plogpK τ and lσ 2 0 1,τ Z ptq :" lµ 2,τ Z ptq ´lµ 2 1,τ Z ptq, we have
						lµ q,τ Z ptq "	#	pq!qτ q Z t ´q p1 `op1qq pq!qt ´q plogptqq ´q p1 `op1qq if τ Z " 0. if τ Z ‰ 0,	(B.3)
	We can then prove that (the proof is similar to that of Lemma 5 in Gardes et al. (2011))
			?	kLA 1,n	d ÝÑ N p0, 1q	where	LA 1,n :"	lθ n,1 pE n´k q ´lµ 1,τ Z pE n´k q lσ 1,τ Z pE n´k q	.	(B.4)
	Concerning now the denominator, we prove in Lemma 3 (stated in Appendix E.2) that
			D k,0 :"	1 k n	kn j"1 ÿ	logp ΛnX pZ n´j`1,n qq ´logp ΛnX pZ n´kn,n qq " Λ´1 k pk `R1,n ,	(B.5)
			HH k,n :"	1 k n	kn ÿ j"1	log logpZ n´j`1,n q ´log logpZ n´kn,n q " LM n `RR n, l	(B.1)
	where
	LM n :"	1 k	k ÿ j"1						k ÿ j"1	log	¨1 1 `logp lpexppKτ Z pEn´j`1,nqqqq θ Z Kτ Z pEn´j`1,nq

•

  In order to prove that σ ´1 n w ´1 n Q 3 " o P p1q, we obtain via a Taylor expansion thatˇˇ2where Tk is a value between Λ k and Λk . The fact that? k|Λ k ´Λ k | " O P p1q (see Lemma 7 in[START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF]) and assumption p11q yields the result.• Let us prove that σ ´1 n w ´1 n Q 4 " o P p1q : as above (see treatment of Q 2 ) Q 4 " σ n log L nk K τ X pΛ k qδ n ,where δ n converges to a gaussian distribution. Moreover K τ X pΛ k q d " aK τ Z pL nk qp1`o P p1qq (see Lemma 4 piq). Hence Assumption p11q yields the result.• Let us finally prove that σ ´1 n w ´1 n Q 5 " o P p1q : recall that Q 5 " log ´l X pexppKτ X pΛ k qqq lX pexppKτ X p´logppnqqq " log ´l X pexppKτ X pL k qqq lX pexppKτ X p´logppnqqq ¯`log ´l X pexppKτ X pΛ k qqq lX pexppKτ X pL k qqq bX px n qK θ X ρ X pλ n qp1 `oP p1qq,where| bX | P RV θ X ρ X , λ n " exppKτ X p´logppnqqq exppKτ X pL k qqand x n " exppK τ X pL k qq. Moreover, since λ n tends to `8, as n tends to infinity (becauseKτ X pL k q

	σ ´1 n w ´1 n Q 4	d " a	K τ Z pL nk q log L nk ş ´logppnq L k u τx´1 log u du	δ n p1 `oP p1qq.
					"
	Q p1q 5	`Qp2q 5 .
	Concerning Q p1q 5 , we know that lX satisfies the SR2 condition (see Remark 10). Hence
	´Qp1q 5	lX pxnq " log ´l X pλnxnq	"
	σ ´1 n w ´1			v n w n

n |Q 3 | " θX,τ X ? k|Λ k ´Λ k | ˇˇˇK 1 τX p Tk q

the original dataset contains 591 data lines, but 17 of them were not workable because of missing vital status or missing survival time.

Appendix

Let us first summarize the contents of the Appendix. It is composed of 5 main parts.

Part A is devoted to the proof of Theorem 1.

Part B is devoted to the proof of Theorem 2.

Part C is devoted to the proof of Theorem 3 Part D is devoted to the proof of Theorem 4.

Part E contains different technical aspects. In particular, the important Lemma 1 and Lemma 2.

The Supplementary Material file contains the proofs of all the Lemmas, and of Propositions 1, 2 and 3.

Recall that L nk is the notation for logpn{kq. Let us introduce the following notations : Λ k " Λ F pZ n´k,n q and Λk " ΛnX pZ n´k,n q.

Appendix A. Proof of Theorem 1

This section details how the asymptotic normality of θX,τ X stems from the combination of properties of the Hill estimator H k,n (relations (A.1), (A.2) and (A.4) below) and of the proportion pk of uncensored data in the tail (Proposition 2 stated next page), via the important decomposition (A.6). Some details are postponed to other sections, in particular the crucial technical Lemma 2 (stated in Appendix E.1) which states the second order properties of the function ppxq " Ppδ " 1|Z " xq. The behavior of the (numerous) remainder terms is detailed in Proposition 3 below.

First, recall that θX,τ X "

According to Proposition 1, we have Z i " H Ź pexppK τ Z pE i qqq, where E 1 , . . . , E n are n independent standard exponential random variables and (see relation ( 2)) H Ź pxq " x θ Z lpxq, l being RV 0 . Hence

where

By the Renyi representation, we have E n´j`1,n ´En´k d " F k´j`1,k , where F 1 , . . . , F k are k independent standard exponential random variables. As was done in [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF] (and borrowing their notations), we have

Introducing, for q P N ˚, (see Lemma 2 of [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF] µ q,τ Z ptq :" Epθ n,q ptqq " ż 8 0 pK τ Z px `tq ´Kτ Z ptqq q e ´x dx " pq!q t qpτ Z ´1q p1 `op1qq (as t Ñ `8) (A.3) and σ 2 1,τ Z ptq :" µ 2,τ Z ptq ´µ2 1,τ Z ptq, it is proved in Lemma 5 of [START_REF] Gardes | Weibull tail-distributions revisited: A new look at some tail estimators[END_REF]

Moreover, we prove in Lemma 3 (stated in Appendix E.2) via Taylor's formula that

where pk denotes the proportion of uncensored data among the k upper data values (see Lemma 3 for the definition of the remainder term R 1,n ). Formulas pA.1q and pA.5q thus easily entail the following important

In the case where α " τ Z p1 ´rq, we have K θ X ρ X pλ n q " Op1q exppcst.L rτ Z nk p1 `op1qqq, where here the constant is positive. Moreover, for some small δ ą 0, bX px n q " expppθ X ρ X `δqK τ X pL k qqop1q " exp ppθ X ρ X `δq.cst.L τ Z nk p1 `op1qqq op1q, where the constant above is positive. Consequently, ? kv n bX px n qK θ X ρ X pλ n q tends to 0 according to the appropriate assumption among H 2 , . . . , H 5 . To conclude, we proceed as in the previous case. l

Appendix E. Technical aspects

The Lemmas stated in this section are proved in the Supplementary Material document.

Appendix E.1. Details about second order conditions and censoring probabilities Recall that s F X pxq " expp´K τX plogpH X pxqqqq and s

where H Ź pxq " x θ Z lpxq and l is slowly varying. This implies that H X ˝HŹ pxq " x a lpxq, with l a slowly varying function and a " θ Z {θ X .

Lemma 1 stated below provides details about the second order properties of the functions H Ź and H X ˝HŹ (and therefore, on the behavior of the variables Z i and Λ X pZ i q). These properties not only depend on the position of the parameters τ X and τ C with respect to each other, but on their proximity through the parameter r defined by

(if either τ X " 0 or τ C " 0, indeed consider that r " ´8). Its proof can be found in the Supplementary Material document.

Lemma 1. Let conditions pA 1 q and pA 2 q hold.

piq For different slowly varying functions generically noted v, we have l X pxq " c X p1 ´xρ X vpxqq and l C pxq " c C p1 ´xρ C vpxqq lX pxq " c ´θX X p1 ´xθ X ρ X vpxqq and lC pxq "

piiq The slowly varying functions l and l associated to H Ź and H X ˝HŹ satisfy a second order condition SR2 : as t Ñ `8,

in the other cases , and | b| P RV ρ and |b| P RV ρ . When ρ " 0, both bptq and bptq are (as t Ñ `8) of the order Opplog tq r´1 q when r ‰ 0, and of the order Opplog tq ´2q when r " 0. piiiq The slowly varying function l Z associated to H Z satisfies

where in particular c Z " c X if τ X ă τ C and r ă 0, and c Z " c C if τ C ă τ X and r ă 0. Moreover we have (with the convention p`8q ´θ " 0 when θ ą 0) lptq Ñ c :" c ´θZ Z and lptq Ñ c :" c X c1{θ X , as t Ñ `8.

Let assumptions pA 1 q and pA 2 q hold (the asymptotics below are x Ñ `8 and t Ó 0).

piq We have

and, more precisely,

piiq When τ Z ą 0 and τ X ‰ τ C , as t Ó 0 we have rptq ´p " Dpθ Z {τ Z q r´1 p´log tq ´τZ p1´rq ´1

`O ´p´log tq ´τZ mint1,1´ru ¯¯, in particular, when

When τ Z ą 0 and τ X " τ C , we have

When τ Z " 0, if τ `" maxpτ X , τ C q we have rptq ´p " cstp´log tq ´1plog logp1{tqq

with the constant being equal to τ

Technical Lemmas

The proofs of the following Lemmas can be found in the Supplementary Material document.

Lemma 3. The denominator of the estimator θX,τ X satisfies the relation

with, for each j " 1, . . . , k, ∆j,k :" ΛnX pZ n´j`1,n q ´Λ nX pZ n´k,n q and the random variable T j,k lies between 0 and ∆j,k Λk .

For the following lemma, recall that pE i q denote the i.i.d. standard exponential variable pE i q satisfying Z i " H Ź pexppK τ Z pE i qq, and that lp¨q denotes the slowly varying function which properties are described in Lemma 1 and which is such that H X ˝HŹ pxq " x a lpxq. Note that in part piiq of this lemma, the results also hold when one replaces E n´k,n by L nk , or replaces Z n´k,n and E n´k,n by Z n´j`1,n and E n´j`1,n (this will occasionally prove useful).

Lemma 4. piq For every i " 1, . . . , n, and whether τ Z ą 0 or is equal to 0, we have

piiq When τ Z ą 0, we have

for some constant β and exponent α " " τ Z when either τ X " τ C , or τ X ‰ τ C and r ď 0, τ Z p1 ´rq when τ X ‰ τ C and r Ps0, 1r.

When 0 " τ X ă τ C , we have Λ X pZ n´k,n q " E n´k,n lpE n´k,n q " E n´k,n p1 `oP p1qq. When 0 " τ C ă τ X , we have Λ X pZ n´k,n q " paτ X q 1{τ X plog E n´k,n q 1{τ X `1 `βplog E n´k,n q ´1p1 `oP p1qq ˘.

Note that the constant β is negative in the case τ X ‰ τ C and r Ps0, 1r.

Lemma 5. Let E 1 , . . . , E n be i.i.d. standard exponential random variables.

Lemma 6. Let E 1 , . . . , E n be i.i.d. standard exponential random variables.

n´k,n pk ˆp1 ´E´τ Z n´k,n q ´1 τ X ´1 `1`τ X log lpexppKτ Z pE n´k,n qqq aτ X Kτ Z pE n´k,n q ¯´1 τ X ´1˙, if 0 ă τ X ă 1 and τ C ‰ 0 pk E n´k,n ´1 lpE n´k,n q ´1¯, if 0 " τ X ă τ C ă 1 paτ X q ´1 τ X plogpE n´k,n qq ´1 τ X pk ˆ´1 `1`τ X log lpE n´k,n q aτ X logpE n´k,n q

Lemma 7. Under the assumptions of Theorem 1, we have, as n Ñ 8, if τ X ‰ 0 and τ C ‰ 0, logpΛ k q " τ Z τ X log L nk p1 `oP p1qq if τ X " 0, logpΛ k q " a log L nk p1 `oP p1qq if τ X ‰ 0, and τ C " 0 logpΛ k q " 1 τ X log log L nk p1 `oP p1qq

Lemma 8. Under the assumptions of Theorem 4, we have, as n tends to infinity,

O P plog L nk q if τ X ‰ 0, τ C ‰ 0 and pτ X " τ C or r ď 0q, O P pL rτ Z nk log L nk q if τ X ‰ 0, τ C ‰ 0, τ X ‰ τ C and r Ps0, 1r, O P plog log L nk q if τ X ‰ 0 and τ C " 0, o P plog L nk q if τ X " 0.


Marcin Trzmielewski

Representation of Students' Mental Health in Information Resources and in Medical Knowledge Organization Systems

Introduction

Mental health issues, such as depression, anxiety, eating disorders, suicidal ideation, and selfinjury, are commonly experienced by university students [START_REF] Zivin | Persistence of mental health problems and needs in a college student population[END_REF]. These disorders are a real burden for this specific population, and they may impair their academic productivity, social life and even hamper their future career [START_REF] Davies | Computer-Delivered and Web-Based Interventions to Improve Depression, Anxiety, and Psychological Well-Being of University Students: A Systematic Review and Meta-Analysis[END_REF]. As mental health is an important and complex issue for public health, according to the WHO's Mental Health Action Plan 2013-2020, a multidisciplinary approach is needed (World Health Organization 2013). The present paper was a part of a study in Information and Communication Sciences (in French: Sciences de l'Information et de la Communication), which is a multidisciplinary academic discipline aiming to study information and communication-related phenomena with an interdisciplinary approach [START_REF] Couzinet | Les connaissances au regard des sciences de l'information et de la communication : sens et sujets dans l'inter-discipline[END_REF]. We mainly exploited epistemology and methods from Library and Information Science and from Knowledge Organization field of research and practice [START_REF] Hjørland | Knowledge organization[END_REF]. Our aim was to enrich reflection on students' mental health by highlighting that documentary mediation processes and tools, and, more precisely, indexing and indexing languages, may be an interest source of information about representation of this issue in scientific and in general-public information.

Medical and health information

Information resources 1 provided by documentation services (e.g. catalogues, bibliographic databases, institutional websites) and social media (e.g. forums, social bookmarking and cataloging websites) seem to have an important role in transmitting information about students' mental health both among specialists and wide public. Our contents assessment on the PubMed 2 and on the National Library of Medicine blog 3 , online services delivered by the National Library of Medicine (NLM), showed that this medical library mostly harbors specialized, validated, and high-quality information on this topic. Moreover, as [START_REF] Walton | Supporting Student Wellbeing in the University Library: A Core Service or a Distraction?[END_REF] highlights, academic libraries have an important role in supporting student's wellbeing by providing physical space, digital support, and collection provision. Concerning social media, the study of [START_REF] Kvasny | African American youth tumbling toward mental health support-seeking positive academic outcomes[END_REF] shows, by using the example of Tumblr®, that microblogging and social networking websites, may provide a powerful mental health support to college students. Furthermore, posts on Twitter® regrouped by hashtag "#StudentMentalHealth" 4 highlight that social media may be spots of exchange of public information on this topic. Documentation and social media services reflect the current coexistence of two kind of information in healthcare domain: medical information and health information [START_REF] Romeyer | TIC et santé : entre information médicale et information de santé[END_REF]. We consider medical information as a type of specialized content focusing on healthrelated phenomena, produced by specialists for specialists, and exchanged for example through scientific articles, books, electronic health records and conversations between health professionals. On the contrary, health information is a kind of information, which is produced by several actors, such as health insurances, pharmaceutical industry, health consumers, intended for the wide public [START_REF] Romeyer | TIC et santé : entre information médicale et information de santé[END_REF][START_REF] Romeyer | Les mutations de l'information publique : le cas de la santé[END_REF]. As [START_REF] Paganelli | S'informer via des médias sociaux de santé. Quelle place pour les experts ?[END_REF] highlight, these two types of information tend to complement themselves. Indeed, as studies on Information Seeking in Health show, whether they are conducted in (library and) information science or in medicine, health professionals willingly share contents with their colleagues and health consumers on microblogging and social networking services, health fora, and blogs [START_REF] Daudigeos | Blogs tenus par des médecins généralistes de la communauté médicale française en 2017 : Quels types de publications ? : Quelles compétences et situations cliniques de médecine générale mises en jeu ?[END_REF][START_REF] Bouarfa | #Doctoctoc / #Docstoctoc : Twitter comme outil d'entraide entre Médecins[END_REF][START_REF] Dor | Les médecins généralistes présents sur les réseaux sociaux : qui sont-ils et que recherchent-ils ? : enquête qualitative à partir d'entretiens semi-dirigés[END_REF]Gonzalez-Pacanowski and Medina-Auguerrebere 2011;[START_REF] Paganelli | S'informer via des médias sociaux de santé. Quelle place pour les experts ?[END_REF]. At the same time, health consumers are interested in specialized information as well: for example, some parents go on bibliographic databases or official websites of public health authorities to inform themselves about vaccination in children [START_REF] Vivion | Web 2.0, parentalité et habitus informationnel[END_REF].

Organization of medical and health knowledge

To mediate and enhance the visibility of medical and health information, both humans (information and documentation professionals or users) and computer programs are used to organize knowledge [START_REF] Trzmielewski | Organisation des connaissances médicales[END_REF], i.e. describe, represent, fil and organize documents and document representations, as well as subjects and concepts [START_REF] Hjørland | Knowledge organization[END_REF]. Medical Knowledge Organization Systems (MKOSs), such as taxonomies, folksonomies, classifications, nomenclatures, thesauri, ontologies, and consumer health vocabularies are structured records made up of concepts and relationships between these concepts, which represent knowledge relating to different domains of medicine and related disciplines [START_REF] Trzmielewski | Organisation des connaissances médicales[END_REF]. MKOSs are used to organize medical and health information and to enable access to documents in physical and digital libraries, in healthcare organizations and on the Web. They often aim to represent the content of documents by expressing their subject (a phenomenon or a perspective) through concepts, represented by terms, which are originated from common-usage words. In addition, with some MKOSs, it is possible to describe documents, collections, and people-related information, according to their information needs and cognition [START_REF] Trzmielewski | Organisation des connaissances médicales[END_REF]. As examples of such systems we may recall the National Library of Medicine Classification, which is extensively used in health libraries to classify books on the shelves [START_REF] Womack | Conformity for Conformity's Sake? The Choice of a Classification System and a Subject Heading System in Academic Health Sciences Libraries[END_REF]; the Medical Subject Headings (MeSH), mobilized often to index contents on library online catalogs and bibliographic databases [START_REF] Boehr | Preparing for the Future: National Library of Medicine's® Project to Add MeSH® RDF URIs to its Bibliographic and Authority Records[END_REF]; the OntoNeo ontology, developed to assist neurosurgeons in retrieving and accessing relevant patient records [START_REF] Raghavan | NeurOn. Modeling ontology for neurosurgery[END_REF]; and folksonomies, which are tags, keywords and hashtags created by social media users through the social tagging process, which may be explained as collaborative indexing of contents [START_REF] Ådland | Tags on healthcare information websites[END_REF]. As all knowledge organization systems (KOSs), we may consider MKOSs as vehicles or systems of communication between authors of documents, professionals of information and documentation, and users, enabling documentary mediation between these actors [START_REF] Couzinet | Les connaissances au regard des sciences de l'information et de la communication : sens et sujets dans l'inter-discipline[END_REF][START_REF] Lazzari | Telematica e basi di dati nei servizi bibliotecari[END_REF][START_REF] Régimbeau | Médiation[END_REF].

Apart from document and information retrieval, MKOSs "can be used to gauge societal thinking" (Huber and Gillaspy 2000, 222), and to increase awareness on the sensitive nature of health in the population, through the mediation of scientific, social, political, and cultural phenomena [START_REF] Trzmielewski | Organisation des connaissances médicales[END_REF]. This point of view inspired for example [START_REF] Huber | An Examination of the Discourse of Homosexuality As Reflected in Medical Vocabularies, Classificatory Structures, and Information Resources[END_REF] to study the discourse of homosexuality transmitted by MKOSs. In the present work, we decided to examine and describe the representation of students' mental health, as reflected in MeSH terms created and used by NLM professionals to index documents on PubMed, and as evidenced by tags created by users on LibraryThing to index books cited on this web application. PubMed 5 is an interface of search in MEDLINE, which is a bibliographical database used to index and abstract scientific books and articles in biomedical domains. LibraryThing 6 is a social cataloging web 2.0 application for storing and sharing book catalogs and various types of book metadata. By taking into consideration paratextual contents communicated through PubMed and LibraryThing, we expect to discover respectively scientific and wide-public discourse on the subject of student' mental health and to highlight goals and values related to this phenomenon associated to the common representation of this topic.

Methodology

Firstly, we searched for scientific articles published on students' mental disorders, abstracted on PubMed, and for books on these themes catalogued on LibraryThing. On PubMed, we used the query: "student mental [Title]" and we filtered publications by "Journal articles". On LibraryThing, we consequently queried the browser with the keywords "student mental health" and "mental health students", and we filtered works by "Books".

Then, we qualitatively gathered information on the authors of the analyzed works (discipline of specialization and affiliation).

Finally, we classified indexing terms on PubMed and on LibraryThing into emerging facets. Facets may have syntactic and semantic function [START_REF] Hudon | Facet[END_REF], but in the present work they are only considered as "semantic containers" regrouping similar terms into main classes. Within the facets, we developed thematic subclasses, and connected them to the content of articles, if the meaning of terms were not entirely explicit. We compared and merged terms from the two sources and we added terms from bibliographic analysis of works to obtain a common representation. Obtained representation was presented in the form of taxonomy sample.

Results

Articles and books cited on PubMed and on LibraryThing

The query used to search for documents on PubMed provided 93 articles published between 1957 and 2020. On LibraryThing, we obtained 86 books, but results were not accurate, and consequently needed to manually select them qualitatively. After the selection, we included 50 books in our analysis, and we collected 53 unique tags created by users to index them. We did not establish temporary limits to collect terms on LibraryThing, considering that the application was launched in 2005, which conducted us to obtain recent data only (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020). After that, we collected 212 unique MeSH terms from the last 15 years (2005-2020), used for indexing 59 articles, to obtain a comparative sample.

Results of our request on PubMed highlight that, while, during several decades, students' mental health topic was often neglected, the number of publications has considerably raised over the last 15 years (2006-2020) (Figure 1). 1960s -7; 1970s -15; 1980s -6; 1990s -3; 2000s -6; 2010s -48; 2020s -7. The articles on PubMed selected for our semantic analysis of indexing terms (59) are written by consultants and researchers in medicine (psychiatry, neurology, addictology, nursing, public health, pediatrics) and social sciences (psychology, anthropology, education, economics, social work). Publications are first of all issued from journals specialized in different disciplines, such as: psychiatry (e.g. Australasian Psychiatry, Early Intervention in Psychiatry, Frontiers Psychiatry), psychology (Psychological Assessment, Psychological Reports, Psychological Services), medical education (BMC Medical Education, Medical Education, American Journal of Pharmaceutical Education), nursing (Journal of Advanced Nursing, Nurse Education Today, Journal of School Nursing) and public health (Journal of American College Health). We may also find articles published in cross-disciplinary journals dedicated to specific phenomena: mental health in general (Journal of Mental Health, International Journal of Mental Health Promotion), mental health of specific populations (Journal of American College Health), depression and anxiety (Depression and Anxiety), cognitive behavior (Cognitive Behaviour Therapy), suicide (Crisis), addiction (Addictive Behaviors). Finally, some articles appear in general medical journals (British Medical Journal, RAND Health Quarterly, Academic Medicine).

Books cited on LibraryThing were written by specialists in medicine (child and adolescent practitioners, researchers or teachers in psychiatry and in mental health nursing) and social sciences (counseling psychologists, researchers or academic teachers in counseling, educational psychology, and social work). In these works, the subject of students' mental health and wellness is often treated and studied by disciplines such as education (teaching), counseling psychology, social work, and mental health nursing. Most books are textbooks and guides including fundamentals and case-based intervention strategies for teachers, parents, students, social workers, and mental health professionals (mental health nurses, psychiatrists, school psychologists and counselors) working with college and university students. This panorama of different disciplines engaged in students' mental health, noticeable on PubMed and on LibraryThing, highlights that this is clearly a multidisciplinary domain and subject, which combines medicine and social science. Journals dedicated to specific phenomena (e.g. suicide, addiction) instead of disciplines also witness that, as a subject, students' mental health has an interdisciplinary nature as well. The information mediated by works cited on these devices is specialized: purely scientific on PubMed and somehow more popularized on LibraryThing (e.g. several books are also dedicated to parents; guides are more accessible in terms of simplicity of writing style than scientific articles).

Indexing on PubMed and on LibraryThing

To accurately index subjects discussed in documents cited on PubMed, indexing professionals from the NLM are assisted by a set of methods and rules and by NLM Medical Text Indexeran indexing tool based on Natural Language Processing technology (National Library of Medicine 2018; 2019; 2020). Accounts on LibraryThing are anonymous and that is why it is difficult to know who is behind them. Its community members refer to themselves as "thingamabrarians", being "a community brought together by the love of books" (The LibraryThing Blog 2013). [START_REF] Rolla | User Tags versus Subject Headings: Can User-Supplied Data Improve Subject Access to Library Collections?[END_REF] clearly describes how the application works: "LibraryThing contains a comprehensive list of books, but, like other Web 2.0 sites, it allows users to interact with the content and supply their own data. Users of LibraryThing can create their own virtual libraries, rate books, and interact with other readers on the site. Also […] users can supply their own tags for their books". Data from this application are interesting because as Hjørland highlights, folksonomies assigned by users of social media (such as LibraryThing) present peoples' beliefs about the world and witness about "un unfiltered representation of their living culture" (Hjørland 2018, 619).

Terms on PubMed are homogenous because they come from MeSH (controlled vocabulary) used to index contents on MEDLINE. They often start with a capital letter (e.g. "Education, Nursing"), while qualifiers names7 start with lowercase (Figure 2). Terms indicate concepts that may be stratified into two of eight Gnoli's dimensions of knowledge organization [START_REF] Gnoli | Introduction to knowledge organization[END_REF]: phenomena (e.g. "Stress Disorders", "Universities", "Smartphone"), and perspectives ("Education, Nursing", "Follow-Up Studies", "epidemiology"). These observations led us to notice that the terminology mediated by MeSH terms is entirely specialized and the manner of subject indexing is professional. Tags assigned by users of LibraryThing are mostly heterogenous (non-controlled vocabulary): terms start sometimes by capital letters, sometimes by lowercases without any preconceived typographical rule (Figure 3). Concepts recalled by tags are phenomena ("suicide", "youth", "psychoactives"), disciplines ("nursing education", "psychology", "Mental health nursing"), documents ("e-bk, "textbook", "non-fiction") and collections ("Shelf 6", "FHCS library", "May"). The spectrum of knowledge organization dimensions is broader than on PubMed, which expresses uncontrolled and individualistic nature of tagging. The terminology transmitted by tags is generalist (simplest than on PubMed) and the manner of indexing is nonprofessional and non-coordinated.

Representation of students' mental health in indexing terms

Semantic analysis of MeSH terms and tags used to index works on PubMed and on LibraryThing shows that students' mental health is an interdisciplinary subject studied by disciplines associated to medical and social sciences. Research methods, applied to study this phenomenon, are heterogeneous and include qualitative and quantitative methods, crosssectional, multicenter, multivariate, self-reporting, and follow-up studies.

Students are stratified in four distinct age groups (childhood, adolescence, young adult age and adult age), highlighting the transition through different life periods. On PubMed, the human genre is always specified, followed by male and female ones, without any quantitative prevalence between two sexes. Transgenders and non-heterosexual subjects are not represented at all. A special focus is attached on students in college and in medical schools, because of a high exposure of this group to mental health issues. Key actors involved in student's mental health issues are health professionals (especially school nurses, and psychiatric and mental health nurses), parents, school staff, and social workers.

Moreover, indexing terms and tags highlight that recurrent student mental health disorders are depression, schizophrenia, substance-related disorders, addiction, ADHD, learning disabilities, psychological trauma, and burnout, while main behaviors include suicide, anxiety, stress, suicidal ideation, substance abuse and emotional problems. Student mental health is associated with social contexts experienced during college and university, such as hostility, discrimination, stigma, stereotyping, violence, permissiveness, bullying, alcohol drinking and sexual activity.

Treatment and initiatives applied to manage these issues are mindfulness, self-help groups, selfcare, cooperative behavior, counseling, psychotherapy, education, anti-drug propaganda, prevention and control. Digital tools (Internet, mobile applications, social media) may be sources of cyber aggression but may also be considered as help-seeking devices (telemedicine).

Types of places related to the student mental disorders are university, faculty, medical school, college, and workplace where students accomplish their internships. Anglophone territories, such as the USA, the UK, and Canada, are mostly represented on PubMed, while other countries are neglected.

Indexing terms on LibraryThing often refer to Mental Health Awareness Month, which shows that this initiative is known for a wider public.

A proposed organization of students' mental health-related knowledge

Gathered terms allowed us to propose a taxonomy sample i.e. a list of organized concepts by hierarchical relationships based on semantic similarity (Gnoli 2020, 77-78). Concepts were classified according to a top-down process of logical division, which split broader concepts into more specific ones (Gnoli 2020, 39). 108 collected terms have been finally chosen and uniformized (terms start with capital letters followed by lowercases) and alphabetically organized within facets (main classes). Proposed taxonomy was validated by a pediatrician (working in a paediatric emergency department), who is used to deal with mental health problems in children, adolescents, and young adults.

Discussion and conclusion

The analysis of information resources and indexing terms highlights that students' mental health is a multidisciplinary domain and subject, dominated by disciplines associated to medicine and social sciences. Students' mental health is investigated through an interdisciplinary approach, which invites to study this object beyond disciplinary boundaries. MeSH indexing terms used on PubMed represent specialistic terminology, are professionally used, and mediate scientific information, while tags on LibraryThinhg are made of nonspecialistic terminology, are non-professionally assigned, and transmit mostly popularizing specialized contents.

The proposed taxonomy sample reveals the complexity of real-life situations and experiences to which students are confronted in their academic stage of life. Special attention is given to the nursing domain and professionals, revealing the importance attributed by specialized discourse to primary care actors and interventions in managing student mental health disorders.

Our taxonomy may also serve as a base to design or update MKOSs to organize and mediate knowledge on students' mental health.

This study has two main limitations. We analyzed two information sources only, that communicate specialized contents. This led us to neglect lay-public genres of discourse and to considerably narrow our field of investigation. Furthermore, we gathered a small sample of tags on LibraryThing, which did not allow us to compare them deeply with MeSH terms from PubMed. Further investigations are therefore needed to obtain a richer and broader representation. Other types of social media, such as Facebook®, Twitter®, or Pinterest®, may probably be a richer source of data to better examine this phenomenon.

Finally, this work shows the equal engagement of information and documentation professionals, and users of social media in the mediation of knowledge dedicated to student's mental health. The significant rise in number of scientific publications over the last 15 years and the interest of wider public on this topic highlight that the awareness on the sensitive nature of students' mental health is increasing and expanding in our society.

Figure 1 -

 1 Figure 1 -Yearly visualization of works found on PubMed. In grey, we can see all available publications (1957-2020) and, in turquoise, our included sample (2005-2020). The number of publications by decade is: 1950s -1; 1960s -7; 1970s -15; 1980s -6; 1990s -3; 2000s -6; 2010s -48; 2020s -7.

Figure 2 -

 2 Figure 2 -An example of subject indexing on PubMed.

Figure 3 -

 3 Figure 3 -An example of tagging on LibraryThing.

PubMed is available at: https://pubmed.ncbi.nlm.nih.gov/.

LibraryThing is available at: https://www.librarything.com/home#.

On PubMed, qualifiers are used with descriptors and afford a means of grouping together those documents concerned with an aspect of a subject (a discipline, a scientific method, etc.) (National Library of Medicine 2015).

Acknowledgment

I thank Dr Davide Paolo Caimmi (Centre Hospitalier Universitaire de Montpellier) for the final revision of the paper and for its work on validating the proposed taxonomy. I also thank Claudio Gnoli (Università di Pavia) for methodological advices. Finally, I thank Céline Paganelli (Université Paul Valéry, Montpellier 3) for her encouragement and full support for my participation to this colloquium.

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