Terje N Andersen

Atle Briskemyr

Frode Telstø

Johnny Bjørnsen

Thomas E Bonnerud

Bjørnar Hernes

Øystein Moldsvor

A 97mW 110MS/s 12b Pipeline ADC Implemented in 0.18µm Digital CMOS

A 12 bit Pipeline ADC fabricated in a 0.18 m pure digital CMOS technology is presented. Its nominal conversion rate is 110MS/s and the nominal supply voltage is 1.8V. The effective number of bits is 10.4 when a 10MHz input signal with 2V P-P signal swing is applied. The occupied silicon area is 0.86mm 2 and the power consumption equals 97mW. A switched capacitor bias current circuit scale the bias current automatically with the conversion rate, which gives scaleable power consumption and full performance of the ADC from 20 to 140MS/s. 1.

Introduction

CMOS technologies moves steadily towards finer geometries, which provide higher digital capacity, lower dynamic power consumption and smaller area. This results in integration of whole systems, or large parts of systems, on the same chip (System on Chip, SoC). For such systems the key parameters for the building blocks are often low power dissipation and small silicon area. Analog to Digital Converters (ADC) are important building blocks in many of the SoC applications. Therefore, ADCs provided as Intellectual Property (IP) blocks goes into a wide range of applications, spanning from imaging to ultrasound and communication systems. This means that the ADC has to occupy small area and have low power dissipation. Further, it has to be designed in the same technology that the rest of the system. This means that the power supply voltage is low and subsequently the maximum signal swing is lowered. Thus, it is necessary to keep the noise and distortion of the ADC itself low. Additionally, to minimize the cost of the total SoC system, the IP block should be implemented in a pure digital process, avoiding use of costly additional analog process options.

This paper addresses these challenges by presenting a 12 bit pipeline ADC which utilizes a 1.8V supply voltage and is fabricated in a 0.18 m pure digital CMOS technology. In section 2 the used pipeline architecture is presented and section 3 explains the design of some key building blocks of the ADC. Section 4 presents the major measurement results, while section 5 concludes the paper.

Pipeline Architecture

The pipeline chain is shown in Fig. 1. The chain consists of 10 succeeding 1.5bit stages and a 2bit flash at the end. The input signal is applied directly to the 1 st stage, which also performs sample-and-hold. Due to the X2-gain in each stage, the 1 st stage has the highest specifications with respect to noise, incomplete settling and distortion. Since the input switches do not utilize bootstrapping, careful design of these is necessary to obtain low distortion. Further, large sampling capacitors (C H ) and high bias currents are necessary to keep the noise low and the incomplete settling within the specifications. Due to relaxed requirements of the stages along the pipeline chain these stages are scaled down regards to C H and bias currents [1]- [START_REF] Cho | A 10 b, 20 Msample/s, 35 mW Pipeline A/D Converter[END_REF], the 2 nd stage with a factor 2/3 and the rest of the stages with 1/3. This result in lower area and lower power consumption with only small degradation in converter performance. The digital output of each stage is passed to a digital circuit, which perform delay and error correction before the digital value appears at the output D OUT . The error correction utilizes the half bit of redundancy in each pipeline stage and corrects for errors in the Analog to Digital Sub-Converter (ADSC) contained in the stage. The operation of ADSC is further explained in section 3 together with the rest of the stage architecture.

The reference voltages, Common-Mode (CM) voltages and the bias currents are supplied to the pipeline chain from on-chip circuitry. The reference voltages are derived from the band-gap voltage and are decoupled by off-chip capacitors. The bias current generator is a Switched-Capacitor (SC) circuit which automatically scale the bias currents to the stages with the conversion rate and the absolute value of an on-chip capacitor. The SC bias current generator is also explained in the next section.

Design of Key Building Blocks

The construction of the pipeline stages is equal for all stages and is shown in Fig. 2

[1]. V INP i -V INN
i is the input signal for i=1 and the residue from the preceding stage for i>1. The operation of the stage can be divided in two phases. In the tracking phase 1 ,

V INP i -V INN
i is applied over the sampling capacitor C H , which is the parallel connection of the parasitic metal capacitors C 1 and C 2 . When 1B goes down, the sampling switch S 1B opens and a sample of

V INP i -V INN i is stored at C 1 and C 2 . At the same time V INP i -V INN
i is also sampled by the ADSC. Further, 1 goes down and 2 goes high and the switches S 1 are opened and S 2 are closed. The opamp, a two-stage Miller opamp with a differential-pair output stage [START_REF] Kelly | A 3V 340mW 14b 75MSPS CMOS ADC with 85dB SFDR at Nyquist[END_REF], is now connected in closed loop and the stage is in its amplification phase. ADSC resolves the input sample and pass its digital value to the Decoder and Switching Block (DSB). Dependent on the digital value, DSB connects the reference voltages (V REFP and V REFN ) or the CM-voltage to the top of C 1 and the output V OUTP i -V OUTN i settles towards the residue. The residue is sampled by the following stage at the end of the amplification phase. At this time instance, the digital output D OUT i is valid and clocked in to the digital circuit.

Because of the low supply voltage it was necessary to pay the switches special attention. The transmission gates S 1 and S 2 experience large voltage swings. This results in large transistors to keep the onresistance of the transmission gate low enough to fulfil the settling requirements. It is especially the PMOS transistor that becomes large due to its lower mobility compared to the NMOS. Therefore, both S 1 and S 2 use bulk switching of the PMOS transistor [START_REF] Hernes | A 1.2V 220MS/s 10b Pipeline ADC Implemented in 0.13µm Digital CMOS[END_REF]. This means that when the switch is on, the bulk (N-well) of the PMOS transistor is switched to source. This results in lower threshold voltage and subsequently lower onresistance of the transistor. When the switch is off, the bulk is connected to the positive supply voltage, increasing the off-resistance due to increased threshold voltage. The sampling switch S 1B is connected to V CM and does not experience large voltage swings. Thus, S 1B consist of NMOS transistors only, providing low onresistance and small parasitic capacitances at the inputs of the opamp.

To obtain low power consumption and low silicon area several actions were taken. First, scaling of the pipeline stages reduce both area and power consumption. Further, in pipeline converters it is common to use non-overlap clocking to ensure that the switches S 2 is not closed before S 1 is opened. In this design the non-overlap clocking is removed and the sequential operation of the switches is ensured by generating these clocks locally in each stage, as Fig. 3 shows. Removing the non-overlap means that the stage has longer time to settle and the gain-bandwidth of the opamp can be lowered, which further results in lower power consumption.

In modern CMOS technologies the spread in the absolute value of capacitors is large. Instead of large fixed bias currents in the opamp that can handle the largest possible capacitive load, the bias currents in this design are made dependent on the absolute value of the capacitances. Fig. 3 shows the SC bias current generator. The OTA, which is connected in unity gain, ensures that the voltage on its output node BIAS is approximately equal to V BIAS . V BIAS is taken from the band-gap voltage circuit and is near independent of variations in process parameters, temperature and supply voltage. The current through the OTA output transistor M0 is set up by V BIAS and the load seen from the node BIAS to ground. This load is the equivalent resistance seen into the SC-circuit. The SC-circuit is clocked by the system clock (CLK) at the conversion rate frequency f CR . Further, the current through M0 is mirrored to I BIAS 1 to I BIAS 10 , which are applied to stage 1 to 10, respectively. The value of these currents are given by (1) (next page), which shows that the bias current depends on the absolute value of the capacitor C B . Thus, by this technique the nominal bias currents in the opamp can be set lower than in a conventional design and the power consumption is Further, [START_REF] Hernes | A 1.2V 220MS/s 10b Pipeline ADC Implemented in 0.13µm Digital CMOS[END_REF] shows that the bias currents to the pipeline stages are also linearly dependent on f CR . This means that the bias currents, and subsequently the power consumption of the ADC, are automatically scaled with the applied clock frequency. This is an important feature in an ADC made as an IP-block due to the wide range of possible applications that this block should fit in to. The effect of the power scaling versus conversion rate is shown in the next section.

1 2 1B 1 2 i INN V i INP V REFP V REFN V 1 S 1 S 1 S 1 S 2 S 2 S i OUTP V i OUTN V 1B S 1B CM V 1 C 1 C 2 C 2 C i OUT D
i BIAS B CR BIAS I C f V (1)

Measurement Results

Fig. 4 shows the power dissipation of the ADC (exluding output drivers) versus conversion rate. As predicted by (1) the bias currents, and subsequently the power dissipation, is linearly scaled versus conversion rate. The plot shows a power dissipation of 97mW at 110MS/s and 110mW at 130MS/s.

The dynamic measurements were done by using RF-sources for the input signal and the clocking of the ADC. Both where filtered using high order passive bandpass filters around the applied frequency to remove harmonics and white noise produced by the sources. Fig. 5 shows measurements of SNR, Signal-to-Noise-and-Distortion Ratio (SNDR) and Spurious-Free-Dynamic-Range (SFDR) versus conversion rate. At 110MS/s, SNR and SNDR equal 67.1dB and 64.2dB, respectively. Further, the plot shows that SNDR is above 64dB from 20MS/s up to 120MS/s and is above 62dB (equals 10 effective number of bits) up to 140MS/s. SFDR is above 69 dB from 5MS/s up to 140MS/s. The signal frequency was 10MHz for these measurements.

Fig. 6 shows measurements of SNR, SNDR and SFDR versus input frequency at a conversion rate of 110MS/s. SNR remains above 66dB up to 100MHz. Above 100MHz, jitter is the main noise contribution and SNR is falling with increasing input frequency. SNDR is larger than 60dB up to 40MHz and is thereafter falling due to decreasing SFDR. The reason why SFDR, and subsequently SNDR, are falling off at high input frequencies is the nonlinearity introduced by the input switches of the ADC. These switches are transmission gates where both the channel resistance and the parasitic capacitances are nonlinear. This problem can be solved by using bootstrapping, but this is not done in this design due to potential lifetime issues. The measurements presented in Fig. 5 and Fig. 6 are done with signal amplitude near full scale (2V P-P ).

In Fig. 7 (next page) the die photograph of the circuit is shown. The layout became compact due to many factors such as strapping the power routing in all metal layers, scaling the power routing for low-power areas and routing above active area. The key measurement results are summarized in Table I (next page). Additional information here is that DNL and INL are ±1.2 LSB and -1.5/+1 LSB, respectively, and the ADC silicon area is 0. included. In (2) ENOB is the effective number of bits of the ADC including distortion, f CR is the conversion rate, A the ADC silicon area, and P SUP the power dissipation.

Conclusions

This paper presents a pipeline ADC with 12 bits of resolution and 110MS/s nominal conversion rate. The effective number of bits is 10.4, the power consumption is 97mW and the silicon area is 0.86mm 2 . The ADC utilizes a SC bias current generator that scale the bias current, and subsequently the power consumption, automatically as a function of the conversion rate. Low area and low and scalable power dissipation results in an ADC IP-block that is well suited for a wide range of application in SoC systems.

Fig. 1 .

 1 Fig. 1. The pipeline architecture.

Fig. 2 .

 2 Fig. 2. The pipeline stage.

Fig. 3 .

 3 Fig.3. The SC-bias current circuit.

Fig. 5 .

 5 Fig.4shows the power dissipation of the ADC (exluding output drivers) versus conversion rate. As predicted by[START_REF] Hernes | A 1.2V 220MS/s 10b Pipeline ADC Implemented in 0.13µm Digital CMOS[END_REF] the bias currents, and subsequently the power dissipation, is linearly scaled versus conversion rate. The plot shows a power dissipation of 97mW at 110MS/s and 110mW at 130MS/s.The dynamic measurements were done by using RF-sources for the input signal and the clocking of the ADC. Both where filtered using high order passive bandpass filters around the applied frequency to remove harmonics and white noise produced by the sources. Fig.5shows measurements of SNR, Signal-to-Noise-and-Distortion Ratio (SNDR) and Spurious-Free-Dynamic-Range (SFDR) versus conversion rate. At 110MS/s, SNR and SNDR equal 67.1dB and 64.2dB, respectively. Further, the plot shows that SNDR is above 64dB from 20MS/s up to 120MS/s and is above 62dB (equals 10 effective number of bits) up to 140MS/s. SFDR is above 69 dB from 5MS/s up to 140MS/s. The signal frequency was 10MHz for these measurements.Fig.6shows measurements of SNR, SNDR and SFDR versus input frequency at a conversion rate of 110MS/s. SNR remains above 66dB up to 100MHz. Above 100MHz, jitter is the main noise contribution and SNR is falling with increasing input frequency. SNDR is larger than 60dB up to 40MHz and is thereafter falling due to decreasing SFDR. The reason why SFDR, and subsequently SNDR, are falling off at high input frequencies is the nonlinearity introduced by the input switches of the ADC. These switches are transmission gates where both the channel resistance and the parasitic capacitances are nonlinear. This problem can be solved by using bootstrapping, but this is not done in this design due to potential lifetime issues. The measurements presented in Fig.5and Fig.6are done with signal amplitude near full scale (2V P-P ).In Fig.7(next page) the die photograph of the circuit is shown. The layout became compact due to many factors such as strapping the power routing in all metal layers, scaling the power routing for low-power areas and routing above active area. The key measurement results are summarized in Table I (next page). Additional information here is that DNL and INL are ±1.2 LSB and -1.5/+1 LSB, respectively, and the ADC silicon area is 0.86mm 2 .Equation (2) (next page) is the Figure of Merit (FM) given in[START_REF] Walden | Analog-to-Digital Converter Survey and Analysis[END_REF] adjusted such that also the ADC area is

Fig. 6 .

 6 Fig. 6. SFDR, SNR and SNDR versus input frequency. The conversion rate and signal swing are 110MS/s and 2V P-P , respectively.

Fig. 4 .

 4 Fig. 4. Power dissipation versus conversion rate. The input frequency and signal swing is 10MHz and 2V P-P , respectively.

  Fig. 8 shows FM (2) versus the inverse of the silicon area of 15 12b ADCs including the ADC presented in this paper (named "This design"). The plot shows that this design has the highest FM and the 2 nd lowest area consumption. Further, this converter is the 2 nd published 12b ADC with 1.8V supply voltage. The ADCs [5]-[7] are closest in FM and also area consumption. The other ADCs in Fig. 8 are taken from IEEE Proc. of ISSCC and IEEE Symposium on VLSI Circuits Digest of Technical Papers over the last 9 years.

Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE'05) 1530-1591/05 $ 20.00 IEEE

Proceedings of the Design, Automation and Test in Europe Conference and Exhibition (DATE'05)

f CR is given in MS/s, A is given in mm 2 and P SUP is given in mW.


Laurent Gardes

Armelle Guillou

Claire Roman

Estimation of extreme conditional quantiles under a general tail first order condition

Keywords: Extreme quantile, local estimation, asymptotic normality

We consider the estimation of an extreme conditional quantile. In a first part, we propose a new tail condition in order to establish the asymptotic distribution of an extreme conditional quantile estimator. Next, a general class of estimators is introduced, which encompasses, among others, kernel's or nearest neighbors' types of estimators. A unified theorem of the asymptotic normality for this general class of estimators is provided under the new tail condition and illustrated on the different well-known examples. A comparison between different estimators belonging to this class is provided on a small simulation study.

Introduction

To describe the dependence between a real-valued random variable Y and an explanatory random vector X of dimension p ∈ N\{0}, different approaches can be used. The most common one is perhaps provided by the conditional mean m(X) := E(Y |X), which gives information on the central part of the conditional distribution. However, depending on the applications in mind, it can be also of interest to consider a conditional quantile instead of m(X) (e.g., median or quartile). To be more specific, denoting by S(•|x 0 ) := P(Y > •|X = x 0 ) the conditional survival function of Y given {X = x 0 } for some x 0 ∈ R p in the support of X, the conditional quantile of level α ∈ [0, 1] of Y given {X = x 0 } is Q(α|x 0 ) := S ← (α|x 0 ) = inf{y ∈ R; S(y|x 0 ) ≤ α} with the convention inf{∅} = +∞. This conditional quantile presents the advantage to be more robust than the classical conditional mean. Given n independent copies (X 1 , Y 1 ), . . . , (X n , Y n ) of (X, Y ), one question of interest is of course the estimation of the conditional quantile Q(α|x 0 ) in a nonparametric way. There exist numerous estimation methods in the literature. The most common one is the indirect method: starting from a suitable estimator S n (•|x 0 ) of S(•|x 0 ), the associated estimator of Q(α|x 0 ) is given by

Q n (α|x 0 ) := S ← n (α|x 0 ) = inf{y ∈ R; S n (y|x 0 ) ≤ α}. (1)
Estimator ( 1) is called indirect since, as pointed by [START_REF] Racine | Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach[END_REF], "one estimates a conditional survival function, and then, one 'backs out' the inferred quantile via inversion". An alternative way to estimate a conditional quantile is by using the so-called check function defined for α ∈ [0, 1] by ρ α (v) := v[α -I (-∞,0] (v)] where for any A ⊂ R, I A (x) = 1 if x ∈ A and 0 otherwise. Indeed, since the conditional quantile is also defined by

Q(α|x 0 ) = arg min τ ∈[0,1] E [ ρ α (Y -τ )| X = x 0 ] ,
the estimation of Q(α|x 0 ) can be achieved by replacing the conditional expectation by a suitable estimator and then by solving the minimization problem. This method of estimation was investigated among others by Koenker and Basset (1978), [START_REF] Koenker | Quantile smoothing splines[END_REF] and [START_REF] He | Quantile splines with several covariates[END_REF]. In this paper, we focus on the so-called indirect method.

In some applications, we are interested in the tail of the conditional distribution rather than on its central part. In this case, instead of looking at the conditional quantile of level α ∈ [0, 1], we consider an extreme conditional quantile, i.e., a conditional quantile of level α n where α n → 0 as the sample size n increases.

To obtain the asymptotic distribution of an indirect conditional quantile estimator, the following two-step procedure can be used. First, we establish the asymptotic distribution of the associated conditional survival function estimator. Next, a delta-type method is used to deduce the result on the conditional quantile estimator from this first step. This requires an additional condition on the conditional survival function. When the level α is fixed, this condition is simply that S(•|x 0 ) is continuously differentiable. However, in case of an extreme level, this condition is much more complicated. In this work, we introduce a new general condition, called Tail First Order Condition, which is the cornerstone to obtain the asymptotic distribution of any indirect conditional quantile estimator. As we will see, this condition is more flexible than the one classically used in extreme value theory.

To understand where the Tail First Order condition comes from, the main ingredients of the proof of the asymptotic normality in case of a fixed level α and of an extreme level α n is outlined in Section 2. In Section 3, this condition is specified and illustrated on many well-known examples of conditional distributions. Section 4 is devoted to the study of a general class of extreme conditional quantile estimators. In particular, a unified theorem for the asymptotic normality is established. A simulation study is provided in Section 5 where several examples of estimators belonging to this class, among them, the kernel's and nearest neighbors' type-estimators, are compared. All the proofs of the results are postponed to Section 6.

Description of the methodology

The aim of this paper is to show the asymptotic normality of a general class of indirect type of conditional quantile estimators when the level is extreme. This requires a condition, which is not usual in the case of a fixed level α.

To understand where this condition comes from we briefly start to present the simple case where the level is fixed, and then, we outline the main differences when it is assumed to be extreme, and we introduce the required condition in that context.

Case where the level is fixed -When the level α is fixed, the asymptotic distribution of (1) can be deduced from the one of the conditional survival function estimator S n (•|x 0 ). More precisely, if we assume that for some y ∈ R, there exists a sequence v n (x 0 ) → ∞ such that for all sequence ε n → 0

v n (x 0 ) S n (y + ε n |x 0 ) -S(y + ε n |x 0 ) d -→ Λ, (2)
where Λ is some non-degenerate distribution, then if

S(•|x 0 ) is a continuously differentiable function with S[Q(α|x 0 )|x 0 ] = α v n (x 0 ) Q n (α|x 0 ) -Q(α|x 0 ) d -→ 1 f (Q(α|x 0 )|x 0 ) Λ, (3)
where

f (•|x 0 ) is the probability density function of Y given X = x 0 with f (Q(α|x 0 )|x 0 ) = 0. The proof of (3) is based on the following remark: for all z ∈ R, letting σ n (x 0 ) := v n (x 0 )f (Q(α|x 0 )|x 0 ), one has P σ n (x 0 ) Q n (α|x 0 ) -Q(α|x 0 ) ≤ z = P[Z n (x 0 ) ≤ z n (x 0 )], (4)
where, [START_REF] Dony | Uniform in bandwidth consistency of kernel regression estimators at fixed point[END_REF]. Note that the asymptotic distribution of indirect estimators for a fixed level α has been treated for instance by [START_REF] Berlinet | Asymptotic normality of convergent estimates of conditional quantiles[END_REF].

Z n (x 0 ) := v n (x 0 ) S n (Q(α|x 0 ) + zσ -1 n (x 0 )|x 0 ) -S(Q(α|x 0 ) + zσ -1 n (x 0 )|x 0 ) and z n (x 0 ) := v n (x 0 )[α -S(Q(α|x 0 ) + zσ -1 n (x 0 )|x 0 )]. From (2) with y = Q(α|x 0 ), Z n (x 0 ) d -→ Λ and since S(•|x 0 ) is continuously differentiable, z n (x 0 ) → z as n → ∞ proving
Case of an extreme level -We consider the situation where the level of the conditional quantile is a sequence α n where α n → 0 as the sample size n increases. Replacing the level α by a sequence α n does not change (at least if α n does not converge too fast to 0) the estimation procedure. We still estimate Q(α n |x 0 ) as in [START_REF] Berlinet | Asymptotic normality of convergent estimates of conditional quantiles[END_REF] just by replacing α by α n . The difference lies in the assumptions required to obtain the asymptotic distribution of Q n (α n |x 0 ). First, instead of (2), the following kind of result for the conditional survival function estimator is required: for some well-chosen sequence

y n (x 0 ) → y * (x 0 ) := Q(0|x 0 ), there exists a sequence v n (x 0 ) → ∞ such that v n (x 0 ) S n (y n (x 0 )|x 0 ) -S(y n (x 0 )|x 0 ) d -→ Λ, (5)
for some non-degenerate distribution Λ. Of course, the sequence v n (x 0 ) depends on the sequence y n (x 0 ). Since y * (x 0 ) is the right endpoint, convergence [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF] focus on the asymptotic behavior of S n (•|x 0 ) in the right tail of the conditional distribution. To obtain the asymptotic distribution of Q n (α n |x 0 ), we start again with (4) where α is replaced by α n . In the extreme level case, the main difficulty is to deal with the non-random sequence z n (x 0 ). More specifically, assuming that S[Q(α|x 0 )|x 0 ] = α at least for α small enough, we need to find a general condition on the conditional distribution ensuring that for a well-chosen sequence σ n (x 0 ) and for a sequence v n (x 0 ) satisfying ( 5) with

y n (x 0 ) = Q(α n |x 0 ) + zσ -1 n (x 0 ) z n (x 0 ) = -α n v n (x 0 ) S[y n (x 0 )|x 0 ] S[Q(α n |x 0 )|x 0 ] -1 → z, (6)
as n → ∞ for all z ∈ R. Obviously, assuming that S(•|x 0 ) is a continuously differentiable function is not relevant here and the sequence

σ n (x 0 ) is not necessar- ily equal to v n (x 0 )f (Q(α n |x 0 )|x 0 ). Since Q(α n |x 0 ) → y * (x 0 ),
Ψ(y) S[y + td(y)|x 0 ] S(y|x 0 ) -1 → φ -1 x0 (t), (7)
where φ -1 x0 ≡ φ -1 is the inverse of a continuous and strictly decreasing function

φ x0 ≡ φ such that φ(t)/t → -1 as t → 0. Indeed, taking σ n (x 0 ) = α n v n (x 0 )/[Ψ(Q(α n |x 0 ))d(Q(α n |x 0 ))] and t -1 n (x 0 ) := σ n (x 0 )d[Q(α n |x 0 )], we obtain z n (x 0 ) = - Ψ[Q(α n |x 0 )] t n (x 0 ) S[Q(α n |x 0 ) + zt n (x 0 )d[Q(α n |x 0 )]|x 0 ] S[Q(α n |x 0 )|x 0 ] -1 .
Under [START_REF] Giné | Mathematical foundations of infinitedimensional statistical models[END_REF] and assuming that t n (x 0 ) → 0, we can show that z n (x 0 ) → z (see Section 3, Proposition 1). Next, the random sequence Z n (x 0 ) is treated by [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF].

To sum up, in the extreme level case, a natural condition on S(•|x 0 ) to establish the asymptotic distribution of the conditional quantile estimator is [START_REF] Giné | Mathematical foundations of infinitedimensional statistical models[END_REF]. Condition [START_REF] Giné | Mathematical foundations of infinitedimensional statistical models[END_REF] is referred in what follows to as the Tail First Order condition. Under this condition and if [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF] holds with y n (x 0 ) := Q(α n |x 0 ) + zσ -1 n (x 0 ), we have

σ n (x 0 )( Q n (α n |x 0 ) -Q(α n |x 0 )) d -→ Λ.
We show in Section 3 that this Tail First Order condition is satisfied by a larger class of conditional distributions than the one satisfying the condition classically used in extreme value theory. Note that while on the fixed level case, the rate of convergence of Q n (α|x 0 ) is proportional to v n (x 0 ) this is no longer the case when estimating an extreme conditional quantile.

A Tail First Order condition

The Tail First Order condition is related to the conditional distribution of Y given {X = x 0 } for some x 0 ∈ R p in the support of X. Since x 0 is fixed, the dependence on x 0 can be omitted. This is what we do in all this section. For a given (conditional) survival function S, we denote by Q = S ← the associated quantile and by x * = S ← (0) the right endpoint.

Definition 1 A survival function S satisfies a Tail First Order (TFO) condition if for some open interval I ⊂ R containing 0, there exist positive functions d and Ψ such that for all t ∈ I,

lim x↑x * Ψ(x) S[x + td(x)] S(x) -1 = φ -1 (t), (8)
where φ -1 is the inverse of a continuous and strictly decreasing function φ : J → I such that φ(t)/t → -1 as t → 0.

Note that convergence [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF] entails that for all t ∈ I and for x large enough, x + td(x) < x * . Consequently, the function Ψ is such that Ψ(x)/S(x) → ∞ as x ↑ x * . Finally, it is easy to check that φ -1 (t)/t → -1 as t → 0. As a consequence of Dini's theorem, we obtain the useful properties gathered in the next proposition.

Proposition 1 If S satisfies a T F O condition, the following statements are true:

1. Convergence in (8) holds locally uniformly on I.

2. For all t 0 ∈ I,

lim (t,x)→(t0,x * ) Ψ(x) t S[x + td(x)] S(x) -1 = lim t→t0 φ -1 (t) t .
We give in the next result some equivalent reformulations of the TFO condition.

Proposition 2

The following statements are equivalent: 1. The survival function S satisfies a TFO condition.

2. There exist positive functions a and g such that for all t ∈ J,

lim α→0 Q[α + tg(α)] -Q(α) a(α -1 ) = φ(t). (9)
3. There exist sequences

a n > 0, b n ∈ R and c n > 0 with nc n → ∞ such that for all t ∈ I, lim n→∞ [nc n S(a n t + b n ) -c n ] = φ -1 (t). (10)
Remarks -1) The relations between the auxiliary functions involved in [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF] and ( 9) are: d(•) = a(1/S(•)) and Ψ(•) = S(•)/g(S(•)).

2) A possible choice for the sequences a n , b n and c n in [START_REF] He | Quantile splines with several covariates[END_REF] is

a n = a(n), b n = Q(1/n) and c n = 1/[ng(1/n)].
It is also easy to check that necessarily g(α) → 0 as α → 0.

3) An interpretation of condition ( 9) is based on the following remark: from the second statement of Proposition 1,

Q[α + tg(α)] -Q(α) tg(α) ∼ - a(α -1 ) g(α) ,
as (t, α) → 0. Hence, one can see the function -a(α -1 )/g(α) as the derivative of Q near 0 and in the direction of g(α). This heuristic is confirmed by the next result which provides a sufficient condition for the TFO condition.

Proposition 3 Assume that Q is a differentiable function and that for some open interval J ⊂ R containing 0, there exists a positive function g such that for all t ∈ J,

lim α→0 Q [α + tg(α)] Q (α) = Θ(t). (11)
If for all t ∈ J, t 0 Θ(s)ds =: θ(t) ∈ R where θ is an increasing function on J such that θ(t)/t → 1 as t → 0 then condition (9) holds with φ(t) = -θ(t) and a(α -1 ) = -Q (α)g(α).

We conclude this section by giving examples of distributions satisfying a TFO condition.

Maximum domain of attraction -In extreme value theory, in order to make inference on the tail of a distribution S, we classically assume that there exist sequences a n > 0 and b n and a non-degenerate distribution function G for which lim

n→∞ [1 -S(a n x + b n )] n = G(x), (12)
for all point of continuity of G. [START_REF] Fisher | Limiting forms of the frequency distribution of the largest or smallest member of a sample[END_REF] and [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF] show that G(x) = G γ (ax + b) for some a > 0 and b ∈ R where

G γ (x) = exp -(1 + γx) -1/γ ,
lim α→0 Q(tα) -Q(α) a(α -1 ) = φ(t). (13)
From de Haan and Ferreira (2006, Theorem B.2.1), the function φ in ( 13) is necessarily of the form φ(t) = c(t -γ -1)/γ for some c = 0 and where γ ∈ R is always the extreme value index. The aim of the next result is to show that the TFO condition introduced in this paper (see Definition 1) is weaker than [START_REF] Koenker | Quantile smoothing splines[END_REF].

Proposition 4 If S satisfies a TFO condition with an auxiliary function g in ( 9) such that α/g(α) → c ≥ 0 as α → 0 (with g continuous and strictly increasing if c = 0) then S satisfies [START_REF] Koenker | Quantile smoothing splines[END_REF].

As a consequence of this result, if a survival function S satisfies the TFO condition with a function g as in Proposition 4, then S also satisfies the TFO condition with g(α) = α and in this case the TFO condition coincides with the classical extreme value condition. Remark also that in this situation (i.e., g(α) = α), condition [START_REF] Koenker | Regression quantiles[END_REF] is equivalent to assume that

lim α→0 Q (tα) Q (α) = t -γ-1 ,
for some γ ∈ R. This condition coincides with condition (1.1.33) in de Haan and Ferreira (2006, Corollary 1.1.10).

At this step, a natural question is: "Can we find survival functions that satisfy the TFO condition but not the classical extreme value one ?" Roughly speaking, this is equivalent to find survival functions S such that (9) holds with a function g such that α/g(α) → ∞. An example of such survival functions is given by super heavy-tailed distributions.

Super heavy-tailed distributions -The term super heavy-tailed is often attached in the literature to a distribution with a slowly varying survival function S, i.e., such that for all t > 0,

lim x→∞ S(tx) S(x) = 1. (14)
It can be shown that these survival functions do not satisfy the classical first order condition [START_REF] Koenker | Quantile smoothing splines[END_REF]. Note that a heavy-tailed distribution corresponds to a survival function satisfying for all t > 0, S(tx)/S(x) → t -1/γ as x → ∞, for some γ > 0. Hence, roughly speaking, a super heavy-tailed distribution is a heavy-tailed distribution with γ = +∞.

Unfortunately, condition ( 14) is not precise enough for the study of super heavytailed distribution. To define more precisely the class of super heavy-tailed distribution, we start by remarking that for heavy-tailed distributions, there exists γ > 0 such that for all s > -1,

lim α→0 Q[(1 + s)α] Q(α) = (1 + s) -γ .
Since super heavy-tailed distribution can be seen as a heavy-tailed distribution with γ = +∞, we propose to replace in the previous limit γ by γ(α) where

γ(α) → ∞ as α → 0. Since lim α→0 (1 + s) -γ(α) = +∞ if s ∈ (-1, 0), 0 if s > 0,
we must replace s by t/γ(α) with t ∈ R to obtain a non-degenerate limit:

lim α→0 Q[(1 + t/γ(α))α] Q(α) = e -t .
The class of super heavy-tailed distributions can thus be defined by the set of distributions for which there exists a positive function g with g(α)/α → 0 as α → 0 such that for all t ∈ R

lim α→0 Q[α + tg(α)] Q(α) = e -t . (15)
It appears that convergence [START_REF] Racine | Nonparametric conditional quantile estimation: A locally weighted quantile kernel approach[END_REF] coincides with the TFO condition with a(α -1 ) = Q(α) and φ(t) = e -t -1. As shown in Proposition 5 below, this definition is equivalent to the one introduced for instance in Fraga [START_REF] Alves | A test procedure for detecting super-heavy tails[END_REF] where the class of super heavy-tailed distributions is defined as the set of distributions for which there exists a positive function b such that

lim x→∞ U [x + tb(x)] U (x) = e t (16)
with 16) implies [START_REF] Owen | Nonparametric conditional estimation[END_REF]. Furthermore, the function b is such that b(x)/x → 0 as x → ∞. Remark finally that the right endpoint of a distribution satisfying ( 16) is necessarily infinite. As examples of super heavy-tailed distribution satisfying [START_REF] Van Der Vaart | Weak convergence and empirical processes with applications to statistics[END_REF], one can cite the standard log-Pareto distribution given by S(x) = [log(x)] -ξ with ξ > 0 and the log-Weibull distribution for which S(x) = exp(-ξ log θ x), with ξ > 0 and θ ∈ (0, 1). For these two distributions, one can take b ∼ U/U .

U (•) := Q(1/•).
Proposition 5 Conditions ( 15) and ( 16) are equivalent. The relation between the involved functions is b(x) = x 2 g(x -1 ).

Extreme conditional quantile estimation

Let (X, Y ) be a random vector taking its values in R p ×R. In all what follows, we assume that (X, Y ) admits a probability density function (pdf). The marginal pdf of X is denoted by f . As in the introduction, for all x 0 ∈ R p , let S(•|x 0 ) and Q(•|x 0 ) be the survival function and the quantile function of the conditional distribution of Y given {X = x 0 }, respectively. Given n independent copies

(X 1 , Y 1 ), . . . , (X n , Y n ) of (X, Y )
, the first part of this section is dedicated to the presentation of a large class of estimators of Q(•|x 0 ). In the second part, we show that under a TFO condition, the proposed estimators computed with an extreme level α n → 0 are asymptotically Gaussian.

A class of conditional quantile estimators

As mentioned in the introduction we focus in this paper on indirect estimators of Q(•|x 0 ). The first step is thus the estimation of the conditional survival function S(•|x 0 ). We consider estimators of the form

S n (y|x 0 ) := n i=1 W n,i (x 0 )I (y,∞) (Y i ). ( 17
)
The set of weights {W n,i (x 0 ), 1 ≤ i ≤ n} is a triangular array of positive random variables depending on the data X 1 , . . . X n as well as on x 0 such that

n i=1 W n,i (x 0 ) = 1.
These properties on the random weights ensure that S n (•|x 0 ) is a well-defined distribution function. This is crucial to estimate the conditional quantile by inverting estimator (17). This class of estimators encompasses various classical estimators of the conditional distribution function, see below for some examples.

The indirect estimator of the conditional quantile of level α ∈ (0, 1) is thus defined as in (1) by

Q n (α|x 0 ) := S ← n (α|x 0 ) = inf{y ∈ R; S n (y|x 0 ) ≤ α}.
Of course, the main feature of the weights

{W n,i (x 0 ), 1 ≤ i ≤ n} is to select a set
W NW n,i (x 0 , h n ) := K X i -x 0 h n n j=1 K X j -x 0 h n , ( 18
)
where K is a density on R p and h n is a positive non-random sequence satisfying h n → 0 as n → ∞. Typically, the probability density function K has a unique mode at 0 in order to give the largest values of the weights for the observations close to x 0 .

Another possibility to select the observations is to take the k n observations which are closest to the reference position x 0 . This approach is called the k n -Nearest Neighbors' (k n -NN) method. More specifically, for some norm • on R p , let {D i (x 0 ) := X i -x 0 , i = 1, . . . , n} be the distances between each observation and x 0 and let D (1) (x 0 ) ≤ . . . ≤ D (n) (x 0 ) the corresponding order statistics. Denoting by {r(i), i = 1, . . . , n} the ranks of these distances (i.e., D (i) (x 0 ) = D r(i) (x 0 ) for i = 1, . . . , n), the k n -NN estimator is obtained by using the following random weights in (17):

W NN n,i (x 0 , k n ) := [(k n -r(i) + 1) + ] kn j=1 j , (19)
where (•) + stands for the positive part function and ∈ N. For instance, by taking = 0 (with the convention 0 0 = 0), we affect the same weight to the k n closest observations. The corresponding weights are referred to as uniform-k n -NN weights. The choice = 1 (resp., = 2) leads to triangular-k n -NN weights (resp., quadratic-k n -NN weights). Roughly speaking, the main difference between these two sets of weights is that the kernel based estimator averages over all observations which are within a fixed distance, whereas the k n -NN approach averages over a fixed number of observations which might be arbitrarily far away. Of course, one can also think about a linear combination of ( 18) and ( 19). For instance, we can consider the random weights defined for τ ∈ (0, 1) by

W LC n,i (x 0 , τ, h n , k n ) := τ M n I [0,1] X i -x 0 h n ∞ + 1 -τ k n I [0,1] r(i) k n , (20)
where M n is the random number of random variables among {X 1 , . . . , X n } that belong to B x0 (h n ), the closed ball with respect to • ∞ centered at x 0 and with radius h n .

Main results

Under general conditions on the random weights {W n,i (x 0 ), i = 1, . . . , n}, we want to establish the convergence in distribution of a normalized version of Q n (α n |x 0 ) for a level α n converging to 0 as n → ∞. As outlined in Section 2, we first need to find a sequence v n (x 0 ) → ∞ and a non-degenerate distribution Λ such that (under some additional assumptions)

v n (x 0 ) S n (y n (x 0 )|x 0 ) -S(y n (x 0 )|x 0 ) d -→ Λ,
for some sequence y n (x 0 ) ↑ y * (x 0 ). This is done in Theorem 1 where the following notation is used

n x0 := n i=1 W 2 n,i (x 0 ) -1
.

Note that the random variable n x0 corresponds, roughly speaking, to the number of observations used in the estimation procedure. For instance, for the Nadaraya-Watson's weights with the uniform kernel

K(•) ∝ I [0,1] ( • ∞ ), it is easy to check that n x0 is exactly the number of points in B x0 (h n ).
For the uniform-k n -NN weights, one has n x0 = k n , the number of nearest neighbors.

Theorem 1 Let x 0 ∈ R p and y n (x 0 ) be a sequence converging to the right endpoint y * (x 0 ) of the conditional distribution of Y given {X = x 0 }. Assume that there exists a sequence m n (x 0 ) such that n x0 /m n (x 0 ) a.s.

-→ 1 and let v 2 n (x 0 ) := m n (x 0 )/S(y n (x 0 )|x 0 ). Under the conditions

v n (x 0 ) max 1≤i≤n W n,i (x 0 ) a.s.
-→ 0 (21) and

v n (x 0 ) n i=1 W n,i (x 0 ) |S(y n (x 0 )|X i ) -S(y n (x 0 )|x 0 )| P -→ 0, (22)
we have that

v n (x 0 ) S n (y n (x 0 )|x 0 ) -S (y n (x 0 )|x 0 ) d -→ N (0, 1).
To understand the usefulness of conditions ( 21) and ( 22), we provide below the main ideas of the proof of Theorem 1, the complete proof being postponed to Section 6. Let

Y x0 i := Q(U i |x 0 )
where U 1 , U 2 , . . . are independent standard uniform random variables, independent of the X i . The random vectors {(X i , Q(U i |X i )), i = 1, . . . , n} are thus independent and distributed as (X, Y ), which implies that

S n (y n (x 0 )|x 0 ) d = n i=1 W n,i (x 0 )I (yn(x0),∞) (Q(U i |X i )).

In other words, one can work as if

Y i = Q(U i |X i ).
The starting point of the proof is the decomposition

S n (y n (x 0 )|x 0 ) -S (y n (x 0 )|x 0 ) = S x0 n (y n (x 0 )) -S (y n (x 0 )|x 0 ) + S n (y n (x 0 )|x 0 ) -S x0 n (y n (x 0 )) ,
where for all y ∈ R,

S x0 n (y) := n i=1 W n,i (x 0 )I (y,∞) (Y x0 i ).
Since E[ S x0 n (y n (x 0 ))] = S(y n (x 0 )|x 0 ), the first term corresponds to the variance term and the second one to the bias term. The first part of the proof consists in establishing the asymptotic normality of the normalized variance term given by:

v n (x 0 ) S x0 n (y n (x 0 )) -S (y n (x 0 )|x 0 ) ,
see Section 6, Proposition 7. This is obtained mainly by applying the Lindeberg's theorem and only condition (21) is required. This condition is in fact equivalent to the Lindeberg's condition.

In the second part of the proof, we show that the bias term given by

B n (x 0 ) := v n (x 0 ) S n (y n (x 0 )|x 0 ) -S x0 n (y n (x 0 ))
converges to 0 in probability (see Section 6, Proposition 8). The proof is based on the following remark. Let W n,x0 be the discrete random measure define for all A ∈ B(R p ) by

W n,x0 (A) := n i=1 W n,i (x 0 )δ Xi (A).
Straightforward calculation leads to

S n (y n (x 0 )|x 0 ) -S x0 n (y n (x 0 )) = n i=1 W n,i (x 0 )I (yn(x0),∞) (Q(U i |•)) (dW n,x0 -dδ x0 ) .
To control the bias term we need to measure the discrepancy between the two probability measures W n,x0 and δ x0 . A useful distance between probability measures is the Wasserstein distance defined for all probability measures P 1 and

P 2 by W 1 (P 1 , P 2 ) = inf {[E(|X 1 -X 2 |)], X 1 ∼ P 1 , X 2 ∼ P 2 }. Condition (22)
can in fact be written in term of the Wasserstein distance as follows

v n (x 0 )W 1 (W * n,x0 , δ * x0 ) P -→ 0, (23)
where W * n,x0 and δ * x0 are the pushforward measures of W n,x0 and δ x0 by the measurable function S(y n (x 0 )|•).

We have now all the ingredients to establish the asymptotic distribution of the conditional quantile estimator of level α n obtained by inverting the estimator S n (•|x 0 ). This requires the following first order condition on the conditional distribution of Y given X = x 0 .

Let a(1/•) ≡ a x0 (1/•) = d[Q(•|x 0 )] and g(•) ≡ g x0 (•) = •/Ψ[Q(•|x 0 )]. From Proposition 2, condition (H) is equivalent to assume that for some open interval J x0 = J ⊂ R containing 0, one has for all t ∈ J lim α→0 Q(α + tg(α)|x 0 ) -Q(α|x 0 ) a(α -1 ) = φ x0 (t),
where φ x0 ≡ φ is a continuous and strictly decreasing function such that φ(t)/t → -1 as t → 0.

Theorem 2 Let x 0 ∈ R p and assume that condition (H) holds. Assume that there exists a sequence m n (x 0 ) such that n x0 /m n (x 0 ) a.s.

-→ 1 and let

v 2 n (x 0 ) := m n (x 0 )/α n . If α n m n (x 0 ) → ∞, v n (x 0 )g(α n ) → ∞, v n (x 0 ) max 1≤i≤n W n,i (x 0 ) a.s. -→ 0 and [α n m n (x 0 )] 1/2 sup |β/αn-1|≤ξ n i=1 W n,i (x 0 ) S[Q(β|x 0 )|X i ] β -1 P -→ 0,
for some ξ ∈ (0, 1) then

v n (x 0 ) g(α n )Q(α n |x 0 ) a(α -1 n ) Q n (α n |x 0 ) Q(α n |x 0 ) -1 d -→ N (0, 1).
Recall that if g(α) = α (or equivalently Ψ ≡ 1), condition (H) coincides with the classical first order condition (13) used in extreme value theory. In this case, φ(t) ∝ (t -γ(x0) -1)/γ(x 0 ) where the function γ is referred to as the conditional extreme value index. Under [START_REF] Nadaraja | On estimating regression[END_REF] and if the conditions of Theorem 2 are satisfied,

[α n m n (x 0 )] 1/2 Q(α n |x 0 ) a(α -1 n |x 0 ) Q n (α n |x 0 ) Q(α n |x 0 ) -1 d -→ N (0, 1).
Moreover, we know from de Haan and Ferreira (2006, Lemma 1.2.9) that under [START_REF] Nadaraja | On estimating regression[END_REF],

Q(α n |x 0 )/a(α -1 n ) → 1/γ + (x 0 )
, where γ + (x 0 ) = max(γ(x 0 ), 0). So, under the first order condition [START_REF] Nadaraja | On estimating regression[END_REF], the worst rate of convergence is achieved when γ(x 0 ) > 0. This was expected since the case γ(x 0 ) > 0 corresponds to heavy-tailed distributions. Let us now focus on the rate of convergence in Theorem 2 for conditional super heavy-tailed distribution. Taking the definition of super heavy-tailed distributions given in Fraga [START_REF] Alves | A test procedure for detecting super-heavy tails[END_REF] into account, we have in this case a(α -1 ) = Q(α|x 0 ) and g(α)/α → 0 as α → 0. Hence, for these distributions,

[α n m n (x 0 )] 1/2 g(α n ) α n Q n (α n |x 0 ) Q(α n |x 0 ) -1 d -→ N (0, 1).
Not surprisingly, this rate is worse than the one for heavy-tailed distributions.

Theorem 2 is proved under general conditions on the random weights used to define the conditional survival estimator (17). We close this section by applying Theorem 2 to particular weights.

Nadaraya-Watson's weights -Taking the weights defined in (18) leads to the well-known Nadaraya-Watson's estimator of the conditional survival function:

S NW n (y|x 0 ) := n i=1 K X i -x 0 h n I (y,∞) (Y i ) n i=1 K X i -x 0 h n . ( 24
)
The corresponding conditional quantile estimator is denoted by Q NW n (α n |x 0 ). In order to apply Theorem 2, we need to check that the Nadaraya-Watson's weights satisfy the required conditions. To this aim, we assume the following on the kernel function K:

(K) the kernel K is either an indicator function on a cell of R p or such that K(x) = L( x ) where L is of bounded variation, continuous on (0, ∞) and with support [0, 1].

It is very easy to check that (K) is satisfied for a large range of usual kernels such as the uniform kernel (K(t

) ∝ I [0,1] ( t ∞ )), triangular (with L(t) ∝ 1 -t), Epanechnikov kernel (L(t) ∝ 1 -t 2 ), biweight kernel (L(t) ∝ (1 -t 2 ) 2 ), etc.
We can now state the convergence in distribution of the conditional survival estimator (24). Recall that f is the pdf of X.

Corollary 1 Let x 0 ∈ R p such that f is continuous at x 0 and f (x 0 ) > 0 and let K be a kernel satisfying (K). Under (H), for sequences h n → 0 and α n ∈ (0, 1)

such that nh p n [α n ∧ (log log n) -1 ] → ∞, α -1 n nh p n g 2 (α n ) → ∞ and sup |β/αn-1|≤ξ x-x0 ≤hn S[Q(β|x 0 )|x] β -1 2 = o 1 nh p n α n (25)
for some ξ ∈ (0, 1) we have

g(α n ) α n Q(α n |x 0 ) a(α -1 n ) (nh p n α n ) 1/2 Q NW n (α n |x 0 ) Q(α n |x 0 ) -1 d -→ N 0, K 2 2 f (x 0 ) .
Note that under the classical first order condition (13) (i.e., when g( Nearest Neighbors' approach -Now, let us consider the k n -NN random weights defined in (19) and leading to the conditional survival function estimator

α n ) = α n in (
S KN N n (y|x 0 ) := n i=1 [(k n -r(i) + 1) + ] I (y,∞) (Y i ) kn j=1 j ,
with k n ∈ {1, . . . , n}, ∈ N and r(i) is the rank of X i -x 0 among the random variables X 1 , . . . , X n . The asymptotic normality of the k n -NN conditional quantile estimator

Q KNN n (α n |x 0 ) is established in the following result. Corollary 2 Let x 0 ∈ R p . Under (H), for sequences k n → ∞ and α n ∈ (0, 1) such that k n α n → ∞, α -1 n k n g 2 (α n ) → ∞ and (k n α n ) sup |β/αn-1|≤ξ x-x0 ≤D (kn ) (x0) S[Q(β|x 0 )|x] β -1 2 P -→ 0,
for some ξ ∈ (0, 1), we have

g(α n ) α n Q(α n |x 0 ) a(α -1 n ) (k n α n ) 1/2 Q KNN n (α n |x 0 ) Q(α n |x 0 ) -1 d -→ N 0, ( + 1) 2 2 + 1 .
The asymptotic variance ( + 1) 2 /(2 + 1) is an increasing function of and thus the best choice (at least in term of variance) seems to be = 0, i.e., when the same weight 1/k n is affected to the k n observations closest to x 0 .

Linear combination of weights -We finally focus on the estimator Q LC n (α n |x 0 ) of Q(α n |x 0 ) obtained by using the linear combination of weights introduced in (20).

Corollary 3 Let x 0 ∈ R p such that f is continuous at x 0 and f (x 0 ) > 0. Let h n → 0, k n → ∞ and α n be sequences such that nh p n / log log n → ∞, n α n → ∞ with n := (nh p n ∧ k n ), α -1 n n g 2 (α n ) → ∞ and ( n α n ) sup |β/αn-1|≤ξ x-x0 ≤(hn∨D (kn ) (x0)) S[Q(β|x 0 )|x] β -1 2 P -→ 0,
for some ξ ∈ (0, 1). Under (H) and if there exists κ ∈ [0, ∞] such that k n /(nh p n ) → κ, we have

g(α n ) α n Q(α n |x 0 ) a(α -1 n ) ( n α n ) 1/2 Q LC n (α n |x 0 ) Q(α n |x 0 ) -1 d -→ N 0, C 2 (κ) 2 p f (x 0 ) .
In practice, one can take k n = κnh p n with κ > 0. The parameter κ is thus a tuning parameter that has to be chosen by a data-driven procedure (see Section 5.1).

Simulation study

In this section, we are interested in the finite sample behavior of the estimator Q n (α n |x 0 ) defined in (1) for a given value of x 0 . The random weights {W n,1 (x 0 ), . . . , W n,n (x 0 )} used in the expression of the estimator (17) of the conditional survival function S(•|x 0 ) often depend on an hyperparameter λ n ∈ R d , d ∈ N \ {0}, useful in order to control the smoothness of the estimator. This is the case for instance for the Nadaraya-Watson's weights, the k n -NN random weights or the LC-weights defined in (18), ( 19) and (20), where λ n is equal to h n , k n and (h n , κ), respectively. In the next section, we propose an adaptive procedure to select λ n in practice.

Choice of the hyperparameter

For t ∈ R p , let us denote by Q n (α n |t, λ n ) an estimator of Q(α n |t) depending on an hyperparameter λ n and by Q n,-i (α n |t, λ n ) the estimator computed without the random pair (X i , Y i ). Our procedure of selection is based on the following simple remark: for a good choice of λ n , the random value S[ Q n,-1 (α n |X 1 , λ n )|X 1 ] should be close to α n at least when the observed value of X 1 is close to x 0 . We thus propose to define our optimal value of λ n as λ opt := arg min{Λ 2 n (λ), λ ∈ R d }, with

Λ n (λ) := E W n,1 (x 0 ) E [W n,1 (x 0 )] S[ Q n,-1 (α n |X 1 , λ)|X 1 ] -α n .
Note that the proximity of X 1 and x 0 is controlled by the random weight W n,1 (x 0 ). Of course, the function Λ n is unknown in practice and should be estimated. We propose to use the following estimator

Λn (λ) := n i=1 W n,i (x 0 )I {Yi> Qn,-i(αn|Xi,λ)} -α n . ( 26
)
The estimated optimal value of the hyperparameter λ n is thus given by

λ n,opt := arg min{ Λ2 n (λ), λ ∈ R d }. ( 27
)
The estimator (26) can be motivated by the following result.

Proposition 6 If there exists a function ϕ : R p × R p×(n-1) → [0, ∞) such that for all i = 1, . . . , n, W n,i (x 0 ) = ϕ(X i , X -i ) where the matrix

X -i is given by [X 1 , . . . , X i-1 , X i+1 , . . . , X n ] then E[ Λn (λ)] = Λ n (λ) for all λ ∈ R d .
Note that the assumption of Proposition 6 is satisfied for the nearest neighbors' approach with the function ϕ defined for t ∈ R p and u

= [u 1 , . . . , u n-1 ] ∈ R p×(n-1) by ϕ(t, u) = λ -1 I { t-x0 <d (λ) (x0)} ,
where

d i (x 0 ) = u i -x 0 , i = 1, . . . , n -1 and d (1) (x 0 ) ≤ . . . ≤ d (n-1) (x 0
) are the corresponding ordered values. This is also the case for the Nadaraya-Watson's weights by using the function

ϕ(t, u) = K[(t -x 0 )/λ] n-1 i=1 K[(u i -x 0 )/λ] + K[(t -x 0 )/λ] .

Finite sample behavior

Using a sample of size n from the random vector (X, Y ), we are interested in estimating an extreme conditional quantile in the situation where the quantile level α n is not too small. We consider the situation where X is a real-valued random variable (p = 1). In a theoretical point of view, we assume that the conditions of Theorem 2 are satisfied for such a sequence α n . In practice, we take α n = 20/n and the quantile Q(α n |x 0 ) is estimated using [START_REF] Berlinet | Asymptotic normality of convergent estimates of conditional quantiles[END_REF]. Three sets of random weights are considered: i) Nadaraya-Watson's weights with the Epanechnikov kernel given by

K(u) = 3 4 (1 -u 2 )I [0,1] (|u|),
ii) the k n -NN weights with = 1 (triangular k n -NN weights), iii) the linear combination of weights given in (20) with τ = 1/2 and k n = κnh n . We generate X from a standard uniform distribution and the four following models have been considered for the conditional survival distribution function of Y given X:

M1 -Conditional Burr distribution:

S(y|X) = 1 + y -ρ/γ(X) 1/ρ , y > 0,
where ρ < 0 and for all x ∈ [0, 1], γ(x) = 2x(1 -x).

It is well-known that for this model, condition ( 13) holds (i.e., condition (H) with g(α|x 0 ) = α). The parameter ρ is referred in the literature to as the second order parameter and it affects the bias of the estimator.

M2 -Conditional Beta distribution with parameters θ 1 > 0 and θ 2 (X) where for all x ∈ [0, 1],

θ 2 (x) = 1/[2x(1 -x)].
This conditional distribution satisfies condition [START_REF] Nadaraja | On estimating regression[END_REF] with a conditional extreme value index given by γ(x) = -1/θ 2 (x) < 0.

M3 -Conditional Gaussian distribution with mean µ(X) = 2X(1 -X) and variance σ 2 .

Under this model, condition ( 13) is satisfied with γ(X) = 0.

We finally consider a model for which condition [START_REF] Nadaraja | On estimating regression[END_REF] does not hold.

M4 -Conditional super heavy-tailed distribution:

S(y|X) = exp -ξ[ln(y)] θ(X) , y > 1, with ξ > 0 and θ(x) = 19(x + 1/2)(3/2 -x)/20 ∈ [0, 0.95].
One can check that this conditional distribution satisfies condition (H) with

a(α -1 ) = Q(α|x) = exp ln(1/α) ξ 1/θ(x)
and g(α) = αθ(x)ξ ln(1/α) ξ

1-1/θ(x)
.

For each model, N = 500 samples of size n = 1000 have been generated. The hyperparameter λ n is chosen according to (27) and the minimization is achieved

• over a regular grid H of 20 points evenly spaced between 0.05 and 0.3 for the Nadaraya-Watson's weights,

• over a grid K of 20 points evenly spaced between 100 and 600 for the Nearest-Neighbors' weights,

• over the grid H × F where F is a grid of 5 evenly spaced points between 0.9 and 1.1.

The accuracy of the estimators is measured by the error

ERROR := 1 N N i=1 Q •,i n (α n |x 0 ) Q(α n |x 0 ) - 1 2
, where • has to be replaced by NW (Nadaraya-Watson's weights), NN (k n -NN approach) or LC (Linear combination of weights) and the index i refers to the i-simulation run. The estimation of Q(α n |x 0 ) is done at three different positions:

x 0 := x (1) 0 = (1 -1/3)/2 ≈ 0.211, x 0 = x (2) 0 = 1/2 and x 0 = x (3)
0 = (1 + 1/2)/2 ≈ 0.854. The results are gathered in Tables 1 to 4. Based on these simulations, we can draw the following conclusions:

• The three methods, NW, NN and LC, perform similarly for the models M1-M3;

• Concerning model M1, the ERRORs increase as |ρ| decreases. This is expected since the estimation is much more difficult when ρ is close to 0 where a bias in the estimation appears. Also the ERRORs increase in general when γ(•) increases;

• Concerning model M2, the ERRORs increase with θ 2 , i.e., when γ(•) = -1/θ 2 (•) increases, and decreases with θ 1 . Compared to the model M1, the ERRORs are considerably smaller, but this is not surprising since the conditional extreme value index is negative in model M2, which means that the observations are bounded;

• Concerning model M3, the ERRORs are not too much sensitive on the values of σ, nor on x 0 . In general the orders of the ERRORs are intermediate between those obtained in the case γ(•) > 0 (model M1) and γ(•) < 0 (model M2);

• Concerning model M4, the ERRORs depend a lot on the value of ξ. Indeed, if ξ is too small, the ERRORs increase drastically and in that case the variability of the results is probably too large to allow a more precise interpretation of the results. For larger values of ξ (ξ = 1 or 3/2), the ERRORs are more reasonable, although larger than for the others models. In that case, a slight increase in θ(•) implies in general a decrease in the ERROR.

6 Proofs

Proof of the results given in Section 3

Proof of Proposition 1 -1. Since S is decreasing and φ -1 is a continuous function, statement 1. is a direct consequence of Dini's theorem. 2. It suffices to remark that from the first statement, one has for all t 0 ∈ I,

lim (t,x)→(t0,x * ) Ψ(x) S[x + td(x)] S(x) -1 = lim t→t0 φ -1 (t).
Proof of Proposition 2 -We first prove that condition (9) implies condition [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF]. From de Haan and Ferreira (2006, Lemma 1.1.1), one has

lim α→0 S[Q(α) + ta(α -1 )] -α g(α) = φ -1 (t),
for all t ∈ I. Replacing α by S(x) leads to

lim x→x * S[x + ta(1/S(x))] -S(x) g[S(x)] = φ -1 (t).
Taking

a n = d(Q(1/n)), b n = Q(1/n) and c n = Ψ(Q(1/n)), we easily show that 1. ⇒ 3.
Finally, let us prove that 3. ⇒ 2. From de Haan and Ferreira (2006, Lemma 1.1.1), we have that for all t ∈ J,

lim n→∞ Q[n -1 (1 + tc n )] -b n a n = φ(t). (28)
Hence, since Q is decreasing and

α -1 ≤ α -1 < α -1 + 1, Q 1 + tc α -1 α -1 ≤ Q[α(1 + tc α -1 )] ≤ Q 1 + tc α -1 α -1 + 1 . (29)
Using (28), we know that

1 a α -1 Q 1 + tc α -1 α -1 -b α -1 → φ(t). (30)
Moreover,

Q[( α -1 + 1) -1 (1 + tc α -1 )] = Q{ α -1 -1 [1 + tc α -1 ξ t ( α -1 )]},
where for all m ∈ N,

ξ t (m) := m 1 + m 1 - 1 tmc m .
Since mc m → ∞, we have ξ t (m) → 1 as m → ∞ . Dini's theorem together with (28) entail that

1 a α -1 Q 1 + tc α -1 α -1 + 1 -b α -1 → φ(t). (31)
Hence, by collecting (29), ( 30) and (31) we obtain

Q[α + tg(α)] -b(α) a(α -1 ) → φ(t), with g(α) = αc α -1 , b(α) = b α -1 and a(α -1 ) = a α -1 .
We conclude the proof by remarking that

Q[α + tg(α)] -Q(α) a(α -1 ) = Q[α + tg(α)] -b(α) a(α -1 ) + Q(α) -b(α) a(α -1 ) → φ(t) + φ(0) = φ(t).

Proof of Proposition 3 -It suffices to remark that

Q[α + tg(α)] -Q(α) a(α -1 ) = Q (α)g(α) a(α -1 ) t 0 Q [α + sg(α)] Q (α) ds.
The local uniform convergence (11) concludes the proof.

Proof of Proposition 4 -From Proposition 2, the TFO condition entails that nc n S(a

n t + b n ) -c n → φ -1 (t) as n → ∞ with c n = 1/[ng(1/n)], a n = a(n) and b n = Q(1/n).
First assume that α/g(α) → c as α → 0 with c > 0. We have that c n → c > 0 as n → ∞ and thus nS(a n t + b n ) → 1 + φ -1 (t)/c. In particular, we have that S(a n t + b n ) → 0 and thus that, letting

F := 1 -S, -nS(a n t + b n ) ∼ ln F n (a n t + b n ) as n → ∞. Hence, lim n→∞ F n (a n t + b n ) = G(t) = exp -1 + φ -1 (t) c ,
showing that condition ( 12) is satisfied. Now, let us consider the case c = 0. From Proposition 2, we have nc n S(a

n t + b n ) → φ -1 (t). Let m n := nc n = 1/g(1/n) =: g(n). Since g(α) → 0 as α → 0, m n → ∞ as n → ∞.
Since g is a continuous and increasing function, we have that g-1 (m) → ∞ as m → ∞.

Letting ãm := a g-1 (m) and bm := b g-1 (m) we obtain the convergence

lim m→∞ mS(ã m t + bm ) = φ -1 (t).
The end of the proof is similar to the one in the case c > 0.

Proof of Proposition 5 -Let us show that ( 16) implies ( 15), the converse being similar. Let g(α) = α 2 b(α -1 ). Since g(α)/α → 0 as α → 0, one has for all t ∈ R ∆(α, t) := α g(α)

1 + t g(α) α -1 -1 → -t,
as α → 0. Hence,

Q[α + tg(α)] Q(α) = U [α -1 + b(α -1 )∆(α, t)] U (α -1
) .

From Dini's theorem, the convergence ( 16) is locally uniform leading to (15).

Proof of Theorem 1

As explained in Section 4.2, the asymptotic normality of the conditional survival estimator is established in two steps: a) prove the asymptotic normality of the variance term and b) show that the bias term is negligible. The first step is a direct consequence of the following lemma.

Lemma 1 Let {V n,1 , V n,2 , . . . , V n,n } be a triangular array of independent copies of a centered random variable

V n . Assume that E(V 2 n ) = 1 and E(|V n | 3 ) < ∞. Let T n := {T n,i , 1 ≤ i ≤ n}
be a triangular array of positive random variables independent of the V n,i and such that T 2 n,1 + . . .

+ T 2 n,n = 1. For T n := max{T n,i , 1 ≤ i ≤ n}, if E(|V n | 3 )T n a.s. -→ 0 then n i=1 T n,i V n,i d -→ N (0, 1).
Proof -Let {t n,i , i = 1, . . . , n} be a triangular array of real numbers satisfying min (t n,i ; i = 1, . . . , n) ≥ 0 and

n i=1 t 2 n,i = 1. ( 32
)
Let t n := max (t n,i ; i = 1, . . . , n) and ν n := E(|V n | 3 ). In a first step, let us show that if ν n t n → 0 as n → ∞ then, for all z ∈ R,

lim n→∞ P n i=1 t n,i V n,i ≤ z = Φ(z), ( 33
)
where Φ is the cumulative distribution function of a N (0, 1) distribution. Since the V n,i are independent and centered random variables, it suffices to prove that the Lindeberg condition is satisfied, i.e., that

lim n→∞ n i=1 t 2 n,i E V 2 n,i I {tn,i|Vn,i|>ε} = 0,
for all ε > 0. Since t n,i ≤ t n for all i ∈ {1, . . . , n},

n i=1 t 2 n,i E V 2 n,i I {tn,i|Vn,i|>ε} ≤ n i=1 t 2 n,i E V 2 n,i I {tn|Vn,i|>ε} = E V 2 n I {tn|Vn|>ε} ,
since the V n,i are identically distributed and under (32). Hölder's inequality entails that

E V 2 n I {tn|Vn|>ε} ≤ ν 2/3 n [P (t n |V n | > ε)] 1/3 . Chebyshev's inequality ensures that P (t n |V n | > ε) ≤ t 2 n /ε 2 and thus E V 2 n I {tn|Vn|>ε} ≤ [ν n t n /ε] 2/3 → 0,
as n → ∞, by assumption. Convergence (33) is thus proved for all triangular array {t n,i , i = 1, . . . , n} satisfying (32) with ν n t n → 0. Now, remark that for all ω ∈ {ν n T n → 0}, convergence (33) entails that

P n i=1 T n,i V n,i ≤ z {T n,i = T n,i (ω); i = 1, . . . , n} -Φ(z) = P n i=1 T n,i (ω)V n,i ≤ z -Φ(z) → 0,
as n → ∞. Note that the last equality is true since the T n,i are independent of the V n,i . Hence, since P[ν n T n → 0] = 1,

lim n→∞ P n i=1 T n,i V n,i ≤ z {T n,i ; i = 1, . . . , n} -Φ(z) = 0 a.s. ( 34
)
To conclude the proof, let us remark that

P n i=1 T n,i V n,i ≤ z -Φ(z) ≤ E P n i=1 T n,i V n,i ≤ z {T n,i ; i = 1, . . . , n} -Φ(z) =: P n . Since P n i=1 T n,i V n,i ≤ z {T n,i ; i = 1, . . . , n} -Φ(z) ≤ 1,
the dominated convergence theorem entails that

lim n→∞ P n = E lim n→∞ P n i=1 T n,i V n,i ≤ z {T n,i ; i = 1, . . . , n} -Φ(z) = 0, from (34).
We can now establish the asymptotic normality of the variance term. Let σ 2 n (x 0 ) := S(y n (x 0 )|x 0 )[1 -S(y n (x 0 )|x 0 )] and recall that m n (x 0 ) is a sequence such that n x0 /m n (x 0 ) a.s.

-→ 1 and that v 2 n (x 0 ) = m n (x 0 )/S(y n (x 0 )|x 0 ). Proof -Remark that

n x0 σ 2 n (x 0 ) 1/2 S x0 n (y n (x 0 )) -S (y n (x 0 )|x 0 ) = n i=1 T n,i (x 0 )V n,i (x 0 ), with T n,i (x 0 ) := (n x0 ) 1/2 W n,i (x 0 ) and V n,i (x 0 ) := [σ n (x 0 )] -1 I {Y x 0 i >yn(x0)} -S(y n (x 0 )|x 0 ) .
It thus suffices to apply Lemma 1 after remarking that n x0 /σ 2 n (x 0 ) a.s.

∼ v 2 n (x 0 ) and that

E(|V n,1 (x 0 )| 3 ) = 1 σ 3 n (x 0 ) [S(y n (x 0 )|x 0 )] 3 [1 -S(y n (x 0 )|x 0 )] + [1 -S(y n (x 0 )|x 0 )] 3 S(y n (x 0 )|x 0 ) = σ -1 n (x 0 ) [S(y n (x 0 )|x 0 )] 2 + [1 -S(y n (x 0 )|x 0 )] 2 ∼ σ -1 n (x 0 ),
as n → ∞, since S(y n (x 0 )|x 0 ) → 0.

The second step of the proof is treated in the following result. Proof -Let U 1 , . . . , U n be independent uniform random variables independent of the X i . Since

Y x0 i = Q(U i |x 0 ) and Y i d = Q(U i |X i ) for all i ∈ {1, . . . , n}, B n (x 0 ) d = v n (x 0 ) n i=1 W n,i (x 0 ) I (-∞,S(yn(x0)|Xi)) -I (-∞,S(yn(x0)|x0)) (U i ).
From Owen (1987, Lemma 3.4.5), one has for all ε > 0,

P(|B n (x 0 )| > ε) ≤ ε + P E |B n (x 0 )| X > ε 2 ≤ ε + P v n (x 0 ) n i=1 W n,i (x 0 )E ∆ n,i (x 0 ) X > ε 2 ,
where X := (X 1 , . . . , X n ) and

∆ n,i (x 0 ) := I (-∞,S(yn(x0)|Xi)) -I (-∞,S(yn(x0)|x0)) (U i ).
Introducing the quantity D n,i (x 0 ) :

= |S(y n (x 0 )|X i ) -S(y n (x 0 )|x 0 )|, it is easy to check that E ∆ n,i (x 0 ) X ≤ E I [S(yn(x0)|x0)-Dn,i(x0),S(yn(x0)|x0)+Dn,i(x0)] (U i ) X ≤ 2D n,i (x 0 ). Remarking that n i=1 W n,i (x 0 )D n,i (x 0 ) = W 1 W * n,x0 , δ * x0 leads to P(|B n (x 0 )| > ε) ≤ ε + P v n (x 0 )W 1 W * n,x0 , δ * x0 > ε 2 /2 .
The result is thus proved by using assumption (23) (or equivalently (22)).

Theorem 1 is thus proved by gathering Propositions 7 and 8.

Proof of Theorem 2

The proof follows the lines described in Section 2. Let us introduce the sequences t

-1 n (x 0 ) := -v n (x 0 )g(α n ) and σ -1 n (x 0 ) = a(α -1 n )t n (x 0 ). It is easy to check that for all z ∈ R, P σ n (x 0 )[ Q n (α n |x 0 ) -Q(α n |x 0 )] ≤ z = P {Z n (x 0 ) ≤ z n (x 0 )} , where y n (x 0 ) := Q(α n |x 0 ) + σ -1 n (x 0 )z, z n (x 0 ) = v n (x 0 )[α n -S(y n (x 0 ) x 0 )] and Z n (x 0 ) := v n (x 0 )[ S n (y n (x 0 )|x 0 ) -S(y n (x 0 )|x 0 )]. From Proposition 1, condi- tion (H) entails that for all t 0 ∈ I, lim (t,y)→(t0,y * (x0)) Ψ(y) t S[y + td(y)|x 0 ] S(y|x 0 ) -1 = lim t→t0 φ -1 (t) t . ( 35
) Since y n (x 0 ) = Q(α n |x 0 ) + a(α -1 n )t n (x 0 )z = Q(α n |x 0 ) + d(Q(α n |x 0 ))t n (x 0 )z with t n (x 0 ) → 0 as n → ∞, (35) entails that as n → ∞ z n (x 0 ) ∼ -zv n (x 0 )t n (x 0 )g(α n ) = z. (36)
Now, to prove that Z n (x 0 ) d -→ N (0, 1), it suffices to show that conditions (21) and ( 22) hold for y n (x 0 ). From (36),

1 - S[y n (x 0 )|x 0 ] α n ∼ zα -1 n v -1 n (x 0 ) = z(α n m n (x 0 )) -1/2 → 0, (37)
as n → ∞ and thus S[y n (x 0 )|x 0 ] ∼ α n . This entails that condition

v n (x 0 ) max 1≤i≤n W n,i (x 0 ) a.s.
-→ 0 is equivalent to condition (21) with y n (x 0 ). It remains to prove condition (22). From (37), there exists ξ > 0 such that for n large enough, S(y

n (x 0 )|x 0 ) ∈ [(1 -ξ)α n , (1 + ξ)α n ]. Hence, for n large enough, n i=1 W n,i (x 0 ) S(y n (x 0 )|X i ) S(y n (x 0 )|x 0 ) -1 ≤ sup |β/αn-1|≤ξ n i=1 W n,i (x 0 ) S[Q(β|x 0 )|X i ] β -1 ,
and the proof is complete.

Proof of Corollaries 1, 2 and 3

We first recall a useful result dealing with the almost sure convergence of the statistic fn (

x) := 1 nh p n n i=1 K X i -x 0 h n ,
which is the kernel estimator of the density f of the random value X.

(x) = L( x )
where L is of bounded variation (see Giné and Nickl, 2015, Exercice 3.6.13).

For the sake of simplicity, we have preferred to replace in Lemma 2 all the conditions involving the kernel function by the stronger (but simpler to check) condition (K).

Corollaries 1, 2 and 3 are direct consequences of Theorem 2 and of the three following lemmas establishing the asymptotic distribution of the corresponding conditional survival function estimators.

Lemma 3 Let x 0 ∈ R p such that f is continuous at x 0 and f (x 0 ) > 0 and let K be a kernel satisfying (K). For sequences h n → 0 and y

n (x 0 ) ↑ y * (x 0 ) such that nh p n [S(y n (x 0 )|x 0 ) ∧ (log log n) -1 ] → ∞ and sup x-x0 ≤hn S(y n (x 0 )|x) S(y n (x 0 )|x 0 ) -1 2 = o 1 nh p n S(y n (x 0 )|x 0 ) , one has
(nh p n S(y n (x 0 )|x 0 )) 1/2 S NW n (y n (x 0 )|x 0 ) S(y n (x 0 )|x 0 ) -1 d -→ N 0, K 2 2 f (x 0 ) . Proof of Lemma 3 -Let K := K 2 / K 2 2 where K 2 2 := Up K 2 (y)dy.
It is easy to check that K also satisfy condition (K). Hence, Lemma 2 entails that almost surely,

lim n→∞ K 2 2 nh p n n x0 = lim n→∞ f 2 n (x 0 ) 1 nh p n n i=1 K x 0 -X i h n = f (x 0 ).
Hence, almost surely, n x0 ∼ f (x 0 )nh p n / K 2 2 =: m n (x 0 ). It is easy to infer that, as soon as nh p n S(y n (x 0 )|x 0 ) → ∞, we have m n (x 0 ) S(y n (x 0 )|x 0 ) max

1≤i≤n W NW n,i (x 0 , h n ) 2 ≤ f (x 0 ) K 2 2 1 nh p n S(y n (x 0 )|x 0 ) K 2 ∞ f 2 n (x 0 )
a.s.

-→ 0.

Similarly, using Assumption (K), we have

I [0,1] X i -x 0 h n ∞ I [0,1] r(i) k n = k n ∧ M n .
Then, straightforward calculation shows that

n -1 x0 = τ 2 M n + 2τ (1 -τ ) k n ∨ M n + (1 -τ ) 2 k n .
Next, since by assumption nh p n / log log n → ∞ and since the uniform kernel satisfies condition (K), Lemma 2 ensures that (2h n ) -p n -1 M n a.s.

-→ f (x 0 ) as n → ∞. Hence, as a first conclusion, n x0 ∼ n 2 p f (x 0 )C -2 (κ) =: m n (x 0 ) almost surely. Furthermore,

max 1≤i≤n W LC n,i (x 0 , τ, h n , k n ) ≤ τ M n + 1 -τ k n .
Hence, using again the almost sure convergence (2h n ) -p n -1 M n → f (x 0 ), lim

Proof of Proposition 6

Proposition 6 is a consequence of the following lemma.

Lemma 6 Let (X, Y, Z) be a random vector for which (X, Y ) and Z are independent. Let g be a measurable function such that g(X, Y, Z) is integrable. One Proof -Since (X, Y ) and Z are independent E[g(X, Y, Z)] = g(x, y, z)P (X,Y ) (dx, dy)P Z (dz).

Hence E[g(X, Y, Z)] = g(x, y, z)P Y (dy|X = x) P X (dx)P Z (dz) = Ψ(x, z)P X (dx)P Z (dz).

The conclusion follows since X and Z are independent.

Proof of Proposition 6 -First remark that the assumption on the weights entails that the W n,i (x 0 ) are identically distributed. Furthermore, since the W n,i (x 0 ) sum to 1, it is clear that E[W n,1 (x 0 )] = . We apply Lemma 6 with X = X 1 , Y = Y 1 , Z = X -1 and g(t, y, u) = ϕ(t, u)I {y>φ(αn,t,u)} where the function φ is such that

Q n,-1 (α n |X 1 , λ) = φ(α n , X 1 , X -1 ).
The conclusion is straightforward since, with the notation of Lemma 6, Ψ(t, u) = ϕ(t, u)S(φ(α n , t, u)|t).

( H )

 H The conditional survival function S(•|x 0 ) satisfies a TFO condition with positive auxiliary functions Ψ x0 ≡ Ψ and d x0 ≡ d.

  H)), the asymptotic normality of the Nadaraya-Watson conditional quantile estimator has already been obtained in Daouia et al. (2013, Corollary 1). This last result also requires the use of condition (25) which controls the oscillations of the function Q(α n |•). Of course, the proof of Daouia et al. (2013, Corollary 1) uses arguments adapted to the Nadaraya-Watson's estimator while Theorem 2 can be used for a large range of weighted conditional survival estimators. As a consequence, conditions on h n and α n involved in Daouia et al. (2013, Corollary 1) and in our Corollary 1 are slightly different. More precisely, the conditions in Daouia et al. (2013, Corollary 1) are nh p n α n → ∞ and nh p+2 n α n → 0 while in our Corollary 1 it is required that nh p n α n → ∞ and nh p n (log log n) -1 → ∞. Hence, if α n log log n → 0 as n → ∞ (i.e., for large quantiles), conditions on the sequences h n and α n are weaker in Corollary 1 than in Daouia et al. (2013, Corollary 1).

Proposition 7

 7 For x 0 ∈ R p , let y n (x 0 ) be a sequence converging to the right endpoint y * (x 0 ) of the conditional distribution of Y given {X = x 0 }. If condition (21) holds then v n (x 0 ) S x0 n (y n (x 0 )) -S (y n (x 0 )|x 0 ) d -→ N (0, 1).

Proposition 8

 8 Let x 0 ∈ R p and y n (x 0 ) be a sequence converging to the right endpoint y * (x 0 ) of the conditional distribution of Y given {X = x 0 }. If condition (22) holds then v n (x 0 ) S n (y n (x 0 )|x 0 ) -S x0 n (y n (x 0 )) P -→ 0.

1 from which Lemma 3 follows according to Theorem 1 . 4 2 = o 1 k 1 d- 1 .i 2 → 1 , 2 = 2 + 1 Lemma 5 2 = o 1 n 2 S 1 d-→ N 0, C 2

 114211121221521212 (x 0 , h n ) S(y n (x 0 )|X i ) S(y n (x 0 )|x 0 ) -1 ≤ sup x-x0 ≤hn S(y n (x 0 )|x) S(y n (x 0 )|x 0 ) -Lemma Let x 0 ∈ R p . For sequences k n and y n (x 0 ) such that, as n → ∞, y n (x 0 ) ↑ y * (x 0 ), k n S(y n (x 0 )|x 0 ) → ∞ and sup x-x0 ≤D (kn ) (x0) S(y n (x 0 )|x) S(y n (x 0 )|x 0 ) -1 n S(y n (x 0 )|x 0 ) , with D (kn) (x 0 ) = X r(kn) -x 0 , one has (k n S(y n (x 0 )|x 0 )) (x 0 )|x 0 ) S(y n (x 0 )|x 0 ) -Proof of Lemma 4 -First, remark that since k n → ∞ as n → ∞,as n → ∞. Thus, n x0 ∼ m n (x 0 ) with m n (x 0 ) = (2 + 1)/( + 1) 2 k n . As soon as k n S(y n (x 0 )|x 0 ) → ∞, we havem n (x 0 ) S(y n (x 0 )|x 0 ) max 1≤i≤n W NN n,i (x 0 , k n ) k n S(y n (x 0 )|x 0 ) → 0.Using the boundn i=1 W NN n,i (x 0 , k n ) S(y n (x 0 )|X i ) S(y n (x 0 )|x 0 ) -1 ≤ sup x-x0 ≤D (kn) (x0) S(y n (x 0 )|x) S(y n (x 0 )|x 0 ) -1 ,we prove Lemma 4 by applying Theorem 1.Let x 0 ∈ R p such that f is continuous at x 0 and f (x 0 ) > 0. Let h n , k n and y n (x 0 ) ↑ y * (x 0 ) be sequences such that nh p n / log log n → ∞, n S(y n (x 0 )|x 0 ) → ∞ with n := (nh p n ∧ k n ) and sup x-x0 ≤(hn∨D (kn) (x0)) S(y n (x 0 )|x) S(y n (x 0 )|x 0 ) -1 S(y n (x 0 )|x 0 ) . If there exists κ ∈ [0, ∞] such that k n /(nh p n ) → κ then ( n S(y n (x 0 )|x 0 )) 1/LC n (y n (x 0 )|x 0 ) S(y n (x 0 )|x 0 ) -(κ) 2 p f (x 0 ) ,where C 2 (κ) := (1 ∧ κ -1 ) κτ 2 + 2 p f (x 0 )(1 -τ ) 2 + 2τ (1 -τ ) (κ ∧ 2 p f (x 0 )) .Proof of Lemma 5 -We start by remarking that n i=1

1 ≤ 1 ,

 11 n,i (x 0 , τ, h n , k n ) = τ (κ ∧ 1) 2 p f (x 0 ) + (1 -τ )(κ -1 ∧ 1), almost surely for all κ ∈ [0, ∞]. As a consequence, since n S(y n (x 0 )|x 0 ) → ∞, condition(21)is satisfied. Finally, using the bounds obtained in the proofs of Lemmas 3 and 4, one hasn i=1 W LC n,i (x 0 , τ, h n , k n ) S(y n (x 0 )|X i ) S(y n (x 0 )|x 0 ) sup x-x0 ≤hn∨D (kn) (x0)S(y n (x 0 )|x) S(y n (x 0 )|x 0 ) and thus condition (22) holds. Theorem 1 concludes the proof.

  has E[g(X, Y, Z)] = E[Ψ(X, Z)], where Ψ(x, z) := E[g(x, Y, z)|X = x].

1 n n i=1 W

 1i=1 . . = E[W n,n (x 0 )] = 1/n. It thus remains to show that E n,i (x 0 )I {Yi> Qn,-i(αn|Xi,λ)} = E W n,1 (x 0 )I {Y1> Qn,-1(αn|X1,λ)} = E W n,1 (x 0 )S[ Q n,-1 (α n |X 1 , λ)|X 1 ] .

  a natural general condition leading to (6) is to assume that for some open interval I x0 = I ⊂ R containing 0, there exist positive functions d x0 ≡ d and Ψ x0 ≡ Ψ such that for all t ∈ I,

	lim y↑y * (x0)

  The following result can be found for instance in Dony and Einmahl (2009, Corollary 2.1). Let x ∈ R p such that f is continuous at x and f (x) > 0. If the kernel K is a bounded density with support included in the unit ball U p of R p and ifK := {K(γ(t -•)), γ > 0, t ∈ R p },is a pointwise measurable Vapnik-Chervonenkis (VC) type class of functions from R p to R then for a sequence h n → 0 such that nh p n / log log n → ∞, we have that fn (x) Conditions on the family K of functions are not easy to check in practice. Nevertheless, the measurability condition on K is satisfied whenever K is rightcontinuous (see Einmahl and Mason, 2005) or K is an indicator function on a cell of R p (see van der Vaart and Wellner, 1996, Example 2.3.4). Concerning the VC condition, it is satisfied for kernel function K such that K

	Lemma 2

a.s.

-→ f (x).

[17] Watson, G.S. (1964). Smooth regression analysis. Sankhya A, 26, 359-372.

27 Table 2: ERROR based on 500 samples of size n = 1000 according to the model M2, for three different values of θ 1 .

(1) 0 0.00476 0.00501 0.00541 0.00190 0.00198 0.00217

0.00177 0.00177 0.00197 0.00063 0.00061 0.00068 θ (x

0.00956 0.01077 0.01076 0.00436 0.00489 0.00509 θ 2 (x 0.00672 0.00751 0.00607 0.00591 0.00728 0.00542 µ(x