Ouerdia Arkoun

Serguei Pergamenchtchikov

Sequential Robust Estimation for Nonparametric Autoregressive Models

Keywords: Adaptive estimation, Nonparametric autoregression, Robust efficiency, Sequential kernel estimator. Subject Classifications: primary 62G07,62G08, secondary 62G20

We construct a robust truncated sequential estimators for the pointwise estimation problem in nonparametric autoregression models with smooth coefficients. For Gaussian models we propose an adaptive procedure based on the constructed sequential estimators. The minimax nonadaptive and adaptive convergence rates are established. It turns out that in this case these rates are the same as for regression models..

INTRODUCTION

One of the standard linear models in the general theory of time series is the autoregressive model (see, for example, [START_REF] Anderson | The Statistical Analysis of Time Series[END_REF] and the references therein). Natural extensions for such models are nonparametric au-Address correspondence to O. Arkoun, Laboratoire de Mathématiques Raphaël Salem, CNRS, UMR 6085, Avenue de l'université, BP 12, 76801 Saint-Etienne du Rouvray Cedex France, E-mail: Ouerdia.Arkoun@supbiotech.fr S. Pergamenchtchikov is partially supported by the grant of RSF number 14-49-00079 (National Research University "MPEI" 14 Krasnokazarmennaya, 111250 Moscow, Russia) and RFBR Grant 16-01-00121 A toregressive models which are defined by

y k = S(x k )y k-1 + ξ k , 1 ≤ k ≤ n , (1.1)
where S(•) is unknown function, the design x k = k/n, y 0 is a constant and the noise (ξ k ) 1≤k≤n are i.i.d. unobservable centered random variable N (0, 1).

It should be noted that the varying coefficient principle is well known in the regression analysis. It permits the use of a more complex forms for regression coefficients and, therefore, the models constructed via this method are more adequate for applications (see, for instance, [START_REF] Fan | Statistical Methods with Varying Coefficient Models[END_REF], [START_REF] Luo | Nonparametric Estimation of the Production Function with Time-Varying Elasticity Coefficients[END_REF]). In this paper we consider the varying coefficient autoregressive models (1.1). There is a number of papers which consider these models such as [START_REF] Dahlhaus | Maximum Likelihood Estimation and Model Selection for Locally Stationary Processes[END_REF], [START_REF] Dahlhaus | On the Kullback-Leibler Information Divergence of Locally Stationary Processes[END_REF] and [START_REF] Belitser | Recursive Estimation of a Drifted Autoregressive Parameter[END_REF]. In all these papers, the authors propose some asymptotic (as n → ∞) methods for different identification studies without considering optimal estimation issues. To our knowledge, for the first time the minimax estimation problem for the model (1.1) has been treated in [START_REF] Arkoun | Nonparametric Estimation for an Autoregressive Model[END_REF] and [START_REF] Moulines | On Recursive Estimation for Time Varying Autoregressive Processes[END_REF] in the nonadaptive case, i.e. for the known regularity of the function S. Then, in [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF] it is proposed to use the sequential analysis method for the adaptive pointwise estimation problem in the case when the unknown Hölder regularity is less than one, i.e when the function S is not differentiable. In the case of the model (1.1), the adaptive pointwise estimation is possible only in the sequential framework. That is why, in this paper, we study sequential estimation methods for the smooth function S. We consider the pointwise estimation at a fixed point z 0 ∈]0; 1[ in the two cases: when the Hölder regularity is known and when the Hölder regularity is unknown, i.e. the adaptive estimation. In the first case we consider this problem in the robust setting, i.e. we assume that the distribution of the random variables (ξ j ) j≥1 in (1.1) belongs to some functional class and we consider the estimation problem with respect to robust risks which have an additional supremum over all distributions from some fixed class. For nonparametric regression models such risks was introduced in [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF] for the pointwise estimation problem and in [START_REF] Galtchouk | Adaptive Asymptotically Efficient Estimation in Heteroscedastic Nonparametric Regression[END_REF] for the quadratic risks. Later, for the quadratic risks the same approach was used in [START_REF] Konev | Efficient Robust Nonparametric Estimation in a Semimartingale Regression Model[END_REF] for regression model in continuous time. Motivated by this facts, we consider the adaptive estimation problem for the Gaussian models (1.1). More precisely, we assume that the function S belongs to a Hölder class with some unknown regularity 1 < β ≤ 2. Unfortunately, we can not use directly the sequential procedure from [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF] for the adaptive estimation of such functions. Since to obtain an optimal rate for the function with β > 1 we have to take into account the Taylor expansion of the function S at z 0 of the order 1. To study the Taylor expansion for sequential procedures one needs to control the behavior of the stopping time. Indeed, one needs to keep the stopping time near of the number of observations. This can not be done by the procedure from [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF] since one needs to use the unknown function S. In this paper we construct a sequential adaptive estimate for smooth functions and we find an adaptive minimax convergence rate for smooth functions. In Section 2, we present the standard notations used in the sequel of the paper. We describe in details the statement of the problem and main results in Section 3. In Section 4, we will study some properties of kernel estimators for the model (1.1) and in Section 5, we study properties of stopping time for the constructed sequential procedure. Section 6 is devoted to the asymptotic upper bound and lower bound for the risk of the sequential kernel estimators. In Section 7, we illustrate the obtained results by numerical examples. Finally, we give an appendix which contains some technical results.

SEQUENTIAL PROCEDURES

Main Conditions

We assume that in the model (1.1) the i.i.d. random variables (ξ k ) 1≤k≤n have a density p (with respect to the Lebesgue measure) from the functional class P ς defined as

P ς := p ≥ 0 : +∞ -∞ p(x) dx = 1 , +∞ -∞ x p(x) dx = 0 , +∞ -∞ x 2 p(x) dx = 1 and sup k≥1 1 ς k (2k -1)!! +∞ -∞ |x| 2k p(x) dx ≤ 1 , (2.1)
where ς ≥ 1 is some fixed parameter. Note that the (0, 1)-Gaussian density belongs to P. In the sequel we denote this density by p 0 . It is clear that for any q > 0

s * q = sup p∈P ς E p |ξ 1 | q < ∞ , (2.2)
where E p is the expectation with respect to the density p from P ς . To obtain the stable (uniformly with respect to the function S ) model (1.1), we assume that for some fixed 0 < ε < 1 and L > 0 the unknown function S belongs to the εstability set

Θ ε,L = S ∈ C 1 ([0, 1], R) : S ≤ 1 -ε and Ṡ ≤ L , (2.3)
where C 1 [0, 1] is the Banach space of continuously differentiable [0, 1] → R functions and S = sup 0≤x≤1 |S(x)|. Similarly to [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF] and [START_REF] Arkoun | Nonparametric Estimation for an Autoregressive Model[END_REF] we make use of the family of the weak stable local Hölder classes at the point z 0

U (β) n (ε, L, * n ) = S ∈ Θ ε,L : |Ω h (z 0 , S)| ≤ * n h β , (2.4)
where

Ω h (z 0 , S) = 1 -1 (S(z 0 + uh) -S(z 0 )) du ,
the positive parameter h is defined later and β = 1 + α is the regularity parameter with 0 < α < 1.

Moreover, we assume that the weak Hölder constant * n goes to zero, i.e. * n → 0 as n → ∞. Moreover, we define the corresponding strong stable local Hölder class at the point z 0 as

H (β) (ε, L, L * ) = S ∈ Θ ε,L : Ω * (z 0 , S) ≤ L * , (2.5)
where

Ω * (z 0 , S) = sup x∈[0,1] | Ṡ(x) -Ṡ(z 0 )| |x -z 0 | α .
We assume that the regularity β ≤ β ≤ β, where β = 1 + α and β = 1 + α for some fixed parameters 0 < α < α < 1.

Remark 2.1. Note that for the regression models the weak Hölder class was introduced in [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF] for the efficient pointwise estimation. It is clear that it is more large than usual one with the same Hölder constant, i.e.

H (β) (ε, L, * n ) ⊆ U (β) n (ε, L, * n ) .
It should be noted also that for diffusion processes the local weak Hölder class was used in [START_REF] Galtchouk | Nonparametric Sequential Minimax Estimation of the Drift Coefficient in Diffusion Processes[END_REF] and [START_REF] Galtchouk | Asymptotically Efficient Sequential Kernel Estimates of the Drift Coefficient in Ergodic Diffusion Processes[END_REF] for sequential and truncated sequential efficient pointwise estimation respectively. Moreover, in [START_REF] Galtchouk | Adaptive Sequential Estimation for Ergodic Diffusion Processes in Quadratic Metric[END_REF] these sequential pointwise efficient estimators were used to construct adaptive efficient model selection procedures in L 2 for diffusion processes.

Nonadaptive Procedure

First, we study a nonadaptive estimation problem for the function S from the functional class (2.4) of the known regularity β = 1 + α. As we will see later to construct an efficient sequential procedure we need to use S as a procedure parameter. So we propose to use the first ν observations for the auxiliary estimation of S(z 0 ). In this step we use the usual kernel estimate, i.e.

S ν = 1 A ν ν j=1 Q(u j ) y j-1 y j , A ν = ν j=1 Q(u j ) y 2 j-1 , (2.6)
where the kernel Q(•) is the indicator function of the interval [-1; 1]; u j = (x j -z 0 )/h and h is some positive bandwidth. In the sequel for any 0 ≤ k < m ≤ n we set

A k,m = m j=k+1 Q(u j ) y 2 j-1 , (2.7)
i.e. A ν = A 0,ν . It is clear that to estimate S(z 0 ) on the basis of the kernel estimate with the kernel Q we can use the observations (y j ) k * ≤j≤k * , where

k * = [nz 0 -nh] + 1 and k * = [nz 0 + nh] . (2.8)
Here [a] is the integral part of a number a. So for the first estimation we chose ν as

ν = ν(h, α) = k * + ι , (2.9)
where

ι = ι(h, α) = [ nh] + 1 and = (h, α) = h α / ln n .
Next, similarly to [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF], we use a some kernel sequential procedure based on the observations (y j ) ν≤j≤n . To transform the kernel estimator in the linear function of observations and we replace the number of observations n by the following stopping time

τ H = inf{k ≥ ν + 1 : A ν,k ≥ H}, (2.10)
where inf{∅} = n and the positive threshold H will be chosen as a positive random variable measurable with respect to the σ -field {y 1 , . . . , y ν }. Therefore, we get

S * h = 1 H   τ H -1 j=ν+1 Q(u j ) y j-1 y j + κ H Q(u τ H ) y τ H -1 y τ H   1 (A ν,n ≥H) , (2.11)
where the correcting coefficient κ H on the set {A ν,τ H ≥ H} is defined as

A ν,τ H -1 + κ H Q(u τ H ) y 2 τ H -1 = H
and κ H = 1 on the set {A ν,τ H < H}. Now, to obtain an efficient estimate we need to use the all n observations, i.e. asymptotically for sufficiently large n the stopping time τ H ≈ n. Similarly to [START_REF] Konev | Estimate of the Number of Observations in Sequential Identification of the Parameters of Dynamical Systems[END_REF], one can show that τ H ≈ γ(S) H as H → ∞, where γ(S) = 1 -S 2 (z 0 ) .

(2.12)

Therefore, to use the all asymptotically observations we have to chose H as the number of observations divided by γ(S). But in our case we use k * -k * observations to estimate S(z 0 ), Therefore, to obtain optimal estimate we need to define H as (k * -k * )/γ(S). Taking into account that k * -k * ≈ 2nh and that γ(S) is unknown we define the threshold H as

H = H(h, α) = φnh , φ = φ(h, α) = 2(1 -) γ( S ν ) , (2.13)
where S ν is the projection of the estimator S ν in the interval ] -1 + ε, 1 -ε[, i.e.

S ν = min(max( S ν , -1 + ε), 1 -ε) .

In this paper we chose the bandwidth h in the following form

h = h(β) = (κ n ) 1 2β+1 , (2.14)
where the sequence κ n is positive, such that

κ * = lim inf n→∞ nκ n > 0 and lim n→∞ n δ κ n = 0 (2.15)
for any 0 < δ < 1.

Adaptive Procedure

We will construct an adaptive minimax sequential estimation for the function S from the functional class (2.5) of the unknown regularity β. To this end we will use the modification of the adaptive Lepskii method proposed in [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF] based on the sequential estimators (2.11). We set

d n = n ln n and N (β) = (d n ) β 2β+1 . (2.16)
Moreover, we chose the bandwidth h in the form (2.14) with κ n = 1/d n , i.e. we set

ȟ = ȟ(β) = 1 d n 1 2β+1
.

(2.17)

We define the grids on the intervals [β , β] and [α , α] as

β k = β + k m (β -β) and α k = α + k m (α -α) (2.18)
for 0 ≤ k ≤ m with m = [ln d n ] + 1, and we set

N k = N (β k ) and ȟk = ȟ(β k ) .
Replacing in (2.9) and (2.13) the parameters h and α we define

νk = ν( ȟk , α k ) and Ȟk = H( ȟk , α k ) .
Now using these parameters in the estimators (2.6) and (2.11) we set Šk = S * ȟk (z 0 ) and

ωk = max 0≤j≤k | Šj -Šk | - λ N j , (2.19)
where

λ > λ * = 4 √ 2 β -β (2β + 1)(2β + 1) 1/2 .
In particular, if β = 1 and β = 2 we get λ * = 4(2/15) 1/2 . We also define the optimal index as

ǩ = max 0 ≤ k ≤ m : ωk ≤ λ N k . (2.20)
The adaptive estimator is now defined as S a,n = S * ȟk and ȟk = ȟǩ .

(2.21)

Remark 2.2. It should be noted that in the difference from the usual adaptive pointwise estimation (see, for example, [START_REF] Lepskiȋ | A Problem of Adaptive Estimation in Gaussian White Noise[END_REF], [START_REF] Galtchouk | Sequential Nonparametric Adaptive Estimation of the Drift Coefficient in Diffusion Processes[END_REF], [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF]) the threshold λ in (2.19) does not depend on the parameters L > 0 and L * > 0 of the Hölder class (2.5).

MAIN RESULTS

Robust Efficient Estimation

The problem is to estimate the function S(•) at a fixed point z 0 ∈]0, 1[, i.e. the value S(z 0 ). For this problem we make use of the risk proposed in [START_REF] Arkoun | Nonparametric Estimation for an Autoregressive Model[END_REF]. Namely, for any estimate S = S n (z 0 ) (i.e. any measurable with respect to the observations (y k ) 1≤k≤n function) we define the following robust risk

R n ( S n , S) = sup p∈P ς E S,p | S n (z 0 ) -S(z 0 )| , (3.1)
where E S,p is the expectation taken with respect to the distribution P S,p of the vector (y 1 , ..., y n ) in (1.1) corresponding to the function S and the density p from P ς . With the help of the function γ(S) defined in (2.12), we describe the sharp lower bound for the minimax risks with the normalizing coefficient

ϕ n = n β 2β+1 . (3.2) Theorem 3.1. For any 0 < ε < 1 lim n→∞ inf S sup S∈U (β) n (ε,L, * n ) γ -1/2 (S) ϕ n R n ( S n , S) ≥ E|η| , (3.3)
where η is a Gaussian random variable with the parameters (0, 1/2).

Now we give the upper bound for the minimax risk of the sequential kernel estimator defined in (2.11).

Theorem 3.2. The estimator (2.11) with the parameters (2.13) -(2.14) and κ n = n -1 satisfies the following inequality

lim n→∞ sup S∈U (β) n (ε,L, * n ) γ -1/2 (S) ϕ n R n (S * h , S) ≤ E|η| ,
where η is a Gaussian random variable with the parameters (0, 1/2).

Remark 3.1. Theorems 3.1 and 3.2 imply that the estimator (2.11), with the parameters (2.14) is asymptotically robust efficient with respect to class P ς .

Adaptive Estimation

Now we consider the Gaussian model (1.1), i.e. assume that the random variables (ξ j ) j≥1 are N (0, 1).

The problem is to estimate the function S at a fixed point z 0 ∈]0, 1[, i.e. the value S(z 0 ). For any estimate Sn of S(z 0 ) (i.e. any measurable with respect to the observations (y k ) 1≤k≤n function), we define the adaptive risk for the functions S from

H (β) (ε, L, L * ) as R a,n ( Sn ) = sup β∈[β;β] sup S∈H (β) (ε,L,L * ) N (β) E S | Sn -S(z 0 )| , (3.4)
where N (β) is defined in (2. [START_REF] Galtchouk | Adaptive Sequential Estimation for Ergodic Diffusion Processes in Quadratic Metric[END_REF]), E S = E S,p 0 is the expectation taken with respect to the distribution P S = P S,p 0 .

First we give the lower bound for the minimax risk. We show that with the convergence rate N (β) the lower bound for the minimax risk is strictly positive. Theorem 3.3. There exists L * 0 > 0 such that for all L * > L * 0 , the risk (3.4) admits the following lower bound:

lim inf n→∞ inf Sn R a,n ( Sn ) > 0 ,
where the infimum is taken over all estimators Sn .

The proof of this theorem is given in [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF]. Now we give the upper bound for the minimax risk of the sequential kernel estimator defined in (2.11). To obtain an upper bound for the adaptive risk (3.4)

(Υ n (h)) -1 sup S∈H (β) (ε,L,L * ) sup p∈P ς E S,p |S * h -S(z 0 ))| < ∞ where Υ n (h) = h β + (nh) -1/2 .
Using this theorem we can establish the minimax property for the procedure (2.21).

Theorem 3.5. The estimation procedure (2.21) satisfies the following asymptotic property

lim sup n→∞ R a,n ( S a,n ) < ∞ . ( 3

PROPERTIES OF S ν

We start with studying the properties of the estimate (2.6). To this end for any q > 1 we set

* q = (12(1 + κ * )) q (εκ * ) 2q b * q r * q s * q + s * 2q + 1 + 2(1 + L) q r * 2q , (4.1)
where r * q = 2 q-1 |y 0 | q + s * q 1 ε q and b * q = 18 q q 3q/2 (q -1) q/2 . Now we obtain a non asymptotic upper bound for the tail probability for the deviation

∆ ν = S ν -S(z 0 ) . (4.2)
Lemma 4.1. For any q > 1, h > 0 and a > Lh

sup S∈Θ ε,L sup p∈P ς P S,p | ∆ ν | > a ≤ M 1,q (ln n) q h (1-α)q + M 2,q [ι(a -Lh) 2 ] q/2 ,
where M 1,q = 2 q * q and M 2,q = 2 q b * q s * q r * q .

Proof. First, we write the estimation error as follows

∆ ν = B ν + 1 A ν ζ ν ,
where

ζ ν = ν j=1 Q(u j ) y j-1 ξ j and B ν = 1 A ν ν j=1 Q(u j ) (S(x j ) -S(z 0 )) y 2 j-1 .
Note that |B ν | ≤ Lh for any S ∈ Θ ε,L . Putting v = ι/2 we can write

P S,p | ∆ ν | > a = P S,p | ∆ ν | > a , A ν < v + P S,p | ∆ ν | > a , A ν ≥ v ≤ P S,p (A ν < v) + P S,p Lh + |ζ ν | A ν > a, A ν ≥ v ≤ P S,p (A ν < v) + P S,p |ζ ν | A ν > a -Lh, A ν ≥ v . (4.3) Now, for any R → R function f and numbers 0 ≤ k ≤ m -1 we set k,m (f ) = 1 nh m j=k+1 f (u j ) y 2 j-1 - 1 γ(S) 1 nh m j=k+1 f (u j ) . (4.4)
Using this function we can estimate the first term on the left-hand side of (4.3) as

P S,p (A ν < v) = P S,p   k * ,ν (Q) + 1 nh γ(S) ν j=k * +1 Q(u j ) < v nh   = P S,p k * ,ν (Q) < - ι 2nh ≤ P S,p | k * ,ν (Q)| > /2 ≤ 2 q E S,p | k * ,ν (Q)| q .
Therefore, using here Lemma 8.3 we get

P S,p (A ν < v) ≤ 2 q R q * q (ln n) q h (1-α)q ,
where the coefficient * q is defined in (4.1). The last term on the right-hand side of (4.3) can be estimated as

P S,p 1 A ν |ζ ν | > a -Lh , A ν ≥ v ≤ P S,p (|ζ ν | > v(a -Lh)) ≤ 1 v q (a -Lh) q E S,p |ζ ν | q .
Now in view of the Burkhölder inequality, it comes

E S,p |ζ ν | q = E S,p   ν j=1 Q(u j ) y j-1 ξ j   q ≤ b * q E S,p   ν j=1 Q 2 (u j ) y 2 j-1 ξ 2 j   q/2 = b * q E S,p   k * +ι j=k * +1 y 2 j-1 ξ 2 j   q/2
, and after applying the Hölder inequality, we obtain

E S,p |ζ ν | q ≤ b * q ι q/2-1 k * +ι j=k * +1
E S,p y q j-1 ξ q j ≤ b * q s * q r * q ι q/2 . Therefore,

P S,p 1 A ν |ζ ν | > a -Lh, A ν ≥ v ≤ 2 q b * q s * q r * q ι q/2 (a -Lh) q .
Hence Lemma 4.1

Lemma 4.2. Let the bandwidth h be defined by the conditions (2.14)-(2.15). Then, for all m ≥ 1 and

0 < δ < 1 lim n→+∞ sup α≤α≤α h -m sup S∈Θ ε,L sup p∈P ς P S,p | ∆ ν | > δ = 0 .
Proof. By applying Lemma 4.1 for a = δ , we obtain that for sufficiently large n ≥ 1, for which δ > 2Lh, and for any m ≥ 1 and q > m/(1 -α)

h -m P S,p (| ∆ ν | > δ ) ≤ h -m M 1,q (ln n) q h (1-α)q + M 2,q h -m (ι(δ -Lh) 2 ) q/2 ≤ M 1,q (ln n) q κ (1-α)q-m 2β+1 n + 2 q M 2,q δ q h -(m+q/2) n q/2 3q/2 n ≤ M 1,q (ln n) q κ (1-α)q-m 2β+1 n + 2 q M 2,q (ln n) 3q/2 δ q (κ n n) q/2 κ q(1-α/2)-m 2β+1 n .
Taking into account here the conditions (2.15) we come to Lemma 4.2.

PROPERTIES OF STOPPING TIME τ H

First we need to study some asymptotic properties of the term (2.10).

Lemma 5.1. Assume that the threshold H is chosen in the form (2.13) and the bandwidth h satisfies the conditions (2.14) -(2.15). Then for any m ≥ 1 lim sup

n→∞ sup α≤α≤α h -m sup S∈Θ ε,L sup p∈P ς P S,p (A ν,n < H) < ∞ .
Proof. Using the definition of H in (2.13) we obtain

P S,p (A ν,n < H) = P S,p   1 nh n j=ν+1 Q(u j ) y 2 j-1 < H nh   = P S,p   ν,k * (Q) + 1 γ(S) k * j=ν+1 Q(u j ) ∆u j < φ   . Note that k * j=ν+1 Q(u j ) ∆u j = k * -k * -ι nh ≥ 2 - ι + 2 nh .
Taking into account that ε 2 ≤ γ(S) ≤ 1, we obtain

1 γ( S ν ) - 1 γ(S) ≤ 2 ε 4 | ∆ ν | . (5.1)
This yields

P S,p A ν,n < H ≤ P S,p   k * j=ν+1 Q(u j )y 2 j-1 < H , | ∆ ν | ≤ δ   + P S,p | ∆ ν | > δ ≤ P S,p ν,k * (Q) < -2 n + n ε 2 + 4 ε 4 δ + 3 ε 2 nh + P S,p | ∆ ν | > δ .
Therefore for δ < ε 4 /8 and sufficiently large n ≥ 1 we obtain that

P S,p A ν,n < H ≤ P S,p | ν,k * (Q)| > n /2 + P S,p | ∆ ν | > δ .
Lemma 8.3 and Lemma 4.2 imply Lemma 5.1. Now for any weighted sequence (w j ) j≥1 we set

Z n = n j=τ H -1 Q(u j ) w j y 2 j-1 + (1 -κ H ) w τ H Q(u τ H ) y 2 τ H -1 . ( 5
|Z n | ≤ w *   n j=ν+1 Q(u j ) y 2 j-1 -H   = w * (A ν,n -H) ,
i.e. |Z n | ≤ w * (A ν,n -H) + , where (x) + = max(0, x). Therefore,

|Z n | nh ≤ w * n j=ν+1 Q(u j ) y 2 j-1 nh -φ ≤ | ν,n (Q)| + n j=ν+1 Q(u j ) γ(S)nh -φ . Taking into account that n j=ν+1 Q(u j ) = k * -k * -ι ≤ 2nh we obtain E S,p |Z n | nh ≤ E S,p | ν,k * (Q)| + 2 1 γ(S) - 1 γ( S ν ) + 2 ε 2 ≤ E S,p | ν,k * (Q)| + 4 ε 4 E S,p ∆ν (z 0 ) + 2 ε 2 .
Moreover, note that

E S,p ∆ν (z 0 ) = E S,p ∆ν (z 0 ) 1 {| ∆ν (z 0 )|≤ } + E S,p ∆ν (z 0 ) 1 {| ∆ν (z 0 )|> } ≤ + 2P S,p ∆n 0 (z 0 ) > .
Therefore, Lemma 8.3 and Lemma 4.1 imply immediately (5.3).

PROOFS

6.1 Proof of Theorem 3.1

First, similarly to the proof of Theorem 3.1 from [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF] we choose the corresponding parametric functional family S u,δ (•) in the following form

S u,δ (x) = u ϕ n V δ x -z 0 h , (6.1)
with the function V δ defined as

V δ (x) = δ -1 ∞ -∞ Q δ (u)g u -x δ du ,
where Q δ (u) = 1 {|u|≤1-2δ} + 21 {1-2δ≤|u|≤1-δ} with 0 < δ < 1/4 and g is some even nonnegative infinitely differentiable function such that g(z) = 0 for |z| ≥ 1 and 1 -1 g(z) dz = 1. One can show (see [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF]) that for any b > 0 and 0 < δ < 1/4 there exists n * = n * (b, L, δ) > 0 such that for all |u| ≤ b and

n ≥ n * S u,δ ∈ U (β) n (ε, L, * n ) . Therefore, in this case for any n ≥ n * ϕ n sup S∈U (β) n (ε,L, * n ) γ -1/2 (S) R n ( S n , S) ≥ sup S∈U (β) n (ε,L, * n ) γ -1/2 (S)E S,p 0 ψ n ( S n , S) ≥ γ * (n, b) 1 2b b -b E S u,δ ,p 0 ψ n ( S n , S u,δ )du ,
where γ * (n, b) = inf |u|≤b γ -1/2 (S u,δ ). The definitions (2.12) and (6.1) imply that for any b > 0

lim n→∞ sup |u|≤b |γ(S u,δ ) -1| = 0 .
Therefore, by the same way as in the proof of Theorem 3.1 from [START_REF] Galtchouk | Asymptotically Efficient Estimates for Nonparametric Regression Models[END_REF] we obtain that for any b > 0 and

0 < δ < 1/4 lim n→∞ inf S sup S∈U (β) n (ε,L, * n ) γ -1/2 (S) ϕ n R n ( S n , S) ≥ I(b, σ δ ) , (6.2)
where

I(b, σ δ ) = max(1, b - √ b) b σ δ √ 2π √ b - √ b e -σ 2 δ u 2 2 du , with σ 2 δ = 1 -1 V 2 δ (u) du. It is easy to check that σ 2 δ → 2 as δ → 0. Limiting b → ∞ and δ → 0 in (6.
2) yield the inequality (3.3). Hence Theorem 3.1.

Proof of Theorem 3.2

First we set κj =

1 {τ H =j} + κ H 1 {τ H =j} . (6.3)
Then taking this into account we can represent the estimate error as

S * h -S(z 0 ) = -S(z 0 ) 1 (A ν,n <H) + h β B n (h) 1 (A ν,n ≥H) + 1 √ H ζ n (h) 1 (A ν,n ≥H) , (6.4)
where

B n (h) = τ H j=ν+1 κj Q(u j ) (S(x j ) -S(z 0 )) y 2 j-1 h β H and ζ n (h) = τ H j=ν+1 κj Q(u j ) y j-1 ξ j √ H .
First we study the term B n (h). To this end we introduce

B * n = n j=ν+1 Q(u j ) b * j y 2 j-1 , b * j = S(x j ) -S(z 0 ) h 1 {k * ≤j≤k * } .
It is clear that for any S from

U (β) n (ε, L, * n ) sup j≥1 |b * j | ≤ L .
Therefore, using Lemma 5.2 for the sequence (5.2) with w j = b * j we obtain

lim sup n→∞ sup α≤α≤α h -α sup S∈U (β) n (ε,L, * n ) sup p∈P ς E S,p | τ H j=ν+1 κj Q(u j ) b * j y 2 j-1 -B * n | nh = 0 . (6.5)
Moreover, putting

f 1 (u) = Q(u) S(z 0 + hu) -S(z 0 ) h , we obtain B * n nh = 1 γ(S) n j=1 f 1 (u j )∆u j - 1 γ(S) ν j=1 f 1 (u j )∆u j + ν,n (f 1 ) . (6.6)
Using the definition of Ω h (z 0 , S) in (2.4) we can represent the first term as

n j=1 f 1 (u j )∆u j = 1 h Ω h (z 0 , S) - 1 u k * f 1 (u)du + k * j=k * u j u j-1 f 1 (u j ) -f 1 (u) du .
Note now that for any S from

U (β) n (ε, L, * n ) sup -1≤u≤1 |f 1 (u)| ≤ L and sup -1≤u≤1 | ḟ1 (u)| ≤ L ,
and, therefore,

1 u k * f 1 (u)du ≤ L nh and k * j=k * u j u j-1 f 1 (u j ) -f 1 (u) du ≤ 2L nh .
The last bounds imply immediately

lim sup n→∞ sup α≤α≤α h -α sup S∈U (β) n (ε,L, * n ) n j=1 f 1 (u j )∆u j = 0 .
Taking into account Lemma 8.3 in (6.6) we get

lim sup n→∞ sup α≤α≤α h -α sup S∈U (β) n (ε,L, * n ) sup p∈P ς E S,p |B * n | nh = 0 .
Therefore, in view of (6.5)

lim sup n→∞ sup α≤α≤α sup S∈U (β) n (ε,L, * n ) sup p∈P ς E S,p |B n (h) = 0 .
To study the last term in (6.3) note that the definition of the stopping time in (2.10) implies

sup n≥ν+1 sup h * ≤h≤h * sup S∈Θ ε,L sup p∈P ς E S,p |ζ n (h)| 2 ≤ 1 . (6.7)
Therefore, in view of Lemma 8.5 we obtain

lim n→∞ sup S∈Θ ε,L sup p∈P ς E S,p |ζ n (h)|1 (A ν,n ≥H) -E|ζ ∞ | = 0 ,
where ζ ∞ ∼ N (0, 1), i.e. E|ζ ∞ | = 2/π. Moreover, in view of Lemma 4.2 and the bound (5.1) we get

lim n→∞ sup S∈Θ ε,L sup p∈P ς E S,p ϕ 2 n γ(S) H - 1 2 = 0 .
From this and Lemma 5.1 it follows Theorem 3.2.

Proof of Theorem 3.4

First, note that the representation (6.4) implies

E S,p |S * h -S(z 0 )| ≤ P S,p (A ν,n < H) + h β E S,p |B n (h)| 1 (A ν,n ≥H) + E S,p |ζ n (h)| √ H 1 (A ν,n ≥H) . (6.8)
Let us show, that lim sup

n→∞ sup α≤α≤α sup S∈H (β) (ε,L,L * ) sup p∈P E S,p |B n (h)| < ∞ . (6.9)
Indeed, setting

ϑ j = S(x j ) -S(z 0 ) h -Ṡ(z 0 ) u j , (6.10)
we can represent B n (h) as

B n (h) = h -α H Ṡ(z 0 ) B n + h -α H B n , (6.11)
where

B n = τ H j=ν+1 κj Q(u j ) u j y 2 j-1 and B n = τ H j=ν+1 κj Q(u j ) ϑ j y 2 j-1 .
Using now Lemma 5.2 for the sequence (5.2) with w j = u j we obtain that

lim sup n→∞ sup α≤α≤α h -α sup S∈Θ ε,L sup p∈P ς E S,p | V n -B n | nh = 0 ,
where

V n = n j=ν+1 Q(u j ) u j y 2 j-1 .
We can represent this term as

V n = ν,n (Q 1 ) + 1 γ(S) n j=ν+1 Q 1 (u j ) ∆u j = ν,n (Q 1 ) + 1 γ(S)   k * j=k * +1 u j ∆u j - k * +ι j=k * +1 u j ∆u j   , where Q 1 (u) = Q(u) u. Moreover, taking into account here, that k * j=k * +1 u j ∆u j = u k * u k * u du + k * j=k * +1 u j u j-1 u j -u du ≤ 4 nh and k * +ι j=k * +1 u j ∆u j ≤ + 1 nh ,
we obtain

1 nh E S,p | V n | ≤ E S,p | ν,n (Q 1 )| + 1 ε 2 5 nh + . Now Lemma 8.3 yields lim sup n→∞ sup α≤α≤α h -α sup S∈Θ ε,L sup p∈P ς E S,p | V n | nh = 0
and, therefore,

lim sup n→∞ sup α≤α≤α h -α sup S∈Θ ε,L sup p∈P ς E S,p | B n | nh = 0 . (6.12)
To estimate the last term in (6.11) note, that for any function S from H (β) (ε, L, L * ) and for k * ≤ j ≤ k * the coefficients (6.10) can be estimated as

|ϑ j | = u j 0 Ṡ(z 0 + hu) -Ṡ(z 0 ) du ≤ L|u j | h α ≤ L * h α .
Therefore, lim sup

n→∞ sup α≤α≤α h -α sup S∈H (β) (ε,L,L * ) sup p∈P 1 nh E S,p | B n | < ∞ .
Now the property (6.12) implies the inequality (6.9). Therefore, using Lemma 5.1 and the inequality (6.7) in (6.8), we come to Theorem 3.2.

Proof of Theorem 3.5

First of all, note that the coefficient φ defined in (2.13) will be more than one for sufficient large n for which ≤ 1/2. So, using the representation (6.4), we get for any

1 ≤ j ≤ m | Šj -S(z 0 )| ≤ 1 Γ j + ( ȟj ) β |B n ( ȟj )| + 1 n ȟj | ζ n ( ȟj )| , (6.13)
where

Γ j = {A ν,n ( ȟj ) < Ȟj }, ζ n ( ȟj ) = ζ n ( ȟj ) 1 (A ν,n ( ȟj )≥ Ȟj )
, the random functions B n (h) and ζ n (h) are defined in (6.4). We set

i 0 = m(β -β) β -β .
This means that

0 ≤ β -β i 0 < β -β m .
Therefore, taking into account the definition of m in (2.18), we obtain that for any fixed integer

-< ∞ < l < ∞            0 < lim inf n→∞ N (β) N i 0 +l ≤ lim sup n→∞ N (β) N i 0 +l < ∞ ; 0 < lim inf n→∞ h(β) ȟi 0 +l ≤ lim sup n→∞ h(β) ȟi 0 +l < ∞ . (6.14)
These inequalities and Theorem 3.4 imply

lim sup n→∞ sup β≤β≤β N (β) sup S∈H (β) (ε,L,L * ) E S (i 0 , z 0 ) < ∞ , (6.15)
where

(i 0 , z 0 ) = | Ši 0 -1 -S(z 0 )| + | Ši 0 -S(z 0 )| + | Ši 0 +1 -S(z 0 )|. Now considering the estimator S a,n , one has | S a,n -S(z 0 )| ≤ I 1 + I 2 + (i 0 , z 0 ) , (6.16)
where

I 1 = | S a,n -S(z 0 )|1 { ǩ≥i 0 +2} and I 2 = | S a,n -S(z 0 )|1 { ǩ≤i 0 -2} .
We start with the term I 1 .

We have

| S a,n -S(z 0 )|1 { ǩ≥i 0 +2} ≤ | S a,n -Ši 0 |1 { ǩ≥i 0 +2} + | Ši 0 -S(z 0 )|1 { ǩ≥i 0 +2} . Moreover, | S a,n -Ši 0 | 1 { ǩ≥i 0 +2} ≤ ωǩ 1 { ǩ≥i 0 +2} + λ N i 0 ≤ λ N ǩ 1 { ǩ≥i 0 +2} + λ N i 0 ≤ λ N (β) + λ N i 0 .
The inequalities (6.14)-(6.15) imply immediately

lim sup n→∞ sup β≤β≤β N (β) sup S∈H (β) (ε,L,L * ) E S I 1 < ∞ .
Now we study the term I 2 . From (6.13) it follows that

I 2 ≤   1 Γ ǩ + ( ȟǩ ) β |B n ( ȟǩ )| + 1 n ȟǩ | ζ n ( ȟǩ )|   1 { ǩ≤i 0 -2} .
Therefore,

E S I 2 ≤ m I * 1 (S) + I * 2 (S) i 0 -2 j=0 ȟβ j + Ψ n (S) , (6.17)
where

I * 1 (S) = max 0≤l≤m P S (Γ l ), I * 2 (S) = max 0≤l≤m E S |B n ( ȟl )| and Ψ n (S) = 1 √ n i 0 -2 j=0 1 ȟj E S | ζ n ( ȟj )| 1 { ǩ=j} .
Note now, that for any 0 ≤ j ≤ i 0 and for sufficiently large n (for which ln d n ≥ m/2) we get

N (β) ȟβ j ≤ e -β * (i 0 -j) , β * = β(β -β) (2β + 1)(2β + 1) , i.e. N (β) i 0 -2 j=0 ȟβ j ≤ e β * e β * -1 .
So, Lemma 5.1 and the inequality (6.9) yield lim sup n→∞ N (β) sup

S∈H (β) (ε,L,L * )   mI * 1 (S) + I * 2 (S) i 0 -2 j=0 ȟβ j   < ∞ .
Let us consider now the last term (6.17). To this end note that for 0 ≤ j ≤ i 0 -2

{ ǩ = j} ⊆ {ω j+1 ≥ λ/N j+1 } ⊆ ∪ j+1 l=0 {| Šl -S(z 0 )| ≥ λ/N l } .
Therefore,

Ψ n (S) ≤ 1 √ n i 0 -2 j=0 1 ȟj j+1 l=0 E S | ζ n ( ȟj )| 1 {| Šl -S(z 0 )|≥ λ/N l } .
Taking into account that ȟj+1 / ȟj ≤ e, we can rewrite the last inequality as

Ψ n (S) ≤ e √ n i 0 -1 j=1 1 ȟj j l=0 E S | ζ n ( ȟj )| 1 {| Šl -S(z 0 )|≥ λ/N l } . (6.18)
Now, taking into account the inequality (6.7), we get

E S | ζ n ( ȟj )| 1 {| Šl -S(z 0 )|≥ λ/N l } ≤ P S (Γ l ) + E S | ζ n ( ȟj )| 1 { ȟβ l |B n (
E S | ζ n ( ȟj )| 1 { ȟβ l |B n ( ȟl )|≥ λ1 /N l } ≤ (µ * 4 ) 1/4 (I * 2 (S)) 3/4 λ3/4 1 N l ȟβ l 3/4
.

where the term I * 2 (S) is defined in (6.17). Using these bounds in (6.18) we get

Ψ n (S) ≤ e m 2 n ȟ0 I * 1 (S) + e(µ * 4 ) 1/4 (I * 2 (S)) 3/4 λ3/4 1 Υ * n + e m 2 n ȟ0 E S |ζ * | 1 {|ζ * |≥ λ1 √ ln n} , (6.19)
where

Υ * n = 1 √ n i 0 -1 j=1 N j ȟβ j 1/4 N j ȟα j 1/2 υ * j and υ * j = j l=0 N l ȟβ l N j ȟβ j 3/4 .
Let us estimate the term Υ * n . To this end note, that for any 0 ≤ j < i 0 and for sufficiently large n (for which ln d n ≥ m/2)

N j ȟβ j 1/4 = exp β j -β 4(2β j + 1) ln d n ≤ e -β * 1 (i 0 -j) with β * 1 = β -β 8(2β + 1) ,
and

N j ȟα j 1/2 = exp β j -α 2(2β j + 1) ln d n ≤ 1 ȟ(β) = d n N (β)
,

where ȟ(β) is defined in (2.17). Similarly for any 0 ≤ l ≤ j ≤ i 0 we get

N l ȟβ l N j ȟβ j 3/4 = exp 3(β l -β j )(2β +
S∈H (β) (ε,L,L * ) E S |ζ * | 1 {|ζ * |≥ λ1 √ ln n} = 0 .
Hence Theorem 3.5.

NUMERICAL EXAMPLES

Using Scilab, we establish the following simulations to illustrate the obtained results.

Nonadaptive Estimation

In this section we illustrate the results obtained in the case of nonadaptive estimation The purpose is to estimate, at a given point z 0 , the function S defined over [0; 1] by

S(x) = (x -z 0 ) |x -z 0 | α (7.1) n≥1 sup 0≤k≤n sup S∈Θ ε,L sup p∈P ς E S,p |y k | q ≤ r * q , (8.1)
where r * q is defined in (4.1). Proof. From (1.1) we obtain that for any k ≥ 1

y k = y 0 k l=1 S(x l ) + k i=1 k l=i+1 S(x l ) ξ i .
Therefore, for S ∈ Θ ε,L and 1 ≤ k ≤ n,

|y k | q ≤ 2 q-1   |y 0 | q +   k j=1 (1 -ε) k-j |ξ j |   q   .
Moreover, the Hölder inequality gives

  k j=1 (1 -ε) k-j |ξ j |   q ≤   k j=1 (1 -ε) k-j   q-1   k j=1 (1 -ε) k-j |ξ j | q   ≤ 1 ε q-1   k j=1 (1 -ε) k-j |ξ j | q   .
Thus, taking into account the definition (2.2) we get for any p ∈ P ς

E S,p   k j=1 (1 -ε) k-j |ξ j |   q ≤ 1 ε q s * q .
Hence Lemma 8.1. Now we need the following Burkhölder inequality from [START_REF] Shiryaev | Probability[END_REF].

Lemma 8.2. Let (M k ) 1≤k≤n be a martingale. Then for any q > 1

E |M n | q ≤ b * q E   n j=1 (M j -M j-1 ) 2   q/2 , (8.2)
where the coefficient b * q is defined in (4.1).

sup R>0 sup f 1 ≤R 1 (Rh) q sup S∈Θ ε,L sup p∈P ς E S,p k,m (f ) q ≤ * q , (8.3)
where f 1 = f + ḟ and * q is defined in (4.1).

Proof. First of all, note that

m j=k+1 f (u j )y 2 j-1 = T k,m + a k,m , (8.4)
where T k,m = m j=k+1 f (u j )y 2 j and

a k,m = m j=k+1 (f (u j ) -f (u j-1 )) y 2 j-1 + f (u k ) y 2 k -f (u m ) y 2 m .
Moreover, from the model (1.1) we find

(1 -S 2 (z 0 ))T k,m = M k,m + ǎk,m + m j=k+1 f (u j ) where M k,m = m j=k+1 (2S(x j ) y j-1 ξ j + ξ 2 j -1) f (u j ) and ǎk,m = m j=k+1 f (u j )S 2 (x j )y 2 j-1 -S 2 (z 0 )T k,m . (8.5)
Then we can write k,m (f ) as follow where κ = (1 + κ * )/κ * . Therefore, for sufficiently large n we get

k,m (f ) = 1 nh γ(S) M k,m + ǎk,m + a k,m nh . and E S,p | k,m (f )| q ≤ 3 q-1 ε 2q E S,p |M k,m | nh q + |ǎ k,m | nh q + |a k,m | nh q . ( 8
1 nh ≤ ( κ) 2/(2β+1) h ≤ κh , (8.7)
Furthermore, note that (M k,j ) k<j≤m is a martingale. So, by applying the Burkhölder inequality (8.2) and, taking into account that k * -k * ≤ 2nh, we get

E S,p 1 nh M k,m q ≤ b * q R q (nh) q E S,p   k * j=k * +1 2 S(x j ) y j-1 ξ j + ξ 2 j -1 2   q/2 ≤ b * q R q (nh) q/2+1 k * j=k * +1 E S,p 2 S(x j ) y j-1 ξ j + ξ 2 j -1 q ≤ 4 q b * q R q (nh) q/2 r * q s * q + s * 2q + 1 ≤ 4 q b * q κ q R q r * q s * q + s * 2q + 1 h q ,
where the coefficients r * q and s * q are given in (8.1) and (2.2). Note that the term (8.5) can be rewritten as

ǎk,m = S 2 (z 0 )   m j=k+1 (f (u j ) -f (u j-1 )) y 2 j-1 + f (u k )y 2 k -f (u m )y 2 m   + m j=k+1 f (u j )(S 2 (x j ) -S 2 (z 0 ))y 2 j-1 .
We recall, that the function f and its derivative ḟ are bounded by R. Therefore, taking into account that for all S ∈ Θ ε,L and k

* ≤ j ≤ k * the deviation |S(x j ) -S(z 0 )| ≤ L|x j -z 0 | ≤ Lh, we obtain |ǎ k,m | ≤ R   1 nh + Lh m j=k+1 y 2 j + y 2 k + y 2 m   ≤ κ R   (L + 1) h m j=k+1 y 2 j + y 2 k + y 2 m   . Therefore, sup S∈Θ ε,L sup p∈P ς E S,p 1 nh |ǎ k,m | q ≤ 2 q-1 κ q R q r * 2q 2 q (1 + L) q h q + 2 q (nh) q ≤ (4R(1 + L)) q r * 2q h q .
Similarly,

sup S∈Θ ε,L sup p∈P ς E S,p 1 nh |a k,m | q ≤ 3 2 q-1 κ q R q r * 2q h q .
Then, taking this into account in (8.6) we obtain the upper bound (8.3). Hence Lemma 8.3.

Uniform Limit Theorem

In this section we study the following sequence

ζ n (h) = ζ n (h) 1 (A ν,n ≥H) , (8.8)
where ζ n (h) defined by (6.4), the bandwidth h is defined (2.14) and the threshold H is given in (2.13).

We will make use of the following result. Moreover, taking into account that y2 j ≤ y 2 j , we get

E S,p ∆2 2 |G 0 ≤ E p ξ 2 1 Ȟ E S,p   τ H j=ν+1 Q(u j ) y2 j-1 |G 0   ≤ E p ξ 2 1 Ȟ H + δ 4 Ȟ = E p ξ 2 1 1 E p ξ2 1 + δ 4
Taking into account here, that The proof of this Lemma is the same as Lemma A.5 from [START_REF] Arkoun | Sequential Adaptive Estimators in Nonparametric Autoregressive Models[END_REF].

Using this lemma we can obtain that for any q > 2 sup

n≥1 sup h>0 sup S∈C[0,1] E S | ζ n (h)| q ≤ µ * q , (8.11)
where µ q q = 2 q + 2q ∞ 0 t q-1 e -t 2 /2 dt.

  ȟl )|≥ λ1 /N l } + E S |ζ * | 1 {|ζ * |≥ √ ln n λ1 } where λ1 = λ/2 and ζ * = max 1≤j≤m | ζ n ( ȟj )|. In view of the Hölder and Chebyshev inequalities and making use of the upper bound (8.11) we obtain

Lemma 8 . 4 ..Lemma 8 . 5 . 4 δ 8 E p ξ2 1 ( 1 / 2 ζ( 1 - ≤ δ 4 .

 84854811214 (cf.[START_REF] Freedman | Brownian Motion and Diffusion[END_REF], p. 90-91) Let 0 < δ < 1 and r > 0. Assume that (m k ) k≥1 is a martingale difference with respect to the filtration (F k ) k≥1 such that|m k | ≤ δr 1/2 andThere exists a function ρ : (0, +∞) → [0, 2] not depending on distribution of the martingale difference, such that lim x→0 ρ(x) x -Φ(x) ≤ ρ(δ) , where Φ is the standard normal distribution function. The sequence (6.4) satisfies the following limiting property:ζ n =⇒ ζ ∼ N (0, 1) uniformly in p ∈ P ς and S ∈ Θ ε,L .Proof. First for some 0 < δ < 1 we setm j = Q(u j ) yj-1 ξj 1 {ν<j≤n} + δ ξj 1 {j>n} , where yj = y j 1 {|y j |≤δ 2 Ȟ1/2 } , Ȟ = E p ξ2 1 H and ξj = ξ j 1 {|ξ j |≤δ -1 } -E p ξ 1 1 {|ξ 1 |≤δ -1 } .It is clear that the sequence (m ν+j ) j≥0 is a martingale difference with respect to (G j ) j≥0 , where G j is σ -field generated by the observations {y 1 , . . . , y ν+j }. Now we setζ Ȟ = 1 √ Ȟ τH j=1 m ν+j ,where τH = inf k ≥ 1 :k j=1 E (m 2 ν+j |G j-1 ) ≥ Ȟ . Note, that τH = τ H on the set {A ν,n ≥ H} for any H > 0. Lemma 8.4 implies that ζ Ȟ goes in distribution to N (0, 1) uniformly in p ∈ P ς and S ∈ Θ ε,L as δ → 0. Now we set Ωn = ∩ k * j=ν {y j = yj } .Using Lemma 8.1 through the Markov inequality we obtain that sup nh) 2 → 0 as n → ∞ .Moreover, note that on the set Ωn ∩ {A ν,n ≥ H}H Ȟ n -ζ Ȟ = ∆1 + ∆2 , j -1) Q(u j ) yj-1 ξ j , ∆2 = 1 √ Ȟ τ H j=ν+1 Q(u j ) yj-1 ξ jand ξ j = ξ j -ξj = ξ j 1 {|ξ j |>δ -1 } -E p ξ 1 1 {|ξ 1 |>δ -1 } . Note, that E S,p ∆2 1 |G 0 ≤ δ 4 E S,p κ j ) 2 |G 0

  Therefore, Lemma 5.1 and the representation (8.9) yield for any µ > 0 lim δ→0 lim sup n→∞ sup S∈Θ ε,L sup p∈P ς P S,p ζ n -ζ Ȟ > µ = 0 . Hence Lemma 8.5.

8. 3 Lemma 8 . 6 .P

 386 Properties of ζn (h) For all z ≥ S ζ n (h) ≥ z ≤ 2e -z 2 /8 . (8.10)

  of the procedure (2.21) we need to study the family (S * h ) α≤α≤α . Theorem 3.4. The sequential procedure (2.11) with the bandwidth h defined in (2.14) for κ n = ln n/n

	satisfies the following property
	lim sup

n→∞ sup α≤α≤α

  {| ζ n ( ȟj )|≥ λ1

		1) 4(2β l + 1)(2β j + 1)	ln d n ≤ e -β * 2 (j-l) , β * 2 =	3(β -β) 8(2β + 1)	.
	This means that the sequence (υ * j ) j≥1 is bounded, i.e.
				sup j≥1	υ * j ≤	e β * 2 e β * 2 -1	.
	Therefore,
			lim n→∞	N (β) Υ * n = 0 .	(6.20)
	The last term in (6.19) can be estimated through Lemma 8.10, i.e.
							√	ln n}
		≤ 2m λ1	√	ln n e -1 8	λ2 1 ln n + 2 m	+∞ λ1 ln n √	e -z 2 /8 dz .
	Therefore, for sufficiently large n (when λ1	√	ln n ≥ 1) we get that
	E S ζ * 1 {ζ * ≥ λ1	√	ln n} ≤ 2m λ1	√	ln n + 4 n -λ2 1 /8 .	(6.21)
	Now the definition of the parameter λ in (2.19) yields
	lim sup	N (β) m 2			sup
	n→∞	n ȟ0

E S ζ * 1 {ζ * ≥ λ1 √ ln n} ≤ m max 1≤j≤m E S | ζ n ( ȟj )| 1

  .6) where κ = (1 + κ * )/κ * . Now we note, that in view of the first condition in (2.15) for sufficient large n nκ n ≥ κ ,

Acknownledgements. The authors would like to thank the Editor and anonymous reviewers for their valuable comments and suggestions. This research is partially funded by the Academic D.I. Mendeleev Fund Program of the Tomsk State University (research project NU 8.1.55.2015 L) in 2015 -2016.

for z 0 = 1/ √ 2 and α = 0, 3. Taking into account that for this function Ω h (z 0 , S) = 0 we obtain that for any

the function (7.1) belongs to the class U (β) n (ε, L, * n ) with β = 1, 3. The numerical results approximate the asymptotic risk of a estimators defined in (2.11) used due to the calculation of an expectation (it performs an average for M = 30000 simulations) and the finite number of observations n. Here we calculate for the estimator the quantity

For the standard Gaussian random variables (ξ j ) j≥1 in (1.1), and by varying the number of observations n, we obtain different risks listed in the following table:

Numerical Result for Non Sequential Kernel Estimator

Now we give the numerical results for the kernel estimator defined as

For the standard Gaussian random variables (ξ j ) j≥1 in (1.1), and by varying the number of observations n, we obtain different risks listed in the following table :   n 1000 5000 10000 20000 R n 0.046 0.026 0.020 0.010

Adaptive Estimation

In this case we estimate the function (7.1) for z 0 = 1/ √ 2 and α = 0, 7. Obviously, this function belongs to class H (β) (ε, L, L * ) with β = 1, 7, L * = 1 and for any ε and L satisfying the conditions (7.2).

In the adaptive estimation we take the lower regularity β = 1.6 and the higher regularity β = 1.8. We model the sequential adaptive procedure S a,n = S * ȟ defined in (2.21). Numerical results approximate the asymptotic risk for this procedure by the calculation of an expectation via M = 30000 simulations. In this section, we study the deviation (4.4) for the model (1.1). Lemma 8.1. For any q > 1 and 0 < ε < 1 the random variables y k in (1.1) satisfy the following inequality: sup