Andrey Polyakov

Denis Efimov

Wilfrid Perruquetti

Finite-Time and Fixed-Time Stabilization: Implicit Lyapunov Function Approach

Keywords: Lyapunov Methods, Robust Control, Sliding Mode Control

Theorems on Implicit Lyapunov Functions (ILF) for finite-time and fixed-time stability analysis of nonlinear systems are presented. Based on these resutls, new nonlinear control laws are designed for robust stabilization of a chain of integrators. High order sliding mode (HOSM) algorithms are obtained as particular cases. Some aspects of digital implementations of the presented algorithms are studied, it is shown that they possess a chattering reduction ability. Theoretical results are supported by numerical simulations.

Introduction and related works

Many practical applications require severe time response constraints (for security reasons, or simply to improve productivity). That is why, finite-time stability and stabilization problems have been intensively studied, see [START_REF] Roxin | On finite stability in control systems[END_REF], [START_REF] Haimo | Finite time controllers[END_REF], [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF], [START_REF] Orlov | Discontinous systems: Lyapunov analysis and robust synthesis under uncertainty conditions[END_REF]. Time constraint may also appear in observation problems when a finite-time convergence of the state estimate to the real values is required: [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF], [START_REF] Perruquetti | Finite-time observers: Application to secure communication[END_REF], [START_REF] Menard | A global high-gain finitetime observer[END_REF], [START_REF] Bejarano | High order sliding mode observer for linear systems with unbounded unknown inputs[END_REF], [START_REF] Shen | Global finite-time observers for lipschitz nonlinear systems[END_REF].

Let us stress that finite-time stability property is frequently associated with HOSM controls, since these algorithms should ensure finite-time convergence to a sliding manifold [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], [START_REF] Polyakov | Lyapunov function design for finitetime convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF]. Typically, the associated controllers have mechanical and electromechanical applications [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF], [START_REF] Ferrara | On multi-input backstepping design with second order sliding modes for a class of uncertain nonlinear systems[END_REF], [START_REF] Chernousko | Control of nonlinear dynamical systems: methods and applications[END_REF], [START_REF] Dinuzzo | Higher order sliding mode controllers with optimal reaching[END_REF].

The theoretical background of HOSM control systems is very well developed [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF], [START_REF] Levant | Quasi-continuous high-order sliding mode controllers[END_REF], [START_REF]Construction principles of 2-sliding mode design[END_REF]. However, applications of the existing HOSM control algorithms are complicated, since there are a few constructive algorithms for tuning the HOSM control parameters. Most of them are restricted to the second order sliding mode systems [START_REF]Construction principles of 2-sliding mode design[END_REF], [START_REF] Polyakov | Lyapunov function design for finitetime convergence analysis: "twisting" controller for second-order sliding mode realization[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF].

⋆ The preliminary version of this paper was presented at IFAC Symposium on Nonlinear Control Systems 2013 [START_REF] Polyakov | Finite-time stabilization using implicit lyapunov function technique[END_REF]. This work is supported by ANR Grant CHASLIM (ANR-11-BS03-0007)

Email addresses: andrey.polyakov@inria.fr (Andrey Polyakov), denis.efimov@inria.fr (Denis Efimov), wilfrid.perruetti@inria.fr (Wilfrid Perruquetti).

Fixed-time stability, that demands boundedness of the settling-time function for a globally finite-time stable system, was studied in [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF], [START_REF] Cruz-Zavala | Uniform robust exact differentiator[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF]. This property was originally discovered in the context of homogeneous systems [START_REF] Andrieu | Homogeneous approximation, recursive obsertver and output feedback[END_REF]. Fixed-time stability looks promising if a controller (observer) has to be designed in order to provide some required control (observation) precision in a given time and independently of initial conditions.

The main tool for analysis of finite-time and fixed-time stability is the Lyapunov function method (see, for example, [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF]), which is lacking for constructive design in the nonlinear case. This paper deals with ILF method [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF], which relies on Lyapunov functions defined, implicitly, as solutions to an algebraic equation. Stability analysis does not require solving this equation, since the implicit function theorem [START_REF] Courant | Introduction to calculus and analysis[END_REF] helps to check all stability conditions directly from the implicit formulation. The similar approach was presented in [START_REF] Korobov | A general approach to synthesis problem[END_REF] for control design and called the controllability function method (see, also [START_REF] Anan'evskii | Control design for dynamic systems using Lyapunov function method[END_REF]). This paper addresses the problem of a control design for the robust finite-time and fixed-time stabilization of a chain of integrators. The ILF method is used to design the control laws together with Lyapunov functions for closed-loop systems. This method allows us to analyze robustness of the closed-loop system and to design a high order sliding mode control algorithm, which rejects bounded matched exogenous disturbances. Finite-time and fixed-time stability conditions were obtained in the form of Linear Matrix Inequalities (LMI). They provide simple constructive schemes for tuning the control parameters in order to predefine the required convergence time and/or to guarantee stability and robustness with respect to disturbances of a given magnitude.

Through the paper the following notation will be used:

• R is the set of real numbers, R + = {x ∈ R : x > 0}; • dV dt (.)
is the time derivative of a function V along the solution of a differential equation numbered as (.);

• • is the Euclidian norm in R n ; • diag{λ i } n
i=1 is the diagonal matrix with the elements λ i ; • a continuous function σ : R + → R + belongs to the class K if it is monotone increasing and σ(h) → 0 + as h → 0 + ; • for a symmetric matrix P = P T the minimal and maximal eigenvalues are denoted by λ min (P ) and λ max (P ),

• int(Ω) is the interior of the set Ω ⊆ R n .

Problem statement

The paper deals with finite-time and fixed-time stabilization problems for the disturbed chain of integrators. Note that a control design scheme developed for such systems usually admits extension to feedback linearizable nonlinear multi-input multi-output systems [START_REF] Isidori | Nonlinear Control Systems[END_REF]. The problem statement presented below is also typical for high-order sliding mode control design [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF].

Consider a linear single input system of the following form The analysis of such systems requires a special mathematical framework. In this paper we use Filippov theory [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF].

ẋ(t) = Ax(t) + bu(t) + d(t, x(t)), t > 0, (1)
The goal of the paper is to develop control laws such that the origin of the closed-loop system (1) will be globally asymptotically stable and all its trajectories will reach the origin in a finite time or in the fixed time T max ∈ R + .

In addition, the control algorithms to be developed should have effective schemes for tuning the control parameters and assigning of the settling time.

The control design is based of ILF approach to finite-time and fixed-time stability analysis, which is developed in the next section.

Preliminaries

Finite-time and fixed-time stability

Consider the system defined by

ẋ(t) = f (t, x(t)), t ∈ R + , x(0) = x 0 , (2)
where x ∈ R n is the state vector, f : R + × R n → R n is a nonlinear vector field locally bounded uniformly in time.

If f is a locally measurable function that is discontinuous with respect to the state variable x then a solution of the Cauchy problem ( 2) is understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF], namely, as an absolutely continuous function satisfying the differential inclusion

ẋ(t) ∈ K[f ](t, x(t)) for almost all t ∈ [0, t * ), where t * ∈ R + or t * = +∞. The set-valued function K[f ] : R + × R n ⇒ R n is defined for any fixed (t, x) ∈ R + × R n as follows K[f ](t, x) = ε>0 N :m(N )=0 co f (t, B(x, ε)\N ),
where co(M ) defines the convex closure of the set M ⊂ R n , B(x, ε) is the ball with the center at x ∈ R n and the radius ε ∈ R + , the equality m(N ) = 0 means that the Lebesgue measure of the set N ⊂ R n is zero.

Let the origin be an equilibrium of (2), i.e. 0 ∈ K[f ](t, 0).

The system (2) may have non-unique solutions and may admit both weak and strong stability (see, for example, [START_REF] Filippov | Differential equations with discontinuous right-hand sides[END_REF]). This paper deals only with the strong stability properties, which ask for stable behavior of all solutions of the system (2). The next definition of uniform finite-time stability is just a compact representation of Definition 2.5 from [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

Definition 1 The origin of system (2) is said to be globally uniformly finite-time stable if it is globally uniformly asymptotically stable (see, for example, [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF]) and there exists a locally bounded function T : R n → R + ∪ {0}, such that x(t, x 0 ) = 0 for all t ≥ T (x 0 ), where x(•, x 0 ) is an arbitrary solution of the Cauchy problem [START_REF] Roxin | On finite stability in control systems[END_REF]. The function T is called the settling-time function.

Asymptotic stability of the time-independent (autonomous) system always implies its uniform asymptotic stability (see, for example, [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF]). For finite-time stable systems this is not true in general case (see, for example, [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF]), since Definition 1 additionally asks a uniformity of the settling time with respect to initial conditions. The origin of system ẋ(t) = -|x(t)| 0.

) ≤ T max , ∀x 0 ∈ R n .
The presented definition just asks more: strong uniformity of finite-time stability with respect to initial condition. The origin of system ẋ(t) = -|x| 0.5 (t)+|x| 1.5 (t) sign(x(t)), x ∈ R, is globally fixed-time stable, since its settling time function T (x 0 ) = 2 arctan( |x 0 |) is bounded by π ≈ 3.14 .

The uniformity of finite-time and fixed-time stability with respect to system disturbances can also be analyzed. For instance, finite-time stability, which is uniform (in some sense) with respect to both initial conditions and system disturbances, was called equiuniform finite-time stability [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF].

Implicit Lyapunov Function Method

This subsection introduces some stability theorems further used for control design. They refine the known results about global uniform asymptotic, finite-time and fixed-time stability of differential inclusions to the case of implicit definition of Lyapunov function. The next theorem extends Theorem 2 from [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF].

Theorem 3 If there exists a continuous function

Q : R + × R n → R (V, x) → Q(V, x) satisfying the conditions C1) Q is continuously differentiable outside the origin; C2) for any x ∈ R n \{0} there exists V ∈ R + such that Q(V, x) = 0; C3) let Ω= {(V, x) ∈ R + × R n : Q(V, x)=0} and lim x→0 (V,x)∈Ω V =0 + , lim V →0 + (V,x)∈Ω x =0, lim x →∞ (V,x)∈Ω V = + ∞; C4) ∂Q(V,x) ∂V < 0 for all V ∈ R + and x ∈ R n \{0}; C5) sup t∈R+,y∈K[f ](t,x) ∂Q(V,x) ∂x y < 0 for all (V, x) ∈ Ω;
then the origin of (2) is globally uniformly asymptotically stable.

Proof. The conditions C1), C2), C4) and the implicit function theorem [START_REF] Courant | Introduction to calculus and analysis[END_REF] imply that the equation

Q(V, x) = 0 implicitly defines a unique function V : R n \{0} → R + such that Q(V (x), x) = 0 for all x ∈ R n \{0}.
The function V is continuously differentiable outside the origin and

∂V ∂x = -∂Q(V,x) ∂V -1 ∂Q(V,x) ∂x for Q(V, x) = 0, x = 0.
Due to the condition C3) the function V can be continuously prolonged at the origin by setting V (0) = 0. In addition, it is radially unbounded and positive definite. Denote

Z(V, x) = ∂Q ∂V -1 sup t∈R+,y∈K[f ](t,x) ∂Q(V,x) ∂x y and W (x) = Z(V (x), x).
The conditions C4 and C5 imply W (x) > 0 for x = 0. Let x(t, x 0 ) be a solution of the system (2) with initial condition x(0, x 0 ) = x 0 then the function V (x(t, x 0 )) is differentiable for almost all t such that x(t, x 0 ) = 0 and

d dt V (x(t, x 0 )) ≤ -W (x(t, x 0 )).
Finally, we finish the proof using Theorem 4.1 from [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF] and, for example, Lemmas 4, 6 from [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF].

Evidently, the conditions of Theorem 3 mainly repeat (in the implicit form) the requirements of the classical theorem on global asymptotic stability (see, for example, [START_REF] Bacciotti | Lyapunov Functions and Stability in Control Theory[END_REF]). Indeed, Condition C1) asks for smoothness of the Lyapunov function. Condition C2) and the first two limits from Condition C3) provide its positive definiteness. The last limit from Condition C3) implies radial unboundedness of the Lyapunov function. Condition C5) guarantees the negative definiteness of the total derivative of the Lyapunov function calculated along trajectories of the system (2). The only specific condition is C4), which is imposed by implicit function theorem (see, for example, [START_REF] Courant | Introduction to calculus and analysis[END_REF]). This condition is required in order to guarantee that the Lyapunov function is (uniquely) well-defined by the equation Q(V, x) = 0.

Theorem 4 If there exists a continuous function Q : R + × R n → R that satisfies the conditions C1)-C4) of Theorem 3 and the condition C5bis) there exist c > 0 and 0 < µ ≤ 1 such that sup

t∈R+,y∈K[f ](t,x) ∂Q(V,x) ∂x y ≤ cV 1-µ ∂Q(V,x) ∂V for (V, x) ∈ Ω,
then the origin of the system (2) is globally uniformly finite-time stable and

T (x 0 ) ≤ V µ 0 cµ , where Q(V 0 , x 0 ) = 0.
Proof. Theorem 3 implies global uniform asymptotic stability of the origin of (2). The uniform finite-time stablity of the origin follows from the differential inequality dV (x(t,x0)) dt ≤ -cV 1-µ (x(t, x 0 )), which, due to the condition C5bis), holds for almost all t such that x(t, x 0 ) = 0. For a detailed analysis of the obtained differential inequality see, for example, [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF], [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF], [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF].

Theorem 5 Let there exist two functions Q 1 and Q 2 satisfying the conditions C1)-C4) of Theorem 3 and the conditions C6)

Q 1 (1, x) = Q 2 (1, x) for all x ∈ R n \{0}; C7) there exist c 1 >0 and 0 <µ< 1 such that the inequality sup t∈R+,y∈K[f ](t,x) ∂Q1(V,x) ∂x y ≤ c 1 V 1-µ ∂Q1(V,x) ∂V , holds for all V ∈ (0, 1] and x ∈ R n \{0} satisfying the equation Q 1 (V, x) = 0; C8) there exist c 2 >0 and ν>0 such that the inequality sup t∈R+,y∈K[f ](t,x) ∂Q2(V,x) ∂x y ≤ c 2 V 1+ν ∂Q2(V,x) ∂V , holds for all V ≥ 1 and x ∈ R n \{0} satisfying the equa- tion Q 2 (V, x) = 0, then the system (2) is globally fixed-time stable with the settling-time estimate T (x 0 ) ≤ 1 c1µ + 1 c2ν .
Proof. Let the two functions V 1 and V 2 be defined by the equations Q 1 (V, x) = 0 and Q 2 (V, x) = 0 (see, the proof of Theorem 3). Consider the sets

Σ 1 = {x ∈ R n : V 1 (x) > 1}, Σ 2 = {x ∈ R n : V 2 (x) > 1} and prove that Σ 1 = Σ 2 . Sup- pose the contrary, i.e. ∃z ∈ R n such that z ∈ Σ 1 and z / ∈ Σ 2 . On the one hand, Q 1 (V 1 , z) = 0 implies V 1 > 1 and Q 1 (1, z) > Q 1 (V 1 , z) = 0 due to Condition C4). On the other hand, Q 2 (V 2 , z) = 0 implies V 2 ≤ 1 and Q 2 (1, z) ≤ Q 2 (V 2 , z) = 0. The contradiction follows from Condition C6).
Therefore, due to C6) and C4) the function V : R n → R defined by the equality

V (x) =        V 1 (x) for V 1 (x) < 1, V 2 (x) for V 2 (x) > 1, 1 for V 1 (x) = V 2 (x) = 1, is positive definite, continuous in R n and continuously dif- ferentiable for x / ∈ {0} ∪ {x ∈ R n : V (x) = 1}.
The function V is Lipschitz continuous outside the origin and has the following Clarke's gradient [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]:

∇ C V (x) = ξ∇V 1 (x) + (1 -ξ)∇V 2 (x), x ∈ R n , where ξ = 1 for 0 < V 1 (x) < 1, ξ = 0 for V 2 (x) > 1, ξ = [0, 1] for V 1 (x) = V 2 (x) = 1 and ∇V i is the gradient of the function V i , i = 1, 2.
Hence, due to conditions C7) and C8), the inequality

dV (x(t,x0)) dt ≤        -c 1 V 1-µ (x(t, x 0 )) for V (x(t, x 0 )) < 1, -c 2 V 1+ν (x(t, x 0 )) for V (x(t, x 0 )) > 1, -min{c 1 , c 2 } for V (x(t, x 0 )) = 1
, holds for almost all t such that x(t, x 0 ) = 0, where x(t, x 0 ) is a solution of the system (2) with the initial condition x(0) = x 0 . This implies the fixed-time stability of the origin of the system (2) with the estimate of settling-time function given above. Please see [START_REF] Polyakov | Nonlinear feedback design for fixed-time stabilization of linear control systems[END_REF] or [START_REF] Polyakov | Stability notions and Lyapunov functions for sliding mode control systems[END_REF] for more details.

Control design using Implicit Lyapunov Function Method

Finite-Time Stabilization

Introduce the implicit Lyapunov function candidate

Q(V, x) := x T D r (V -1 )P D r (V -1 )x -1, (3)
where

V ∈ R + , x ∈ R n , P ∈ R n×n , P > 0, D r (λ) := diag{λ ri } is the matrix with r = (r 1 , ..., r n ) T ∈ R n + and r i = 1 + (n -i)µ, 0 < µ ≤ 1. Denote H µ := diag{-r i } n i=1 .
The function Q is an implicit analog of the quadratic Lyapunov function. Indeed, for µ = 0 the equality Q(V, x) = 0 gives V (x) = √ x T P x. For µ = 1 it is coincides with the implicit Lyapunov function considered in [START_REF] Adamy | Soft variable-structure controls: a survey[END_REF]. Theorem 6 (Finite-time Stabilization) Let 1) µ ∈ (0, 1], α, β, γ ∈ R + such that α > β and the system of matrix inequalities

AX + XA T + by + y T b T + αX + βI n ≤ 0, -γX ≤ XH µ + H µ X < 0, X > 0 (4)
is feasible for some X = X T ∈ R n×n and y ∈ R 1×n ;

2) the control u has the form

u(V, x) = V 1-µ kD r (V -1 )x, (5)
where

k := yX -1 , V ∈ R + is such that Q(V, x) = 0 and Q is defined by (3) with P = X -1 ;
3) the function d(t, x) satisfy the inequality

β 2 V -2µ ≥ sup t∈R+ d T (t, x)D 2 r (V -1 )d(t, x), ( 6
)
for all V ∈ R + , x ∈ R n such that Q(V, x) = 0;
then the closed-loop system (1) is globally uniformly finitetime stable and T (x) ≤ γV µ µ(α-β) , where Q(V, x) = 0.

Proof. It can be shown that the following inequalities

λmin(P ) x 2 max{V 2+2(n-1)µ ,V 2 } ≤ Q(V, x) + 1 ≤ λmax(P ) x 2 min{V 2+2(n-1)µ ,V 2 } hold for all V ∈ R + and x ∈ R n . The function (3) satisfies conditions C1)-C3) of Theorem 3. The condi- tion C4) of Theorem 3 also holds, since 0 > ∂Q ∂V = V -1 x T D r (V -1 )(H µ P + P H µ )D r (V -1 )x ≥ -γV -1 . Taking into account that D r (V -1 )AD -1 r (V -1 ) = V -µ A and D r (V -1 )bu(x) = V -µ bkD r (V -1 )x we obtain ∂Q ∂x (Ax + bu(x) + d(t, x)) = z 1 z 2 T W z 1 z 2 + β -1 V µ z T 2 z 2 -αV -µ z T 1 P z 1 ,
where [START_REF] Bhat | Finite-time stability of continuous autonomous systems[END_REF] and the Schur complement. Since z T 1 P z 1 = 1 and z T 2 z 2 ≤ β 2 V -2µ by ( 6) then the condition C5bis) of Theorem 4 holds for c = (α -β)/γ.

z 1 = D r (V -1 )x, z 2 = D r (V -1 )d, the matrix W := 1 V µ P (A+bk) + (A+bk) T P + αP P P -V µ β I n is negative semidefinite due to
Let us make remarks about the presented control scheme:

• The practical implementation of the control (5) requires to find the solution V (x) of the equation Q(V, x) = 0, which can be solved numerically and on-line using the current value of the state vector. A simple numerical scheme that can be utilized for this purpose is presented in Section 5. • If n = 2 and µ = 1 the function V (x) can be found analytically. Indeed, the equation

Q(V, x) = 0 becomes V 4 -p 22 x 2 2 V 2 -2p 12 x 1 x 2 V -p 11 x 2 1 = 0,
where {p ij } are elements of the matrix P > 0 and

(x 1 , x 2 ) T = x ∈ R 2 .
The roots of the obtained quartic (with respect to V ) equation can be found using, for example, Ferrari formulas. Due to Theorem 6 the equation has a unique positive root for any

(x 1 , x 2 ) T ∈ R 2 . Denote z(x 1 , x 2 ) = 2p22x 2 2 -C1(x1,x2)-C2(x1,x2) 3
, where

C i (x 1 , x 2 ) = 3 (-1) i √ ∆ 2 1 (x1,x2)-4∆ 3 0 (x1,x2)+∆1(x1,x2) 2 , ∆ 0 (x 1 , x 2 ) = p 2 22 x 4 2 -12p 11 x 2 1 and ∆ 1 (x 1 , x 2 ) = p 3 22 x 6 2 -180p 2 12 x 2 1 x 2 2 .
The roots of the quartic equation coincide with roots of two quadratic equations [START_REF] Carpenter | On the solution of the real quartic[END_REF]:

V 2 + (-1) i z(x 1 , x 2 )V + r i (x 1 , x 2 ) = 0, i = 1, 2,
where

r i (x 1 , x 2 ) = p22x 2 2 +z(x1,x2) 2 + (-1) i p12x1x2 √ z(x1,x2)
.

The explicit representations of roots are rather cumbersome. They are omitted in order to safe the space. • The implicit restriction (6) to the system disturbances and uncertainties takes an explicit form when µ = 1 and the matching condition [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF] e n,n = 1. So, the finite-time control (5) designed for µ = 1 rejects the bounded disturbances realized HOSM algorithm. The HOSM version of (5) has a discontinuity only at zero similarly to the quasi-continuous HOSM algorithm [START_REF] Levant | Quasi-continuous high-order sliding mode controllers[END_REF]. • For µ = 1 the control ( 5) is globally bounded. Indeed, the equality

x T D r (V -1 )P D r (V -1 )x = 1 implies D r (V -1 )x 2 ≤ 1 λmin(P ) and u(x) ≤ k • D r (V -1 )x ≤ k √ λmin(P ) .
Hence the condition u(x) ≤ u 0 , u 0 ∈ R + can be formulated as additional LMI:

X y T y u 2 0 ≥ 0. (7)
Indeed, the inequality

u 2 =x T D r (V -1 )k T kD r (V -1 )x ≤ u 2 0 = u 2 0 x T D r (V -1 )P D r (V -1 )
x holds for all V ∈ R + and x ∈ R n such that Q(V, x) = 0. Hence k T k ≤ u 2 0 P and Schur complement gives [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF] for X = P -1 , y = kX.

• The advantage of the control design scheme presented in Theorem 6 is related to simplicity of tuning the control parameters, which is based on LMI technique. The parameters γ and α are introduced in (4) in order to tune the settling time of the closed-loop system.

Corollary 7 Let the conditions of Theorem 6 hold and the control law is modified as u(x) = u(V 0 , x), where V 0 ∈ R + is an arbitrary fixed number, then the ellipsoid

Π(V 0 )= x ∈ R n : x T (D r (V 0 )XD r (V 0 )) -1 x ≤ 1 (8)
is strictly positively invariant set of the closed-loop system (1), i.e. x(t 0 ) ∈ Π(V 0 ) ⇒ x(t) ∈ int Π(V 0 ), t > t 0 , where x(•) is any trajectory of the closed-loop system (1) and t 0 ≥ 0 is an arbitrary instant of time.

Proof. Rewrite the matrix inequality (4) in the form:

P A + A T P + P bk + k T b T P + αP + βP 2 ≤ 0. Hence we derive D r (V -1 0 )(P A + A T P + P bk + k T b T P + αP + βP 2 )D r (V -1 0 ) ≤ 0. Denoting P 0 = D r (V -1 0 )P D r (V -1 0 ) > 0 and taking into account D -1 r (V -1 0 )AD r (V -1 0 ) = V µ 0 A, D -1 r (V -1 0 )b = V 0 b
we obtain the following matrix inequality:

P 0 A + A T P 0 + P 0 bk 0 + k T 0 b T P 0 + αP0+βP0D 2 r (V0)P0 V µ 0 ≤ 0,
where

k 0 = V 1-µ 0 kD r (V -1
0 ), α > β > 0 and P 0 > 0. This means that the matrix A + bk 0 is Hurwitz, i.e. u(x) = u(V 0 , x) = k 0 x is a stabilizing linear feedback control for the system (1) with the Lyapunov function Ṽ (x) = x T P 0 x. Note that using Schur Complement the obtained matrix inequality can be transformed into the form W 0 ≤ 0, where

W 0 =   P 0 (A+bk 0 )+(A+bk 0 ) T P 0 + α V µ 0 P 0 P 0 P 0 - V µ 0 D 2 (V -1 0 ) β   .
Taking into account the inequality (6) for V = V 0 (or equivalently Ṽ = 1) we derive

d dt Ṽ (1) = x T (P 0 A+A T P 0 +P 0 bk 0 +k T 0 b T P 0 )x+2d T P 0 x = x d T W 0 x d + V µ 0 β d T D 2 r (V -1 0 )d - α Ṽ V µ 0 ≤ β -α V µ 0 < 0 for Ṽ (x) = 1.
The obtained inequality implies that the ellipsoid ( 8) is strictly positively invariant.

The next corollary proves stability of the sampled-time realization for the ILF-based control algorithm (5).

Corollary 8 Let {t i } ∞ i=0 be a strictly increasing sequence of arbitrary time instants, 0 = t 0 < t 1 < t 2 < ... such that lim i→∞ t i = +∞. Let all conditions of Theorem 6 hold and the control u is applied as follows:

u(t) = u(V i , x(t)) for t ∈ [t i , t i+1 ), where V i > 0 : Q(V i , x(t i )) = 0.
Then the origin of the system (1) is globally asymptotically stable.

Proof. Let V (x) be a positive definite function implicitly defined by the equation Q(V, x) = 0 and x(t) be a trajectory of the closed-loop system (1) with the sampled control application described above. Let us prove that the sequence {V (x(t i ))} ∞ i=1 is monotone decreasing and tends to 0. This, obviously, implies convergence of x(t) to the origin.

Consider the time interval [t i , t i+1 ) and the function Ṽi (x) = x T P i x, where

P i = D r (V -1 (x(t i )))P D r (V -1 (x(t i )) > 0. The switching control u(x) = u i (x) on this interval takes the form u i (x) = u(V (x(t i )), x) = k i x, where k i := V 1-µ (x(t i ))kD r (V -1 (x(t i ))). Repeating the proof of Corollary 7 we show Ṽi (x(t i )) > Ṽi (x(t)) for all t ∈ (t i , t i+1 ). Since V (x) is such that Q(V (x), x) = 0 then for all t ∈ (t i , t i+1 ) Q(V (x(t i )), x(t)) = x T (t)D r (V -1 (x(t i ))P D r (V -1 (x(t i ))x(t) -1 = Ṽi (x(t)) -1 < Ṽi (x(t i )) -1 = Q(V (t i ), x(t i )) = 0 = Q(V (x(t)), x(t)).
For any given x ∈ R n \{0} the function Q(V, x) is monotone decreasing for all V ∈ R + (see Condition C4) of Theorem 3 and the proof of Theorem 6). Then the obtained in- 0)) for all t > 0, i.e. the origin of the closed-loop system (1) is Lyapunov stable. The proven properties of closed-loop system also imply that x(t) and d(t, x(t)) are bounded by some constants for all t > 0.

equality implies V (x(t)) < V (x(t i )), ∀t ∈ (t i , t i+1 ]. More- over V (x(t)) ≤ V (x(
Since the function V (x) is positive definite then the monotone decreasing sequence {V (x(t i ))} ∞ i=1 converge to some limit. Let us show now that this limit is zero. Suppose the contrary, i.e.

lim i→∞ V (x(t i )) = V * > 0 or equivalently ∀ε > 0 ∃N = N (ε) : 0 ≤ V (x(t i )) -V * < ε, ∀i ≥ N . Let us represent the sampled control law u in the form u(V (x(t i )), x) = u(V * , x) + ∆ i x, ∆ i = V 1-µ (x(t i ))kD r (V -1 (x(t i ))) -V 1-µ * kD r (V -1 *
). Since the control function u = u(V, x) is continuous in R + × R n and then there exists γ ∈ K (possibly depended on µ, k, V * , n) such that ∆ i ≤ γ(ε) for all i ≥ N (ε). This means that for t > t N (ε) the closed-loop system (1) can be presented in the form

ẋ = (A + bk * )x + b∆ i x + d(t, x) (9)
where

k * = V 1-µ * kD r (V -1 * ). Consider the Lyapunov function candidate Ṽ (x) = x T P * x, where P * = D r (V -1 * )P D r (V -1 * ).
Under the assumptions made above we have Ṽ (x(t)) ≥ 1 for all t > 0 and repeating the proof of Corollary 7 we derive

d Ṽ (x(t)) dt (1) ≤ V µ * d T D 2 r (V -1 * )d β -α Ṽ (x(t)) V µ * +2x T (t)P * b∆ i x(t) ≤ -α+β V µ * + 2x T (t)P * b∆ i x(t) + V µ * (d T D 2 r (V -1 * )d -β 2 V -2µ * ) β .
Since x(t) and d(t, x(t)) are bounded by some constants for all t > 0 then for sufficiently small ε > 0 we have

d Ṽ (x(t)) dt (1)
≤ -α+β

2V µ *
< 0 for all t > t N (ε) . This contradicts the condition V (x(t)) ≥ V * for all t > 0. Consequently, lim i→∞ V (x(t i )) = 0. Hence, we derive the asymptotic stability of the closed-loop system (1). The proven corollary shows that the sampled-time realization of the developed "implicit" control scheme preserves the asymptotic stability to the origin of the closed-loop system (1) independently on the sampling period. Between two switching instants the unperturbed system is linear, so analysis of its discrete-time version can be studied using discretization schemes of linear systems (see, e.g. [START_REF] Dorf | Modern Control Systems[END_REF]).

Fixed-time Stabilization

Consider now two functions:

Q 1 (V, x) = x T D r1 (V -1 )P D r1 (V -1 )x -1, Q 2 (V, x) = x T D r2 (V -1 )P D r2 (V -1 )x -1,
where P ∈ R n×n , P = P T > 0,

r 1 = (1 + (n -1)µ, 1 + (n -2)µ, ..., 1 + µ, 1) T ∈ R n , r 2 = (1, 1 + ν, ..., 1 + (n -2)ν, 1 + (n -1)ν) T ∈ R n , 0 < µ ≤ 1 and ν ∈ R + . Denote H ν = diag{-(r 2 ) i } n i=1 .
Theorem 9 (Fixed-time stabilization) If 1) the system of matrix inequalities

             AX + XA T + by + y T b T + αX + βI n ≤ 0, -γ 1 X ≤ XH µ + H µ X < 0, -γ 2 X ≤ XH ν + H ν X < 0, X > 0, (10)
is feasible for some X = X T ∈ R n×n , y ∈ R 1×n and numbers µ ∈ (0, 1], ν, α, β, γ 1 , γ 2 ∈ R + such that α > β;

2) the control u has the form

u(V, x)= V 1-µ kD r1 (V -1 )x for x T P x < 1, V 1+nν kD r2 (V -1 )x for x T P x ≥ 1, (11)
where k = yX -1 , P = X -1 and V defined by

V = V (x) such that Q 1 (V, x) = 0 for x T P x < 1, Q 2 (V, x) = 0 for x T P x ≥ 1;
3) the disturbance function d satisfy

β 2 V -2µ ≥ sup t∈R+ d T (t, x)D 2 r1 (V -1 )d(t, x) if x T P x < 1, β 2 V 2ν ≥ sup t∈R+ d T (t, x)D 2 r2 (V -1 )d(t, x) if x T P x ≥ 1; then the closed-loop system (1) is fixed-time stable with the settling-time estimate: T (x) ≤ (γ 1 /µ + γ 2 /ν)(α -β) -1 . Proof. The functions Q 1 (V, x) and Q 2 (V, x) satisfy the con- ditions C1)-C4) of Theorem 3. Since Q 1 (1, x) = Q 2 (1, x),
then the condition C6) of Theorem 5 also holds.

Repeating the proof of Theorem 6 it can be shown that

∂Q1 ∂x (Ax+bu(x)+d(t, x)) ≤ α-β γ1 V 1-µ ∂Q1 ∂V for x T P x ≤ 1, ∂Q2 ∂x (Ax+bu(x)+d(t, x)) ≤ α-β γ2 V 1+ν ∂Q2
∂V for x T P x ≥ 1. Applying Theorem 5 we finish the proof.

If µ = 1, |d n (t, x)| ≤ β and the matching condition holds then Theorem 9 provides a HOSM control with fixed-time reaching phase.

Let us call the controls ( 5) and ( 11) by the finite-time ILF control and the fixed-time ILF control, respectively. The ILF control with µ = 1 we call the HOSM ILF control.

It is worth to stress that Corollaries 7 and 8 stay true for the case of fixed-time control (11) application. We do not prove the finite-time and fixed-time stability properties for the sampled-time realization of the ILF controls. However, the motion of the sampled system will be close to the original one if the sampling period is sufficiently small.

Aspects of practical implementation

Corollaries 7 and 8 give some remarks on possible implementation of the developed control scheme. A detailed study of sampled-time and discrete-time versions of the presented control algorithms goes beyond the scope of this paper providing the subject for a future research. In this section we provide just some general ideas to be used for ILF control implementation.

The control scheme (5) can be realized in digital control devices, which allow us to solve the equation Q(V, x) = 0 numerically and on-line for any given x ∈ R n \{0} . Rather simple numerical procedures can be utilized for this purpose. The function Q(V, x) satisfy the properties C1)-C4) of Theorem 3. So, for each fixed x ∈ R n \{0} the function Q(V ) = Q(V, x) is monotone decreasing and has the unique zero on the interval (0, +∞). In this case we may use, for example, the bisection method in order to solve the scalar equation Q(V ) = 0.

Let the control u(V, x) be given by ( 5) and the parameter V may change its value at some time instants t 0 = 0, t i > 0, i = 1, 2, .... Namely, let the control signal u(t) be defined as

u(t) = u(V i , x(t)) for t ∈ [t i , t i+1 ) and V i ∈ R + . Recall that u(V i , x) is a linear stabilizing feedback for any V i ∈ R + (see, the proof of Corollary 7). Denote x i = x(t i ). If V i = V (x i
) then stability of the sampling control scheme follows from Corollary 8.

The simplest algorithm for finding the switching control parameter V i is given below.

Algorithm 10 (Implementation of the ILF control)

INITIALIZATION: a = V min ; V 0 > V min ; b = V 0 ; STEP : If x T i D r (b -1 )P D r (b -1 )x i > 1 then a = b; b = 2b; elseif x T i D r (a -1 )P D r (a -1 )x i < 1 then b = a; a = max{ a 2 , V min }; else c = a+b 2 ; If x T i D r (c -1 )P D r (c -1 )x i < 1 then b = c; else a = max{V min , c}; endif; endif; V i = b, where V min ∈ R + is a minimal admissible value of V i .
INITIALIZATION defines a linear feedback u(V 0 , x) on the first sampling interval [t 0 , t 1 ) by means of selection of the value V 0 ∈ R + . STEP of the algorithm is applied at each sampling time instances t i in order to define the value V i (and the linear feedback u(V i , x)) for the next interval [t i , t i+1 ). Let x i ∈ R n is a current state vector. If at the sampling instant t i STEP of the algorithm is repeated many times for the same x i (for example, there exists a loop containing STEP) then Algorithm 10 realizes: 1) a localization of the unique positive root of the equation Q(V, x i ) = 0, i.e. V (x i ) ∈ [a, b]; 2) improvement of the obtained localization by means of the bisection method, i.e. |b -a| → 0. Such an application of Algorithm 10 allows us to calculate V (x i ) with a high precision, however, it requests a high computational capability of a control device. Therefore, it is more reasonable to realize STEP of Algorithm 10 just once or a few (2 or 3) times at each sampling instant. Some additional considerations are needed in order to show that the algorithm will work properly in this case.

Denote Π(V i )={x ∈ R n : x T D r (V -1 i )P D r (V -1 i )x ≤ 1}.
If the set of initial conditions is known then the value V 0 ∈ R + can be always selected such that x 0 ∈ Π(V 0 ). Otherwise, applying Algorithm 10 for sufficiently small sampling period there always exists a finite number i * ∈ {0, 1, 2, ...} such that x i * ∈ Π(V i * ). Corollary 7 implies that the ellipsoid Π(V i ) is strictly positively invariant for the closedloop system (1) with the feedback law u(x) = u(V i , x) of the form [START_REF] Moulay | Finite-time stability and stabilization: State of the art[END_REF]. Monotonicity condition ∂Q(V,x) ∂V < 0 implies that Π(V ′ ) ⊂ Π(V ′′ ) for V ′ < V ′′ . In order to guarantee stability of the sampled-time realization of the developed control, on each STEP of Algorithm 10 the upper estimate (i.e. V i = b) is selected in order to design the linear feedback u(V i , x) for the next interval

[t i , t i+1 ). Such selection ensures V i+1 ∈ (0, V i ] if x i ∈ Π(V i ) and the sequence {V i } ∞
i=i * generated by Algorithm 10 is non-increasing. The bisection procedure will operate until x(t) / ∈ Π(V min ). So, it can be shown that V i → V min as i → ∞.

The parameter V min defines the lower possible value of V and the "minimal" attractive invariant set Π(V min ) for the closed-loop system. This parameter cannot be selected arbitrary small due to finite numerical precision of digital devices. Fixed-time control application can be realized with small changes of Algorithm 10.

Remark 11 An additional advantage of the developed control scheme is related to the possible reduction of the chattering effect for HOSM ILF control application. Indeed, the HOSM control of the form (5) with µ = 1 has the unique discontinuity point x = 0. According Algorithm 10, near discontinuity point the feedback law is defined as follows: u(x) = kD(V min )x, ∀x ∈ Π(V min ), where V min ∈ R + , i.e. we always have a linear continuous control inside the ellipsoid Π(V min ). Such a modification of the control law obviously follows the classical idea of the chattering reduction developed for the first order sliding mode algorithms [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF]. Namely, for practical realization of the sliding mode control, the discontinuous feedback law can be replaced with a high-gain linear feedback if the system state is close to the switching manifold [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF].

6 Numerical examples

HOSM ILF control

Consider the system (1) for n = 4, d 1 (t, x) = d 2 (t, x) = d 3 (t, x) = 0 and |d 4 (t, x)| ≤ β = 0.2. We take d 4 (t, x) = 0.1 sin(t) + 0.1 cos(x 4 ) and x(0) = (2, 0, 0, 0) T .

The numerical solution of ODE for the closed-loop system has been obtained using the explicit Euler method with a fixed step size h ∈ R + . In order to show the effectiveness of the developed control scheme with respect to the chattering reduction, we select quite a large step size h = 0.1.

The Fig. 1(a) shows the simulation results for the closedloop system with the HOSM ILF control (5) (µ = 1) that is restricted by |u(x)| ≤ 1 (see, [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF]) and applied by the scheme presented in Algorithm 10 for V min = 0.1 and h = 0.1.

The parameters of the HOSM ILF control (5) were selected by means of solving the LMI system (4), [START_REF] Engel | A continuous-time observer which converges in finite time[END_REF] In order to compare the obtained results with some existing HOSM controller, the simulations also have been done for the fourth order sliding mode (4-SM) nested controller [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF] of the form

u = -sign x 4 + 0.5 |x 1 | 3 + |x 2 | 4 + |x 3 | 6 1 12 sign(φ) , φ = x 3 + 1 4 (|x 1 | 3 +|x 2 | 4 ) 1 6 sign x 2 +0.15|x 1 | 3 4 sign[x 1 ] .
The effective procedures for parameters adjustment of high order sliding mode control algorithms are not developed yet for n ≥ 3. So, the parameters of this controller have been selected manually. The simulations of the 4-SM nested control algorithm were initially done for the small sampling period: h = 10 -3 . They showed good performances of this control: fast convergence rate and rejection of matched bounded disturbances. The chattering effect appears in the system, when the sampling period becomes larger. The simulation results with h = 0.1 for the 4-SM nested control are depicted on the Fig. 1(b). The Fig. 1(c) shows the control inputs generated by two considered algorithms.

Note that the comparison of the sliding mode control laws for the sufficiently large sampling period h = 0.1 is motivated by practical reasons. For instance, the HOSM control is frequently demonstrated on the control problems for mobile robotics (see, for instance, [START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF]). However, the computational restrictions of autonomous mobile systems does not allow us to spend to much computational power for the generation of control actions. In practice, the reasonable sampling period can be 0.05 -0.1 second [START_REF] Guerra | Finite-time supervisory stabilization for a class of nonholonomic mobile robots under input disturbances[END_REF].

Fixed-time ILF control

Define the fixed-time control u in the form [START_REF] Shen | Global finite-time observers for lipschitz nonlinear systems[END_REF] The numerical simulation has been done using the Euler method with a fixed step size h = 0.01. The control has been applied using Algorithm 10 with V min = 0.001. The simulation results are presented on Fig. 2 for x 0 = (1, 0, 0) T and x 0 = (9, 0, 0) T . They show a "week" dependence of the convergence time on the initial conditions. Definitely, it requires a rather high magnitude of the control.

Conclusion

The paper develops the Implicit Lyapunov Function method for finite-time and fixed-time stability analysis. Using this theoretical framework new algorithms of finite-time and fixed-time stabilization for a chain of integrators and high order sliding mode control design are developed. The obtained control schemes have the following advantages:

• The control design algorithms are constructive. The schemes for tuning the control parameters have LMI representations.

• For µ = 1 the algorithms provide high order sliding mode control rejecting matched bounded disturbances.

• The digital implementation of the developed control scheme admits the effective chattering reduction by means of tuning the parameter V min . • The fixed-time ILF controllers allow us to prescribe the convergence time independently of the initial condition.

and disadvantages:

• The algorithms are applicable only in digital controllers.

• The practical realization of the developed control schemes asks for additional computational power of the digital control device, which is required for on-line computation of the ILF value at the current state.

The developed Implicit Lyapunov Function method for finite-time and fixed-time stability analysis is promising to tackle many other problems such as fast observation and estimation, development of fast adaptation algorithms or fast consensus protocols, etc.

  where x ∈ R n is the state vector, u ∈ R is the control input, and the function d : R + × R n → R n describes the system uncertainties and disturbances. The whole state vector x is assumed to be measured. Let the function d be measurable locally bounded uniformly in time, i.e. sup t∈R+,x∈R n : x <δ d(t, x) < ∞ for any δ > 0. Both the control function u and the function d are admitted to be discontinuous with respect to x. For example, the function d may describe unknown dry friction of a mechanical model.

  for µ = 1 and α = 0.5, β = 0.2, γ = 10, u 0 = 1: -0.0330, -0.2522, -0.7823, -1.1386).

2 = 4 . 5 .

 245 with n = 3, µ = 0.5, ν = 0.1, where the matrix k = (-2.9319, -5.6235, -2.6998) are obtained from the LMI[START_REF] Bejarano | High order sliding mode observer for linear systems with unbounded unknown inputs[END_REF] with β = 0, α = 1, γ 1 = 4.5, γ In this case the settling time estimate provided by Theorem 9 gives T (x 0 ) ≤ 54.

4 (

 4 b) The 4-SM nested control[START_REF] Levant | Homogeneity approach to high-order sliding mode design[END_REF].

  nested control (c) Comparison of control inputs.

Fig. 1 .

 1 Fig. 1. The simulation results for the HOSM ILF control (Vmin = 0.1) and the 4-SM nested control.

  (1, 0, 0) T x 0 =(9, 0, 0) T (c) Control inputs.

Fig. 2 .

 2 Fig. 2. The simulation results for the fixed-time ILF control.

  ∃T max ∈ R + such that T (x 0

	Definition 2 ([23]) The origin of system (2) is said to be
	globally fixed-time stable if it is globally uniformly finite-
	time stable and the settling-time function T is globally
	bounded, i.e.
	5 sign[x(t)], x ∈ R is globally uniformly finite-time stable, since its settling-time
	function T is locally bounded: T (x 0 ) = 2 |x 0 |.

  holds, i.e. d i (t, x) ≡ 0 for i = 1, ..., n -1 and d = (d 1 , ..., d n ) T . In this case the condition (6) becomes |d n (t, x)| ≤ β and the term βI n in (4) can be replaced with βE, where the matrix E = {e ij } ∈ R n×n has only one nonzero element:


J Kappus

G Mabon

ADAPTIVE DENSITY ESTIMATION IN DECONVOLUTION PROBLEMS WITH UNKNOWN ERROR DISTRIBUTION

Keywords: Adaptive estimation, Deconvolution, Density estimation, Mean squared risk, Nonparametric methods, Replicate observations

We investigate the data driven choice of the cutoff parameter in density deconvolution problems with unknown error distribution. To make the target density identifiable, one has to assume that some additional information on the noise is available. We consider two different models: the framework where some additional sample of the pure noise is available, as well as the model of repeated measurements, where the contaminated random variables of interest can be observed repeatedly, with independent errors. We introduce spectral cutoff estimators and present upper risk bounds. The focus of this work lies on the optimal choice of the bandwidth by penalization strategies, leading to non-asymptotic oracle bounds.

Introduction

This paper addresses the problem of the adaptive bandwidth selection via penalization in deconvolution problems with unknown error distribution. We study in parallel two different models. In both models, we assume that the random variables are real-valued.

Model 1. The random variable X of interest is perturbed by some additional additive error ε, independent of X, so the empirically accessible quantity is Y = X + ε. The distribution of the noise is assumed to be unknown. One observes n independent copies of Y :

Y j = X j + ε j , j = 1, . . . , n.
In addition it is assumed that a sample (ε -j ) j=1,...,M of the pure noise, independent of the Y j , is available.

Model 2. The noisy random variable X can be measured repeatedly, with independent errors. The observations are then of the form, Y j,k = X j + ε j,k , j = 1, . . . , n and k = 2.

All the X j are assumed to be independent and identically distributed and all the ε j,k are independent and identically distributed and independent of the X j . Again, it is assumed that the distribution of the ε j,k is unknown. In addition, the error terms are assumed to be centered.

Density deconvolution is a classical topic in nonparametric statistics and a large amount of literature on this subject has been published since the late 80s. Rates of convergence and their optimality have been studied, for example, in [START_REF] Carroll | Optimal rates of convergence for deconvolving a density[END_REF], [START_REF] Stefanski | Rates of convergence of some estimators in a class of deconvolution problems[END_REF], [START_REF] Stefanski | Deconvoluting kernel density estimators[END_REF], [START_REF] Fan | On the optimal rates of convergence for nonparametric deconvolution problems[END_REF] and [START_REF] Efromovich | Density estimation for the case of supersmooth measurement errors[END_REF]. For the study of sharp asymptotic optimality, see [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF], Butucea and Tsybakov (2008a,b). Adaptive estimation for deconvolution problems has then been investigated by [START_REF] Pensky | Adaptive wavelet estimator for nonparametric density deconvolution[END_REF], who apply wavelet techniques, by [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF], who consider the adaptive bandwidth selection for projection estimators and by [START_REF] Butucea | Adaptive estimation of linear functionals in the convolution model and applications[END_REF] for linear functionals. We can also cite [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF] in a multivariate setting.

However, the above mentioned papers have been working under the assumption that the distribution of the errors is perfectly known, which is clearly not realistic in most fields of application. The systematic study of deconvolution with unknown error distribution in presence of an additional noise sample, which corresponds to Model 1, dates back to the late 90s. For the study of convergence rates, see [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], [START_REF] Johannes | Deconvolution with unknown error distribution[END_REF] or [START_REF] Meister | Deconvolution Problems in Nonparametric Statistics[END_REF].

The rigorous study of adaptive procedures in a deconvolution model with unknown errors has only recently been addressed. We are aware of the work by [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF], by [START_REF] Johannes | Adaptive circular deconvolution by model selection under unknown error distribution[END_REF] who consider a model of circular deconvolution and by [START_REF] Dattner | Adaptive quantile estimation in deconvolution with unknown error distribution[END_REF], who deal with adaptive quantile estimation via Lespki's method.

In comparison to the classical deconvolution model with known errors, the research on the model of repeated measurements has not been so intense. For the more general model of repeated measurements with skew error densities, rates of convergence have been studied in [START_REF] Li | Nonparametric estimation of the measurement error model using multiple indicators[END_REF] and recently been improved in [START_REF] Comte | Density deconvolution from repeated measurements without symmetry assumption on the errors[END_REF]. Consistent estimation under minimal a priori assumptions is investigated in [START_REF] Neumann | Deconvolution from panel data with unknown error distribution[END_REF]. For repeated measurements with symmetric error density, we refer to [START_REF] Delaigle | On deconvolution with repeated measurements[END_REF] and Comte et al. (2012). This model has various applications in economics, see, for example, [START_REF] Bonhomme | Generalized non-parametric deconvolution with an application to earnings dynamics[END_REF], but also in a medical context, see Comte et al. (2012).

In the last mentioned paper, as well as in [START_REF] Delaigle | On deconvolution with repeated measurements[END_REF], practical strategies for the adaptive bandwidth selection have been proposed, but a theoretical justification is not given which is a motivation for the rigorous study presented in this paper. Essential tools for our approach rely on considerations presented in [START_REF] Kappus | Adaptive nonparametric estimation for Lévy processes observed at low frequency[END_REF]. There are also many common points with recent contributions by [START_REF] Dattner | Adaptive quantile estimation in deconvolution with unknown error distribution[END_REF]. In comparison to the last mentioned authors, the main difference lies in the fact that their approach is minimax and asymptotic whereas we are interested in non-asymptotic oracle bounds, thus following the model selection paradigma in the sense of [START_REF] Birgé | An alternative point of view on Lepski's method[END_REF] and [START_REF] Massart | Concentration Inequalities and Model selection[END_REF].

Model 1 corresponds, in many respects, to the situation which has been investigated in [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF]. Let us clarify the essential differences: in a deconvolution model with estimated characteristic function of the errors, the risk bounds are determined by the size M of the noise sample, as well as the number n of observations of Y . For sample sizes M ≥ n the risk bounds correspond to the model with known error distribution. However, for M < n the bound on the risk gets worse. The approach by [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] is tailored for the case where M is, by a polynomial factor, larger than n, and cannot be extended to M < n. The additional considerations presented in the present work allow to handle the case of small noise samples. This seems to be of some practical relevance when one turns away from the classical measurement error model and regards, in some context of physics or biology, ε as a signal overlying some other signal X. In such a framework, the size of the noise sample will be determined by some extraneous influence and the assumption that M > n may fail to hold true.

We want to emphasize another important difference between our reasoning and the arguments given in [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF], but also in [START_REF] Dattner | Adaptive quantile estimation in deconvolution with unknown error distribution[END_REF]. The last mentioned papers do always work under the standing assumption that the interesting density and error density belong to certain prescribed classes of functions. More precisely, it is assumed therein that the characteristic functions have an exponential or polynomial decay behavior. We are able to dispose completely of any of such semi-parametric assumptions. This is motivated by arguments given in [START_REF] Birgé | An alternative point of view on Lepski's method[END_REF]. From a model selection point of view, rather than considering a family of parameter sets and aiming at building an estimator which is simultaneously asymptotically minimax, the target is to find the best estimator within a collection leading to non-asymptotic oracle inequalities. It is argued in [START_REF] Birgé | An alternative point of view on Lepski's method[END_REF] that these considerations make sense without specifying any particular family of parameter sets. From this point of view, it is desirable to avoid, as far as possible, any a priori parametric assumptions and provide a fully general treatment.

This paper is organized as follows. In Section 2, we fix the notation and assumptions, introduce the estimators and present upper risk bounds. In Section 3, the data driven choice of the cutoff parameter is investigated. We introduce penalized criteria and derive non asymptotic oracle bounds for the corresponding estimators. In Section 4, we present some data examples to illustrate the practical performance of our estimator. All proofs are postponed to Section 6.

Statistical model, estimation procedure and risk bounds

In the present section we fix the statistical model and assumptions, introduce the estimators and recall, for the readers convenience, the non asymptotic risk bounds which have been presented in earlier publications on the subject. We start by introducing some notation which will be used throughout the rest of the text.

2.1. Notations. For two real numbers a and b, a ∨ b := max(a, b) and a ∧ b := min(a, b). For two functions ϕ, ψ : R → C belonging to L 1 (R) ∩ L 2 (R), ϕ denotes the L 2 -norm of ϕ, defined by ϕ 2 = R |ϕ(x)| 2 dx, and ϕ, ψ the scalar product between ϕ and ψ, defined by ϕ, ψ = R ϕ(x)ψ(x) dx. The Fourier transform ϕ * is defined by

ϕ * (x) = e ixu ϕ(u) du.
Besides, if ϕ * belongs to L 1 (R) ∩ L 2 (R), then the function ϕ is the inverse Fourier transform of ϕ * and can be written ϕ(x) = 1/(2π) e -ixu ϕ * (u) du. The convolution product * is defined as (ϕ * ψ)(x) = ϕ(x -u)ψ(u) du. Lastly, we abbreviate Model 1 by ns-model, where ns stands for noise sample, and Model 2 by rd-model for replicate data.

Statistical model and estimators.

In the situation of Model 1 and Model 2 defined in the introductory part, the target is to recover the density f of X from the data. In the sequel, we limit our considerations to the case where the number k of repeated measurements is equal to two. This setting easily generalizes to a higher dimensional model. However, for sake of clarity, we omit the details.

Throughout the paper, we make the following assumptions:

(A1) X and ε have square integrable densities f and f ε w.r.t. Lebesgue measure.

(A2) ∀x ∈ R, f * ε (x) = 0. (A3) f ε is symmetric around zero. Let f Y denote the density of X + ε. By independence of X and ε, f Y = f * f ε . Under (A2), we have the equality f * = f * Y /f * ε . An estimator f * Y of f Y can be calculated from the data. If f * ε is known, an unbiased plug-in-estimator of f * is then given by f * Y /f * ε .
The inverse Fourier transform is then applied to get an estimate of f . However, since neither f * Y nor 1/f * ε are integrable, it is necessary to apply some regularization technique, for example, a spectral cutoff estimator. In this particular case, the estimator of f would be 1/(2π) |u|≤πm e -iux f * Y (u)/f * ε (u) du. We can notice that this estimator corresponds both to a kernel estimator built with a sinc kernel [START_REF] Butucea | Deconvolution of supersmooth densities with smooth noise[END_REF]) or to a projection type estimator as in [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF].

In the present case, the error distribution is assumed to be unknown. To make the problem identifiable, some additional information on the noise is required. In the ns-model, we introduce the empirical characteristic function of ε,

∀u ∈ R, f * ε,ns (u) = 1 M M j=1
e iuε-j .

In the same way, f * Y is estimated by its empirical version

∀u ∈ R, f * Y,ns (u) = 1 n n j=1 e iuYj .
Secondly the identification of f * ε is also possible in the model of repeated observations. Under the symmetry assumption (A3), we have the following equalities

∀u ∈ R, E e iu(Yn+j,1-Yn+j,2) = E e iu(εn+j,1-εn+j,2) = E e iuεn+j,1 2 = E e iuεn+j,1 2 = (f * ε (u)) 2 . (1)
When (A3) is violated, the model is much more complicated and requires a completely different approach since f * ε is not a real positive function anymore. For further discussion, see [START_REF] Comte | Density deconvolution from repeated measurements without symmetry assumption on the errors[END_REF]. Formula (1) suggests to define the following unbiased estimator of (f * ε ) 2 :

∀u ∈ R, f * 2 ε,rd (u) = 1 n n j=1 cos (u(Y j,1 -Y j,2 )) .
Moreover, an unbiased estimator of f * Y is given by

∀u ∈ R, f * Y,rd (u) = 1 2n n j=1
e iuYj,1 + e iuYj,2 .

When considering the empirical characteristic functions, one has to be careful about the fact that the Y j,k are not independent of each other, nor independent of the ε j,k .

Finally, one has to pay attention to the fact that small values of the empirical characteristic function in the denominator lead to unfavorable effects and a bad performance of the estimator. This is an immediate consequence of the fact that a reasonable estimation of the ratio 1/f * ε is impossible as soon as the denominator is smaller than the standard deviation. This phenomenon has been investigated in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF]. This entails the necessity to consider a regularized version of the empirical characteristic function in the denominator. [START_REF] Delaigle | On deconvolution with repeated measurements[END_REF] propose a ridge-parameter approach. However, this requires a careful discussion of the choice of the ridge parameter. For this reason, we prefer the completely data driven approach proposed in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], which has also been applied by [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] for the ns-model and by Comte et al. (2012) for the rd-model. This approach corresponds to the following estimators of the ratio:

1 f * ε,ns (x) = 1 | f * ε,ns (x)| ≥ M -1/2 f * ε,ns (x) and 1 f * ε,rd (x) := 1 f * 2 ε,rd (x) ≥ n -1/2 f * 2 ε,rd (x)
.

These definitions lead to defining the empirical versions of f * as follows:

f * ns (u) := f * Y,ns (u) f * ε,ns (u) and f * rd (u) := f * Y,rd (u) f * ε,rd (u)
.

Finally, the objects to be estimated are characteristic functions, so the absolute values are bounded by 1.

For this reason, one should prefer the regularized versions

f * ns (u) := f * ns (u) max{ f * ns (u) , 1} and f * rd (u) := f * rd (u) max{ f * rd (u) , 1}
.

Given any positive, real valued m, the above considerations lead to defining the spectral cutoff estimators of f as follows:

fm,ns (x) := 1 2π e -iux f * m,ns (u) du and fm,rd (x) :=

1 2π e -iux f * m,rd (u) du, (2)
with f * m,ns (u) := f * ns (u)1 [-πm,πm] (u) and f * m,rd (u) := f * rd (u)1 [-πm,πm] (u)
. Moreover, we use the notation

f m (x) := 1 2π e -iux f * m (u) du and f * m (u) := f * (u)1 [-πm,πm] (u).
2.3. Upper risk bounds. The following non-asymptotic risk bounds are valid for the estimators defined in the preceding section.

Proposition 2.1. (i) In presence of an additional noise sample, under (A1)-(A2), there exists a universal positive constant

C such that E f -fm,ns 2 ≤ 2 f -f m 2 + C 1 n πm -πm 1 |f * ε (u)| 2 du + 1 M πm -πm |f * (u)| 2 |f * ε (u)| 2 du . (3)
(ii) In the model of repeated measurements, under (A1) -(A3), there exists a universal positive constant

C ′ such that E f -fm,rd 2 ≤ 2 f -f m 2 + C ′ 1 n πm -πm 1 |f * ε (u)| 2 du + 1 n πm -πm |f * (u)| 2 |f * ε (u)| 2 (|f * ε (u)| 2 ∨ n -1/2 ) du . (4)
Remark 1. The first two terms on the right-hand side of Equations ( 3) and (4) correspond to the usual terms when the distribution of the errors is known: the squared bias term and a bound on the variance. The last term is due to the estimation of f * ε which depends on the considered model. These bounds have already been established in the literature on deconvolution, see [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF], [START_REF] Delaigle | On deconvolution with repeated measurements[END_REF] or [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF], Comte et al. (2012). We can hence omit the proof.

In view of the rates of convergence, we observe the following: consider first the ns-model. If M ≥ n, there is no loss in the rate in comparison to a deconvolution problem with known f ε . However, a loss in the rate may occur for M < n. More precisely, if the ratio M/n is small, in comparison to f * /f * ε , the estimator does not achieve the rates of convergence which are known to be optimal for deconvolution with known error distribution. This is intuitive, since f can only be identified through f ε and there is no hope to estimate f with high precision when the information on f ε is not reliable. For a detailed discussion and minimax lower bounds, we refer to [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF]. Next, consider the rd-model. From Equation (4), one derives immediately, that there is no loss in the rate, in comparison to deconvolution with known error distribution, if the decay of f * outbalances the decay of f * ε . If this is no longer true, the following holds: the smoother f ε is, in comparison to f , the worse are the resulting rates of convergence. It can be shown that a loss in the rate is unavoidable in this context (Alexander Meister, personal communication), but to the best of our knowledge, minimax lower bounds have not been published for this particular case.

Data driven bandwidth selection and oracle bounds

The goal of this section is to provide a strategy for the optimal data driven choice of the smoothing parameter. Given a collection M of cutoff parameters, which may vary with M and n, the bandwidth m should ideally outbalance the bias and variance term displayed in Equations ( 3) and ( 4). This trade-off is easier to realize when the variance term is assumed to be known, which is often the case in the literature on model selection. In a deconvolution problem with perfectly known error distribution m should mimic the oracle choice

m th = argmin m∈M -f m 2 + 1 n 1 2π πm -πm (1 -|f * Y (u)| 2 ) |f * ε (u)| 2 du .
In the present framework, the considerations are even more involved since the characteristic function in the denominator is unknown and the variance is hence not feasible to actually compute. Following the model selection paradigm, see [START_REF] Birgé | From model selection to adaptive estimation[END_REF], [START_REF] Birgé | An alternative point of view on Lepski's method[END_REF] or [START_REF] Massart | Concentration Inequalities and Model selection[END_REF], we select m as the minimizer of a penalized criterion such that

m = argmin m∈M -fm 2 + pen(m) .
The penalty term should be chosen large enough to annihilate the fluctuation of fm around its target, for all m in the model collection M simultaneously, but on the other hand, should ideally be as close as possible to the variance term in order to preserve the non-asymptotic risk bounds. In a model selection problem with known variance, the penalty term is deterministic, which is no longer the case in the present situation. Before introducing the stochastic penalty terms, we shall need the following definitions: for δ > 0 and u ∈ R,

w(u) := (log(e + |u|)) -1 2 -δ . Moreover k N (u) := N -1/2 (log N ) 1/2 w(u) -1 , N = n, M.
The weight function w has been introduced in [START_REF] Neumann | Nonparametric estimation for Lévy processes from low frequency observations[END_REF] and the considerations presented in that paper, combined with ideas given in [START_REF] Kappus | Adaptive nonparametric estimation for Lévy processes observed at low frequency[END_REF] play an important role for our arguments. Since the penalty terms will involve an empirical version of the characteristic function in the denominator, the oracle inequalities depend on a precise control of the deviation of f * ε from f ε , simultaneously on the real line. It is shown in [START_REF] Neumann | Nonparametric estimation for Lévy processes from low frequency observations[END_REF] that the distance between both object, weighted by w, is simultaneously small on the real axes. In the penalty terms, there will hence occur a loss of logarithmic order, in comparison to the variance term.

Let us now introduce the stochastic penalty terms. In the ns-model,

pen ns (m) := pen 1,ns (m) + pen 2,ns (m), := 16 3π πm -πm τ 2 Y k 2 n (u) | f * ε,ns (u)| 2 du + 2 3π πm -πm τ 2 ε k 2 M (u)| f * Y,ns(u) | 2 | f * ε,ns (u)| 4 du, (5)
with τ Y = τ ε = √ 6. In the rd-model, pen rd (m) := pen 1,rd (m) + pen 2,rd (m),

:= 16 3π πm -πm τ 2 Y k 2 n (u) | f * ε,rd (u)| 2 du + 4 9π πm -πm τ 2 ε k 2 n (u)| f * Y,rd(u) | 2 | f * ε,rd (u)| 6 du, (6)
with τ Y as previously and τ ε = √ 3.

The cutoff parameters are selected as the minimizers of the following penalized criteria.

τ 2 Y k 2 n (u) |f * ε (u)| 2 du + 2 3π πm -πm τ 2 ε k 2 M (u)|f * (u)| 2 |f * ε (u)| 4 du.
and pen rd (m) := pen 1,rd (m) + pen 2,rd (m)

:= 16 3π πm -πm τ 2 Y k 2 n (u) |f * ε (u)| 2 du + 4 9π πm -πm τ 2 ε k 2 n (u)|f * (u)| 2 |f * ε (u)| 4
du.

We are now ready to formulate the oracle bounds and hence the main result of this paper:

Theorem 3.1. (i) Let M be a collection of cutoff parameters, with max M ≤ √ M ∧ n.
Assume that (A1) and (A2) are satisfied. Let m ns be defined by (7) and f mns,ns according to (2). Then there exists a universal positive constant C ad and a positive constant C depending on the particular choice of τ and δ, but not on any of the underlying distributions, such that

E f -f mns,ns 2 ≤ C ad inf m∈M f -f m 2 + pen ns (m) + C M ∧ n . (ii) Let M a collection of cutoff parameters with max M ≤ √ n. Assume that (A1) -(A3) are satisfied. Let
m rd and f mrd,rd be defined according to (7) and (2). Then there is a universal positive constant C ad and a positive constant C depending on the choice of γ and δ, but not on the underlying distributions, such that

E f -f mrd,rd 2 ≤ C ad inf m∈M f -f m 2 + pen rd (m) + C n .
Remark 2. It is remarkable that we are able to establish non-parametric oracle bounds which make sense without specifying any particular semi-parametric model. Related problems are frequently discussed under specific a priori assumptions on the decay behavior of f * and f * ε , see for example [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF] or [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF]. In the present work, we can completely dispose of any such assumptions, so our approach is as general as possible.

Another interesting point about our considerations is the following: the only assumption imposed on the collection M of cutoff parameters is some upper bound on the largest index. No further specification is necessary and we may work with an arbitrarily fine grid, allowing good approximation properties. This is a consequence of the fact that our proofs rely on one sole application of the Talagrand inequality. Additional applications of the Bernstein inequality and sums over M are not required.

Finally, it is worth emphasizing that, to the best of our knowledge, the non-asymptotic oracle bounds for the rd-model are completely new and the same is true for the bounds in the ns model with M < n.

We have considered the problem from a non-asymptotic perspective, but Theorem 3.1 entails, from the asymptotic and minimax point of view, the following observation: in those cases where minimax rates of convergence are known, the procedure achieves, up to a logarithmic loss, automatic adaptation over prescribed non-parametric function classes, typically Sobolev-spaces or classes of super-smooth functions. ⊲ Choose m such as m = argmin m∈M -fm 2 + pen(m) .

⊲ Compute f m (x) = 1 2π π m -π m e -ixu f * Y (u) f * ε (u) du.
As often in model selection methods, the values of the constants in the penalty, here denoted by τ Y and τ ε , which are obtained from the theory are too large in practice. Therefore a calibration step is required and done: for a small set of densities and different sample sizes the mean integrated squared error (MISE) is computed in order to determine admissible range for the values of the constants (see [START_REF] Baudry | Slope Heuristics: Overview and Implementation[END_REF] for a description of this step). Finally, the penalties are chosen according to Equations ( 5) and ( 6) with τ Y replaced by 0.6 and τ ε by 0.3 for the ns-model and rd-model. Besides we consider the model collection

M = {m = k/10, 1 ≤ k ≤ 25}.
In practice, one can take k in a much larger set and propose larger values for m; the first selected value can be followed by another run of the estimation algorithm with a thinner grid of proposals around the selected value. We limited the set here because the proposed values seemed adequate and allowed less time-consuming repeated experiments.

In the sequel we also use the notations ror and rad define as follows

ror = min m∈M Ê f -fm 2 and rad = Ê f -f m 2 ,
where Ê is the approximation of theoretical expectation computed via Monte-Carlo repetitions.

The whole implementation is conducted using R software. The integrated squared error (ISE) f -f m 2 is computed via a standard approximation and discretization (over 300 points) of the integral on an interval of R denoted by I. Then the MISE E f -f m 2 is computed as the empirical mean of the approximated ISE over 500 simulation samples. [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] and influence of M in the NS-model. We compute different estimators of the signal for different values of M and consider different signal densities and two noises. Following [START_REF] Comte | Penalized contrast estimator for adaptive density deconvolution[END_REF], we study the following densities on the interval I:

Comparison with

⊲ Laplace distribution, f (x) = e - √ 2|x| / √ 2, I = [-5, 5]. ⊲ Mixed Gamma distribution : X = W/ √ 5.48, with W ∼ 0.4Γ(5, 1) + 0.6Γ(13, 1), I = [-1.5, 26]. ⊲ Cauchy distribution : f (x) = π(1 + x 2 ) -1 , I = [-10, 10]. ⊲ Standard Gaussian distribution, I = [-4, 4].
All the densities are normalized with unit variance except the Cauchy density.

We consider the two following noise densities with same variance 1/10.

⊲ Gaussian noise : f ε (x) = 1 σ ε √ 2π exp - x 2 2σ 2 ε , f * ε (x) = exp - σ 2 ε x 2 2 . ⊲ Laplace noise : f ε (x) = 1 2σ ε exp - |x| σ ε , f * ε (x) = 1 1 + σ 2 ε x 2 .
We want to study the influence of the relationship of n and M on the estimation of f in the ns-model. We then consider different values of n and values of M = √ n and M = n.

Results. The results of the simulations are given in Table 1. Table 1 illustrates the case where we can recover a preliminary sample of the noise ε (our so-called ns-model). First we see that the risk decreases when the sample size increases. Likewise the risk increases when the variance increases. Secondly the results are very close to those of [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF]. Nevertheless our procedure is equivalent or better.

The main improvement is that our procedure has better performances when the size of the preliminary sample is small. Moreover we show that the procedure does not need big M since we reach the same performances as [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] for M = n when they take M = n 2 . Indeed if we consider the mixed Gamma distribution for n = M = 500, our risk is 0.232 with a Laplace noise, while [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] with M = n 2 have 0.382. We can make the same remarks for the other distributions (except for the Laplace distribution) with a sample size 100 or 250 and with the two noise distributions. For the Laplace distribution, the results of [START_REF] Comte | Data-driven density estimation in the presence of additive noise with unknown distribution[END_REF] are better but they do not outperform ours.

Nonetheless, in our model, the risk decreases more rapidly when M and n increase.

Effect of the variance. We also test how our procedure behaves when the variance is increased. In Table 2, we present the results of simulations where the variance takes the values 1/4, 1/2 and 1. We only report the case of a Gaussian error distribution since the results of the Laplace error are very similar. Moreover a Gaussian distribution is a case less favorable. Indeed its Fourier transform decays exponentially and it is known to imply possibly slower asymptotic rates. with a Laplace and Gaussian noise of variance 1/10, rad is the risk of the adaptive estimator, ror is the risk of the oracle estimator, M is the size of the noise and n is the size of the Y i 's. with a Gaussian noise, rad is the risk of the adaptive estimator, ror is the risk of the oracle estimator, M is the size of the noise and n is the size of the Y i 's.

σ 2 ε = 1/4 σ 2 ε = 1/2 σ 2 ε = 1 f M n
Results. The results are reported in Table 2. As before the risk decreases when n and M increase.

Similarly when we increase the contamination of the variable of interest by increasing the variance of the error distribution, the risk increases. The procedure performs still well since the adaptive risk is close to the oracle risk around twice bigger.

4.3.

Illustrations in the RD-model. For this model, we use the same signal and error distributions as described in 4.2. We consider different values of n: 200 and 2000, with variance 1/10 and 1/2.

Results. The results are reported in Table 3. We note that the values of the MISE are very close for both error distributions. Moreover the adaptive risk is also close to the oracle risk. It is multiplied by approximately 1.5 when the variance of the error distribution is 1/10 and 2 when the variance equals 1/2. Again the risk decreases when n and M increase. And the risk increases when the variance increases.

4.4.

Comparison with a kernel estimator. Recently, some papers as [START_REF] Comte | Finite sample penalization in adaptive density deconvolution[END_REF] have investigated the necessity of inversion in statistical inverse problem. The idea is to compare the performances of a simple kernel estimator directly applied to the data Y i with the adaptive estimator of the ns-model. This model allows us to choose a non symmetric noise which makes the estimation with a kernel estimator more difficult. That is why we compare our estimator with a kernel in two contexts: one with a symmetric noise and another one with an unsymmetric noise.

σ 2 ε = 1/10 σ 2 ε = 1/2 n =
First, for the symmetric noise, we take the results of Table 2 with variance 1/2 for a Gaussian noise with M = n and compute the kernel estimator for this design.

Secondly we compute the estimator of the signal in the ns-model with M = ⌊ √ n⌋ with an unsymmetric noise. We consider two densities defined in the beginning of this section: mixed Gamma and Gaussian. We choose the error distribution as follows

⊲ Gamma noise: f ε (x) = x α-1 β α e -βx Γ(α) 1 x≥0
with parameters α = 1 and β = 2. The variance of the error distribution is then 1/4. For the kernel estimator, we use the function density of R with a Gaussian kernel where the bandwidth is selected by cross-validation. 4. Comparison of the method with the results of a Gaussian kernel estimator with a symmetric noise (MISE×100 averaged over 500 simulations), rker is the risk of the kernel estimator, rad is the risk of the adaptive estimator, ror is the risk of the oracle estimator, M is the size of the noise and n is the size of the Y i 's. 5. Comparison of the method with the results of a Gaussian kernel estimator with an unsymmetric noise (MISE×100 averaged over 500 simulations), rker is the risk of the kernel estimator, rad is the risk of the adaptive estimator, ror is the risk of the oracle estimator, M is the size of the noise and n is the size of the Y i 's.

f
f n = 100 n = 250 n = 500
Results. The results are reported in Tables 5 and4. In Table 4, we see that the results are very close. For the mixed Gamma of the kernel the results are a little bit better but the risk of our estimator decreases more rapidly. For the Gaussian distribution our results are better and the risk decreases also more rapidly when the sample size increases..

For the unsymmetric noise the results are reported in Table 5. We see that for the mixed Gamma, the kernel estimator performs unexpectedly well. For n = 100, the risks are practically the same as those of the adaptive estimator. When n increases the adaptive estimator performs better but the kernel estimator still gives satisfying results.

On the other hand, the results of the Gaussian distribution illustrate well the importance of inversion in statistical inverse problem. Indeed the risk with a kernel estimator is around 3.10 -2 for the diverse values of n while for the adaptive estimator the risk is divided by 3. More precisely for n = 100 the risk of the kernel estimator is multiplied by 2 compared to the adaptive estimator, by 5 for n = 250 and by 7 for n = 500. So when the error distribution is unknown and the signal to noise ratio (named s2n) is not too large, our procedure is worthy of interest. When the s2n gets larger, we only expect the deconvolution procedure not to deteriorate the results compared to direct estimation. It has been checked to be true for known noise density by [START_REF] Comte | Finite sample penalization in adaptive density deconvolution[END_REF] see section 4.7 and 4.9. As in practice the value of the s2n is unknown, our procedure is always recommended.

Figures 1 and2 illustrate the estimation of a mixed Gamma using both penalized estimators in the cases for n = M = 200 and n = M = 500. The estimation is made with additional Gaussian and Laplace noises with a variance of 1/10. The bimodal specificity of the density is well described. Moreover the precision increases with the sample size.

Concluding remarks

This paper deals with adaptive deconvolution estimation of a density when the noise density is unknown. We have considered two cases: one where a preliminary sample of the noise can be observed and another one where the variable of interest X can be observed repeatedly with independent errors. For both models, we have proposed a theoretical adaptive procedure which automatically makes a data driven bias-variance compromise. Moreover it allows us to not specify rates of convergence since they are mechanically reached. This procedure enables us to treat the problem of adaptive estimation in repeated observation model which is completely new. The estimation procedure relies on the independence of the estimators of the characteristic functions of f * Y and f * ε . Its advantage is to be very general under weak assumptions. Indeed, that procedure takes into account cases where there can be small number of replications which matches realistic applications as in medicine or economics. Besides of its theoretical properties, our procedure has showed good performances in simulation.

At last we think that our procedure can be extended to density estimation of a random effect in linear mixed-effects model. Indeed we are aware of the work of Comte and Samson (2012) who proposed an adaptive procedure based on deconvolution methods in the unknown-error case which is not optimal and Dion ( 2013) who used a Lepski's method in the known-error case. In that model, the noise can also be recovered by successive difference similarly to the repeated model but the characteristic function of the noise would be raised to a greater power. We may then propose in the same spirit an adaptive procedure for the random effect in linear mixed-effects model.

6. Proofs 6.1. Preliminaries. We start by restating, for the reader's convenience, the following version of Talagrand's inequality:

Lemma 6.1. Let I be some countable index set. For each i ∈ I, let Z

|S (i) n | ≤ 3 2 E sup i∈I |S (i) n | + κ ≤ 2 exp - nκ 2 c 1 v 2 + c 2 κ .
A proof of this result is given in [START_REF] Massart | Concentration Inequalities and Model selection[END_REF], see page 170. It follows from the arguments presented therein that for any r, s > 0 with 1 s + 1 r = 1, the constants can be chosen c 1 = 2s 2 and c 2 = 6r.

The following result will be essential for the theoretical justification of the adaptive procedure. The proof was given in [START_REF] Kappus | Adaptive nonparametric estimation for Lévy processes observed at low frequency[END_REF]. It uses a fundamental Lemma shown in [START_REF] Neumann | Nonparametric estimation for Lévy processes from low frequency observations[END_REF] , combined with Lemma 6.1.

Lemma 6.2. Let Z 1 , . . . , Z n be i.i.d. random variables. Assume that E|Z 1 | ≤ m Z for some m Z > 0. Let f * Z denote the empirical characteristic function. For some δ > 0, let w(u) = (log(e + |u|)) -1/2-δ .

Assume that τ > 2 √ p. Then there exists a positive constant C depending on m Z and the choice of γ and δ such that for arbitrary n ∈ N:

P ∃u ∈ R : | f * Z (u) -f * Z (u)| > τ (log(n)) 1/2 w(u) -1 n -1/2 ≤ Cn -p (8)
Remark 3. In the situation of the preceding Lemma: if the Z j are assumed to have a symmetric distribution and the empirical characteristic function is defined to be f * Z (u) = 1/n n j=1 cos(uZ j ), it is enough to assume τ > √ 2p to obtain (8).

The next result has been formulated and proved in [START_REF] Neumann | On the effect of estimating the error density in nonparametric deconvolution[END_REF].

Lemma 6.3. Let Z 1 , . . . , Z n be i.i.d. random variables. Let f * Z be the true and f * Z the empirical characteristic function. Moreover let

1/ f * Z (u) = 1{| f * Z (u)| ≥ n -1/2 }/ f * Z (u).
Then for arbitrary p ∈ N, there exists a positive constant C depending only on p such that

E 1 f * Z (u) - 1 f * Z (u) p ≤ C 1 |f * Z (u)| p ∧ n -p 2 |f * Z (u)| 2p . (9)
Remark 4. In the preceding lemma, one should keep in mind that in the rd-model, Y j,1 -Y j,2 = ε j,1 -ε j,2 plays the role of Z j .

Lemma 6.3 yields the following useful corollary, which allows to compare the stochastic penalty terms to their deterministic counterparts.

Corollary 6.4. Let the penalty terms be defined according to Equations (5) and (6). Then for some C > 0,

E [ pen ns (m)] ≤ Cpen ns (m) and E [ pen rd (m)] ≤ Cpen rd (m).
Proof. It is enough to consider pen rd . Equation ( 9) gives immediately for some C > 0,

E πm -πm τ 2 Y k 2 n (u) | f * ε,rd (u)| 2 du ≤ C πm -πm τ 2 Y k 2 n (u) |f * ε (u)| 2 du.
The Cauchy-Schwarz inequality and the estimate 1

/| f * ε,rd (u)| 2 ≤ n 1/2 imply E πm -πm | f * Y,rd (u)| 2 τ 2 ε k 2 n (u) | f * ε,rd (u)| 6 du ≤2 πm -πm E 1/2 f * Y,rd (u) -f * Y,rd (u) 4 E 1/2    τ 2 ε k 2 n (u) f * ε,rd (u) 12    du + 2 πm -πm E τ 2 ε k 2 n (u) |f * Y (u)| 2 | f * ε,rd (u)| 6 du ≤2 πm -πm τ 2 ε k 2 n (u) |f * ε (u)| 2 du + 2 πm -πm τ 2 ε k 2 n (u)|f * (u)| 2 |f * ε (u)| 2 |f * ε (u)| 2 ∨ n -1/2
du. This completes the proof for the rd-model. The arguments for the ns-model are the same, line for line, so we omit the details. The only difference lies in the fact that, in the definition of f * ε,ns and f * ε,ns , M plays now the role of n as defined in Lemma 6.3. The proof of Theorem 3.1 relies on the following auxiliary result which is, in turn, a consequence of Lemma 6.2.

Proposition 6.5. (i) In the model of repeated measurements, there exists an universal positive constant C such that

E    sup k≥m k,m∈M fk,rd -fm,rd 2 -6 f k -f m 2 - 3 4 pen rd (m, k) +    ≤ C n .
(ii) If an additional sample of the pure noise is available, for some universal positive constant C,

E    sup k≥m k,m∈M fk,ns -fm,ns 2 -6 f k -f m 2 - 3 4 pen ns (m, k) +    ≤ C n ∧ M .
Proof.

(i) Let us introduce the favorable events

E Y,rd := ∀u ∈ R : | f * Y,rd (u) -f * Y (u)| ≤ τ Y k n (u) . and E ε,rd := ∀u ∈ R : | f * 2 ε,rd (u) -f * 2 ε (u)| ≤ τ ε k n (u) . Applying Parseval's equality we can estimate fk,rd -fm,rd 2 = 1 2π f * k,rd -f * m,rd 2 = 1 2π A(m,k) | f * Y,rd (u)| 2 | f * ε,rd (u)| 2 du ≤ 1 π A(m,k) | f * Y,rd (u) -f * Y (u)| 2 | f * ε,rd (u)| 2 du + 1 π A(m,k) |f * Y (u)| 2 | f * ε,rd (u)| 2 du. ( 10
)
We start by dealing with the first summand appearing in the last line of ( 10). The definition of E Y,rd and the fact that

1 π A(m,k) τ 2 Y k 2 n (u) | f * ε,rd (u)| 2 du ≤ 3 8 pen 1,rd (m, k)
immediately imply the following inequality

sup k≥m k,m∈M 1 π A(m,k) | f * Y,rd (u) -f * Y (u)| 2 | f * ε,rd (u)| 2 du - 3 8 pen 1,rd (m, k)) + 1 E Y,rd ≤ sup k≥m k,m∈M 1 π A(m,k) τ 2 Y k 2 n (u) | f * ε,rd (u)| 2 du - 3 8 pen 1,rd (m, k) + 1 E Y,rd = 0.
Consider now the second summand in the last line of (10). Recall that f * ε and f * 2 ε,rd are real valued and (formally, with c/∞ := 0),

| fε,rd (u)| 2 = f * 2 ε,rd (u), if f * 2 ε,rd (u) ≥ n -1/2 ∞, else.
This entails the series of inequalities

1 π A(m,k) |f * Y (u)| 2 | f * ε,rd (u)| 2 du -6 f k -f m 2 = 1 π A(m,k) |f * Y (u)| 2 | f * ε,rd (u)| 2 du - 3 π f * k -f * m 2 = 1 π A(m,k) |f * Y (u)| 2 1 | f * ε,rd (u)| 2 - 3 |f * ε (u)| 2 du ≤ 1 π A(m,k) |f * Y (u)| 2 1 | f * ε,rd (u)| 2 - 3 |f * ε (u)| 2 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } du ≤ 2 3π A(m,k) | f * Y,rd (u)| 2 f * 2 ε (u) -3 f * 2 ε,rd (u) | f * ε,rd (u)| 4 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } du + 2 π A(m,k) |f * Y (u) -f * Y,rd (u)| 2 f * 2 ε (u) -3 f * 2 ε,rd (u) | f * ε,rd (u)| 2 |f * ε (u)| 2 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } du. ( 11
)
For the expression appearing in the last line of formula (11), we observe that

2 π A(m,k) |f * Y (u) -f * Y,rd (u)| 2 f * 2 ε (u) -3 f * 2 ε,rd (u) | f * ε,rd (u)| 2 |f * ε (u)| 2 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } du ≤ 2 π A(m,k) |f * Y (u) -f * Y,rd (u)| 2 f * ε (u) 2 | f * ε,rd (u)| 2 |f * ε (u)| 2 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } du = 2 π A(m,k) |f * Y (u) -f * Y,rd (u)| 2 | f * ε,rd (u)| 2 du.
The definition of pen 1,rd and E Y,rd readily imply that

sup k≥m k,m∈M 2 π A(m,k) |f * Y (u) -f * Y,rd (u)| 2 | f * ε,rd (u)| 2 du - 3 8 pen 1,rd (m, k) + 1 E Y,rd = 0
Consider now the second to last line in Equation ( 11). We observe that 3| f *

ε,rd (u)| 2 ≤ |f * ε (u)| 2 implies | f * ε,rd (u)| 2 = f * 2 ε,rd (u) ≤ (1/2)|f * ε (u) 2 -f * 2 ε,rd (u)|.
From this, we derive that

f * ε (u) -3 f * 2 ε,rd (u) | f * ε,rd (u)| 4 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } ≤ 1 2 f * ε (u) 2 -f * ε,rd (u) 2 | 2 | f * ε,rd (u)| 6
.

Using this and the definition of E ε,rd and pen 2,rd , we arrive at This completes the proof of part (i).

sup k≥m k,m∈M A(m,k) | f * Y,rd (u)| 2 |f * ε (u) 2 -f * ε,rd (u) 2 | 3/2π| f * ε,rd (u)| 4 1 {|f * ε (u)| 2 ≥3| f * ε,rd (u)| 2 } du - 3 4 pen 2,rd (m, k) + 1 Eε,rd ≤ sup k≥m k,m∈M A(m,k) | f * Y,rd (u)| 2 |f * ε (u) 2 -f * ε,rd (u) 2 | 2 3π| f * ε,rd (u)| 6 du - 3 4 pen 2,rd (m, k) + 1 Eε,rd ≤ sup k≥m k,m∈M A(m,k) | f * Y,rd (u)| 2 τ 2 ε k 2 n (u) 3π f * ε,rd (
(ii) The proof of the second part uses essentially the same arguments as the proof of the first part, so we content ourselves with sketching the important steps.

E Y,ns := ∀u ∈ R : | f * Y,ns (u) -f * Y (u)| ≤ τ Y k n (u) and E ε,ns := ∀u ∈ R : | f * ε,ns (u) -f * ε (u)| ≤ τ ε k M (u) .
In analogy with (10) and ( 11), we find that

f * k,ns -f * m,ns 2 -6 f * k -f * m 2 ≤ 2 π A(m,k) | f * Y,ns (u) -f * Y (u)| 2 | f * ε,ns (u)| 2 du + 1 3π A(m,k) | f * Y,rd (u)| 2 |f * ε (u)| 2 -3| f * ε,ns (u)| 2 | f * ε,ns (
+ = 0.
Finally, we find that P E c Y,ns ∪ E c ε,ns ≤ C(n -3/2 + M -3/2 ), leading to the desired bound on the complement set. 6.3. Proof of the oracle bounds. Proposition 6.5 was the essential technical step towards proving the oracle bounds. It shows how the penalty terms control the fluctuation of f * m 2 around its target, be it in the model of repeated measurements or in the deconvolution model with additional noise sample. Once this result is fixed, the considerations for both models run exactly along the same line. For this reason, we may henceforth drop all subscripts relative to one particular model. We have

f -f m 2 ≤ 2 f -fm ⋆ 2 + 2 fm ⋆ -f m 2 .
(12) Taking expectation and applying Proposition 2.1 gives for the first summand on the right hand side

E f -fm ⋆ 2 ≤ 2 f -f m ⋆ 2 + pen(m ⋆ ),
since the variance term is a fortiori bounded from above by the penalty term. So there remains to consider the second summand on the right hand side of (12). + pen(m ⋆ ).

Thus fm ⋆ -f m 2 1 G = fm ⋆ 2 -f m 2 1 G ≤ pen(m ⋆ ).
Taking expectation and applying Corollary 6.4 yields for some positive constant C,

E f -f m 2 1 G ≤ 2E f -fm ⋆ 2 + 2E [ pen(m ⋆ )] ≤ 2 f -f m ⋆ 2 + 2Cpen(m ⋆ ).
We have thus proved the desired result on G:

E f -f m 2 1 G ≤ C inf m∈M f -f m 2 + pen(m) .
• Next, we consider the set

G c = { m > m ⋆ }. It holds that f m -fm ⋆ 2 1 G c = 4 f m -fm ⋆ 2 - 3 4 f m -fm ⋆ 2 1 G c .

m

  ns = argmin m∈M -fm,ns 2 + pen ns (m) and m rd = argmin m∈M -fm,rd 2 + pen rd (m) . (7) Before formulating the oracle bound for the corresponding estimators, let us introduce the deterministic counterparts of the stochastic penalty terms: pen ns (m) := pen 1,ns (m) + pen 2,ns (m)

  Practical estimation procedure. Let us describe first the adaptive procedures as it is implemented for the both models:⊲ For m ∈ M = {m 1 , . . . , m n }, compute -fm 2 + pen(m).

Figure 1 .

 1 Figure 1. Estimations for n = M = 200 of a mixed Gamma (bold black line), in blue dashed line for the ns estimator and in red longdashed for the rd estimator.

Figure 2 .

 2 Figure 2. Estimations for n = M = 500 of a mixed Gamma (bold black line), in blue dashed line for the ns estimator and in red longdashed for the rd estimator.

n

  be independent and identically distributed random variables with values in [-1, 1], defined on the same probability space. Let v 2 := sup i∈I Var[Z there are universal positive constants c 1 and c 2 such that for any κ > 0, P sup i∈I

6. 2 .

 2 A technical auxiliary result. In the sequel, for arbitrary k > m, we use the notation pen ns (m, k) := pen ns (k) -pen ns (m) and pen rd (m, k) := pen rd (k) -pen rd (m), as well as pen ℓ,ns (m, k) := pen ℓ,ns (k) -pen ℓ,ns (m) and pen ℓ,rd (m, k) := pen ℓ,rd (k) -pen ℓ,rd (m), ℓ = 1, 2. Moreover, A(m, k) := {u ∈ R : m ≤ |u| ≤ k}.

  consider the exceptional set E c Y,rd ∪ E c ε,rd . Using the fact that for arbitrary m, the absolute value of f * m,rd is, by definition, bounded by 1, as well as the fact that max M ≤ √ n, we can estimate ε,rd ∩ E Y,rd ) c ) ≤ √ n P E c Y,rd + P E c ε,rd . Thanks to Lemma 6.2, it holds that √ n P E c Y,rd + P E c

  Proof of Theorem 3.1. We denote by m ⋆ the oracle cutoff,m ⋆ = argmin m∈M -fm 2 + pen(m) .

•

  Consider first the set G = { m ≤ m ⋆ }. Let us notice that on G fm ⋆ -Besides according to the definition of m, one has the following inequality:

Table 1 .

 1 Values of approximated MISE E f -f mns,ns 2 ×100 averaged over 500 samples

				n = 100	n = 250	n = 500
	f Laplace M = ⌊	f ε √ n⌋ rad 2.779 Lap. ror (2.223) (1.370) (1.342) (0.828) (0.988) (0.544) Gaus. Lap. Gaus. Lap. Gaus. 2.714 1.790 1.933 1.304 1.403
		M = n	rad 2.606	2.620	1.543	1.626	1.136	1.339
	Mixed Gamma	M = ⌊	ror (1.848) (1.357) (1.107) (0.786) (0.796) (0.539) √ n⌋ rad 0.787 0.777 0.377 0.385 0.247 0.247 ror (0.671) (0.710) (0.331) (0.337) (0.205) (0.209)
		M = n	rad 0.751	0.725	0.360	0.365	0.232	0.234
	Cauchy M = ⌊	ror (0.642) (0.682) (0.325) (0.319) (0.194) (0.197) √ n⌋ rad 0.891 0.888 0.433 0.465 0.311 0.325 ror (0.721) (0.731) (0.386) (0.418) (0.361) (0.275)
		M = n	rad 0.817	0.806	0.402	0.416	0.285	0.301
	Gaussian M = ⌊	ror (0.687) (0.687) (0.361) (0.379) (0.232) (0.254) √ n⌋ rad 0.774 0.777 0.450 0.446 0.275 0.303 ror (0.657) (0.655) (0.388) (0.415) (0.190) (0.211)
		M = n	rad 0.666	0.644	0.406	0.392	0.247	0.272
				ror (0.512) (0.540) (0.345) (0.325) (0.127) (0.126)

Thus it is more difficult to recover the target density f .

Table 2 .

 2 Values of approximated MISE E f -f mns,ns 2 ×100 averaged over 500 samples

	Mixed Gamma	⌊	100 √ n⌋ rad 1.767 ror (0.808) (0.473) (0.299) (0.940) (0.652) (0.456) (1.042) (0.865) (0.692) 250 500 100 250 500 100 250 500 0.889 0.471 2.170 1.229 0.673 2.798 1.973 1.232
		n	rad 1.231	0.620	0.387	1.483	0.835	0.554	1.594	1.135	0.951
	Gaussian ⌊	ror (0.791) (0.439) (0.265) (0.895) (0.581) (0.386) (0.979) (0.776) (0.585) √ n⌋ rad 1.663 0.920 0.543 1.951 1.185 0.867 3.163 2.139 1.898 ror (0.913) (0.531) (0.397) (1.489) (0.988) (0.747) (2.621) (1.988) (1.542)
		n	rad 1.245	0.669	0.522	1.884	1.053	0.690	2.458	1.432	0.982
				ror (0.578) (0.280) (0.180) (0.832) (0.444) (0.292) (1.400) (0.848) (0.613)

Table 3 .

 3 Values of approximated MISE E f -f mrd,rd 2 × 100 averaged over 500 samples, rad is the risk of the adaptive estimator, ror is the risk of the oracle estimator, M is the size of the noise and n is the size of the Y i 's.

				200	n = 2000	n = 200	n = 2000
	f	f ε	Lap.	Gaus.	Lap.	Gaus.	Lap.	Gaus.	Lap.	Gaus.
	Laplace rad 1.706	1.683	0.655	0.725	3.901	3.539	1.938	2.007
		ror (1.163) (1.216) (0.369) (0.438) (2.904) (2.710) (1.234) (1.536)
	Mixed	rad 0.558	0.548	0.075	0.076	0.857	0.799	0.248	0.226
	Gamma ror (0.381) (0.391) (0.061) (0.060) (0.583) (0.550) (0.144) (0.133)
	Cauchy rad 0.535	0.595	0.127	0.127	1.172	1.027	0.312	0.242
		ror (0.358) (0.377) (0.068) (0.069) (0.599) (0.617) (0.181) (0.174)
	Gaussian rad 0.338	0.335	0.045	0.050	0.703	0.586	0168	0.155
		ror (0.231) (0.235) (0.034) (0.035) (0.441) (0.399) (0.092) (0.076)

  Using the fact that for a > 0, x, y ∈ C, |x + y| 2 ≤ (1 + a)|x| 2 + (1 + 1/a)|y| 2 holds, we find that1 3π A(m,k) | f * Y,rd (u)| 2 |f * ε (u)| 2 -3| f * ε,ns (u)| 2 | f * ε,ns (u)| 4 du ≤ 1 2π A(m,k) | f * Y,rd (u)| 2 |f * ε (u) -f * ε,ns (u)| 2 | f * ε,ns (u)| 4

								du.
	Consequently, on E ε,ns , sup k≥m k,m∈M	1 3π A(m,k)	| f * Y,rd (u)| 2 |f * ε (u)| 2 -3| f * ε,ns (u)| 2 | f * ε,ns (u)| 4	du -	3 4	pen 2,ns (m, k)	+
	≤ sup k≥m k,m∈M	1 2π A(m,k)	| f * Y,rd (u)| 2 τ 2 ε k 2 M (u) | f * ε,ns (u)| 4	du -	3 4	pen 2,ns (m, k)
								u)| 4	du.
	Again, on E Y,ns ,	sup k≥m k,m∈M	2 π A(m,k)	| f * Y,ns (u) -f * Y (u)| 2 | f * ε,ns (u)| 2	du -	3 4	pen 1,ns (m, k)

+ = 0.

We realize that, by definition of f m, see ( 13), on G c ,

It follows from there that

Taking expectation and applying Proposition 6.5, as well as the monotonicity of the bias term, we conclude that

with N = n in the rd-model and N = M ∧ n in the ns-model. This is the desired oracle bound for G c .

acknowledgments The authors would like to thank F. Comte for her advice, suggestions and her understanding support during this work. They are also grateful to two anonymous referees for detailed and instructive remarks and comments which helped them to substantially improve this paper.