Non-parametric estimation in a semimartingaleregression model. Part 1. Oracle inequalities.pdf AMS 2000 Subject Classifications: Primary: 62G08; Secondary: 62G051. IntroductionConsider a regression model in continuous time( ) 0 t tdy = S t dt + dƒ , ≤ t ≤ n, (1)where S is an unknown 1 -periodic R  R function, 2S  L [0,n] ; 0 ( ) t t≥ ƒ is a squareintegrable unobservable semimartingale noise such that for any function f from2L [0,n] the stochastic integral0( )nn s sI f =  f dƒ (2)is well defined with2 20E ( ) 0 and E ( )nn n sI f I f f ds∗= ≤ƒ  , (3)where ∗σ is some positive constant.An important example of the disturbance 0 ( ) t t≥ ƒ is the following processt 1 t 2 tξ = _ w +_ z , (4)where _1 and _2 are unknown constants, 1 2| _ | + | _ |> 0 , 0 ( ) t tw≥ is a standardBrownian motion, 0 ( ) t tz≥ is a compound Poisson process defined as1t Nt jjz Y==Σ, (5)where 0 ( )t tN≥ is a standard homogeneous Poisson process with unknown intensityλ > 0 and 1 ( ) j jY≥ is an i.i.d. sequence of random variables with2E 0 and E 1j jY = Y = . (6)1 The paper is supported by the RFFI - Grant 09-01-00172-a.24 V. Konev, S. PergamenshchikovLet 1 ( ) k kT≥ denote the arrival times of the process 0 ( )t tN≥ , that is,inf{ 0 }k tT = t ≥ :N = k . (7)As is shown in Lemma A.2, the condition (3) holds for the noise (4) with2 21 2∗σ = _ +_ λ .The problem is to estimate the unknown function S in the model (1) on the basis ofobservations 0 쨀( )yt ≤t≤n .This problem enables one to solve that of functional statistics which is stated asfollows. Let observations 0 ( ) kk n x≤ ≤ be a segment of a sequence of independentidentically distributed random processes 0 1 ( ) k kt t x xЎВ ЎВ= specified on the interval (0 1) , ,which obey the stochastic differential equations0 0 ( ) 0 1 k k kt tdx = S t dt + dѓМ , x = x , ≤ t ≤ , (8)where 1 ( ) k≤k≤n ξ is an i.i.d sequence of random processes 0 1 ( ) k kt ≤t≤ ξ = ξ with the samedistribution as the process (4). The problem is to estimate the unknown function2 f (t)Ѓё L [0,1] on the basis of observations 1 ( ) kk n x≤ ≤ . This model can be reduced to(1), (4) in the following way. Let 0 ( ) t t n y y ≤ ≥= denote the process defined as :11 1 0if 0 1if 1 2ttkk t kx tyy x x k t k k n − − +⎧⎪ , ≤ ≤ ;= ⎨+ − , − ≤ ≤ , ≤ ≤ . ⎪⎩This process satisfies the stochastic differential equation_( ) t tdy = S t dt + dξ ,_where_S(t) = S({t}) and11 1if 0 1if 1 2tt kk t ktk t k k n− +−⎧⎪ ѓМ , ≤ ≤ ;ѓМ = ⎨+ ѓМ , − ≤ ≤ , ≤ ≤ ; ѓМ ⎪⎩__{t} = t −[t] is the fractional part of number t .In this paper we will consider the estimation problem for the regression model (1) in2L [0 1], with the quality of an estimate_S being measured by the mean integratedsquared error (MISE)_ _2 ( ) E ( )SR S S S S, := − , (9)where ES stands for the expectation with respect to the distribution PS of the process(1) given S ;12 20|| f || := Ѓз f (t)dt .It is natural to treat this problem from the standpoint of the model selectionapproach. The origin of this method goes back to early seventies with the pioneeringpapers by Akaike [1] and Mallows [16] who proposed to introduce penalizing in a loglikelihoodtype criterion. The further progress has been made by Barron, Birge andMassart [2, 17] who developed a non-asymptotic model selection method which enabledone to derive non-asymptotic oracle inequalities for a gaussian non-parametricNon-parametric estimation in a semimartingale regression model 25regression model with the i.i.d. disturbance. An oracle inequality yields the upper boundfor the estimate risk via the minimal risk corresponding to a chosen family of estimates.Galtchouk and Pergamenshchikov [6] developed the Barron - Birge - Massarttechnique treating the problem of estimating a non-parametric drift function in adiffusion process from the standpoint of sequential analysis. Fourdrinier andPergamenshchikov [5] extended the Barron - Birge - Massart method to the modelswith dependent observations and, in contrast to all above-mentioned papers on themodel selection method, where the estimation procedures were based on the leastsquares estimates, they proposed to use an arbitrary family of projective estimates in anadaptive estimation procedure, and they discovered that one can employ the improvedleast square estimates to increase the estimation quality. Konev and Pergamenshchikov[14] applied this improved model selection method to the non-parametric estimationproblem of a periodic function in a model with a coloured noise in continuous timehaving unknown spectral characteristics. In all cited papers the non-asymptotic oracleinequalities have been derived which enable one to establish the optimal convergencerate for the minimax risks. Moreover, in the latter paper the oracle inequalities havebeen found for the robust risks.In addition to the optimal convergence rate, an important problem is that of theefficiency of adaptive estimation procedures. In order to examine the efficiencyproperty one has to obtain the oracle inequalities in which the principal term has thefactor close to unity.The first result in this direction is most likely due to Kneip [13] who obtained, for agaussian regression model, the oracle inequality with the factor close to unity at theprincipal term. The oracle inequalities of this type were obtained as well in [3] and in[4] for the inverse problems. It will be observed that the derivation of oracle inequalitiesin all these papers rests upon the fact that by applying the Fourier transformation onecan reduce the initial model to the statistical gaussian model with independentobservations. Such a transform is possible only for gaussian models with independenthomogeneous observations or for the inhomogeneous ones with the known correlationcharacteristics. This restriction significantly narrows the area of application of suchestimation procedures and rules out a broad class of models including, in particular,widely used in econometrics heteroscedastic regression models (see, for example, [12]).For constructing adaptive procedures in the case of inhomogeneous observations oneneeds to amend the approach to the estimation problem. Galtchouk andPergamenshchikov [7, 8] have developed a new estimation method intended for theheteroscedastic regression models. The heart of this method is to combine the Barron-Birge-Massart non-asymptotic penalization method [2] and the Pinsker weighted leastsquare method minimizing the asymptotic risk (see, for example, [18, 19]). Combiningof these approaches results in the significant improvement of the estimation quality (seenumerical example in [7]). As was shown in [8] and [9], the Galthouk -Pergamenshchikov procedure is efficient with respect to the robust minimax risk, i.e.the minimax risk with the additional supremum operation over the whole family ofaddmissible model distributions. In the sequel [10, 11], this approach has been appliedto the problem of a drift estimation in a diffusion process. In this paper we apply thisprocedure to the estimation of a regression function S in a semimartingale regressionmodel (1). The rest of the paper is organized as follows. In Section 2 we construct themodel selection procedure on the basis of weighted least squares estimates and state themain results in the form of oracle inequalities for the quadratic risks. Section 3 gives theproofs of all theorems. In Appendix some technical results are established.26 V. Konev, S. Pergamenshchikov2. Model selectionThis Section gives the construction of a model selection procedure for estimating afunction S in (1) on the basis of weighted least square estimates and states the mainresults.For estimating the unknown function S in model (1), we apply its Fourierexpansion in the trigonometric basis 1 ( ) j j≥ φ in 2 L [0 1], defined as11 ( ) 2 (2 [ 2] ) 2j jѓУ = , ѓУ x = Tr ѓО j/ x , j ≥ , (10)where the function ( ) cos( ) jTr x = x for even j and ( ) sin( ) jTr x = x for odd j ; [x]denotes the integer part of x . The corresponding Fourier coefficients10( ) ( ) ( ) j j jθ = S,φ = Ѓз S t φ t dt (11)can be estimated as_01( )nj n j tt dyn ,θ = Ѓз φ . (12)In view of (1), we obtain_ 1 1( ) j n j j n j n n jIn n , , ,= θ + ξ , ξ = φθ , (13)where nI is given in (2).For any sequence 1 ( ) j jx x≥= , we set2 2{ 0}1 1and ( ) 1j x jj jx x # xЃ‡ Ѓ‡| |>= =| | =Σ =Σ . (14)Now we impose the additional conditions on the noise 0 ( ) t t≥ ξ .1C ) There exists some positive constant σ > 0 such that the sequence2E 1j n j n j, ,ς = ѓМ −ѓР, ≥ ,for any n ≥1, satisfies the following inequality1 1( )( ) sup ( ) nx H # x nc n B x∗,Ўф , ЎВ= | | 0 and 4EjY < Ѓ‡ . Then the estimation procedure (26), for any n ≥1 and0 < ѓП ≤ 1/3 , satisfies the oracle inequality (27) with2 21 2 1c (n) 0∗ ∗σ = σ = _ + λ_ , = ,and ( ) 2 42 2 11sup ( ) 4 Enc n Y∗ ∗ ∗ЎГ≤ ѓР ѓР +_ .The proofs of Theorems 1, 2 are given in Section 3.Corollary 3. Let the conditions of Theorem 1 hold and the quantity σ in 1C ) beknown. Then, for any n ≥1 and 0 < ѓП < 1/3 , the estimator (26) satisfies the oracleinequality_ _2 1 3 2 1( ) min ( ) ()1 3 nR S S R S Sn∗ ҐгҐгЎфҐГ+ ѓП− ѓП, ≤ , + ѓµ ѓП,− ѓПwhere ( ) nΨ ρ is given in (28).1 . E s t ima t i o n o f σNow we consider the case of unknown quantity σ in the condition 1C ) . One canestimate σ as_ _ 2with [ ] 1nn jnj ll n,=σ =Σ θ = + . (29)Non-parametric estimation in a semimartingale regression model 29Proposition 4. Suppose that the conditions of Theorem 1 hold and the unknownfunction S(t) is continuously differentiable for 0 ≤ t < 1 such that110| S | = Ѓз | S(t) | dt < +Ѓ‡. _ _(30)Then, for any n ≥1,_ ( )E nS nSnѓИ| ѓР − ѓР |≤ , (31)where 2 1 11 2 1 4 1 24 ()( ) 4 ( )nS c nS S c nn n∗ ∗∗/ /| | σκ = | | +σ + + + .__The proof of Proposition 4 is given in Section 3. Theorem 1 and Proposition 4 implythe following result.Theorem 5. Suppose that the conditions of Theorem 1 hold and S satisfies theconditions of Proposition 4. Then, for any n ≥1 and 0 < ρ < 1/3 , the estimate (26)satisfies the oracle inequality_ _2 1 3 2 1( ) min ( ) ()1 3 nR S S R S S Dn∗ ҐгҐгЎфҐГ+ ѓП− ѓП, ≤ , + ѓП,− ѓП(32)where2 (1 ) ( )( ) 2 ( )(1 3 )nn nSDnρ −ρμκρ = Ψ ρ + .− ρ2 . S p e c i f i c a t i o n o f w e i g h t si n t h e s e l e c t i o n p r o c e d u r e ( 2 6 )Now we will specify the weight coefficients 1 ( ( ))j j γ ≥ in a way proposed in [7] for aheteroscedastic discrete time regression model. Consider a numerical grid of the form1 {1 } { } n mA … k t …t∗= , , × , , ,where it = iѓГ and 2m = [1/ε ] . We assume that both parameters k 1 ∗ ≥ and 0 < ѓГ ≤ 1 arefunctions of n , i.e. k k (n) ∗ ∗= and ε = ε(n) , such thatlim ( ) lim ( ) ln 0lim ( ) 0 and lim ( )n nn nk n k n nn n n∗ ∗ЎжЎД ЎжЎДҐдЎжЎД ЎжЎД⎧ = +Ѓ‡, = ,⎪⎨ѓГ = ѓГ = +Ѓ‡ ⎪⎩(33)for any ѓВ > 0 . One can take, for example,1( ) and ( ) ln( 1)ln( 1)n kn nn∗ε = = ++for n ≥1.For each ( ) nѓї = ѓА, t Ѓё A , we introduce the weight sequence1 ( ( ))j j ѓї ѓї ≥ γ = γgiven as ( ) {1 0} { 0 } ( ) 1 1 ( ) 1 j j j j j jαѓАγѓї = ≤ ≤ + − /ωѓї < ≤ѓЦ , (34)where [ ] 0 0j j ( ) ln n α= α = ω / ,30 V. Konev, S. Pergamenshchikov1 (2 1)2( 1)(2 1)( t n) and/ β+α β β ββ+ β+ω = τ τ = .π βWe set{ }nAαѓЎ= ѓБ ,ѓїЃё . (35)It will be noted that in this case k m ∗ ѓЛ = .Remark 1. It will be observed that the specific form of weights (34) was proposedby Pinsker [19] for the filtration problem with known smoothness of regression functionobserved with an additive gaussian white noise in the continuous time. Nussbaum [18]used these weights for the gaussian regression estimation problem in discrete time.The minimal mean square risk, called the Pinsker constant, is provided by the weightleast squares estimate with the weights where the index α depends on the smoothnessorder of the function S . In this case the smoothness order is unknown and, instead ofone estimate, one has to use a whole family of estimates containing in particular theoptimal one.The problem is to study the properties of the whole class of estimates. Below wederive an oracle inequality for this class which yields the best mean square risk up to amultiplicative and additive constants provided that the the smoothness of the unknownfunction S is not available. Moreover, it will be shown that the multiplicative constanttends to unity and the additive one vanishes as nЃЁЃ‡ with the rate higher than anyminimax rate.In view of the assumptions (33), for any ѓВ > 0 , one haslim 0n nѓВЃЁЃ‡ν= .Moreover, by (34) for any nѓїЃё A{ ( ) 0}11jjαЃ‡ѓБ > ѓї=Σ ≤ ѓЦ .Therefore, taking into account that 1A A 1β≤ < for ѓА ≥ 1, we get1 3 ( ) nn/ѓК = ѓК ≤ /ѓГ .Therefore, for any ѓВ > 0 ,1 3lim 0 nn n/ +ѓВЃЁЃ‡μ= .Applying this limiting relation to the analysis of the asymptotic behavior of theadditive term ( ) nD ρ in (32) one comes to the following result.Theorem 6. Suppose that the conditions of Theorem 1 hold and 1S Ѓё L [0,1]_. Then,for any n ≥1 and 0 < ρ < 1/3 , the estimate (26) with the weight coefficients (35)satisfies the oracle inequality (32) with the additive term ( ) nD ρ obeying, for any ѓВ > 0 ,the following limiting relation( )lim 0 nnDnѓВЃЁЃ‡ρ= .Non-parametric estimation in a semimartingale regression model 313. Proofs1 . P r o o f o f T h e o r e m 1Substituting (22) in (20) yields for any ѓБЃёѓЎ2 _1Err ( ) ( ) 2 ( ) || || ( )n n j n njJ j S PЃ‡,=ѓБ = ѓБ + ΣѓБ ѓЖЃЊ + −ѓП ѓБ , (36)where _ __1 1 1nj n j n j j n j j n j n j nn n n n, , , , , ,ѓР−ѓРѓЖЃЊ = ѓЖ −ѓЖ ѓЖ = ѓЖ ѓМ + ѓМ + ς +_and the sequences 1 ( ) j n j , ≥ς and 1 ( )j,n j≥ ξ_are defined in conditions 1C ) and 2 C ) .Denoting1 11( ) ( ) ( ) ( ) j jnj jL j M jnЃ‡ Ѓ‡,= =γ =Σ γ , γ = Σγ θ ξ , (37)and taking into account the definition of the "true" penalty term in (24), we rewrite (36) as_12Err ( ) ( ) 2 ( ) 2 ( ) ( ) nn n nJ L M Bn n,σ − σγ = γ + γ + γ + γ +2 2 _ ( ( ))2 () || || () nn nB eP S Pn,γ+ γ + −ρ γσ, (38)where e(γ) = γ/ | γ | , the functions 1 nB,and 2 nB,are defined in (15) and (16).Let 0 0 1 ( ( ))j j γ = γ ≥ be a fixed sequence in Γ and γ_be as in (25). Substituting γ0and γ_in the equation (38), we consider the difference__ _ __ _0 0 12 2 00 02Err ( ) Err ( ) ( ) ( ) 2 ( ) ( ) 2 ( )( ) ( )2 ( ) 2 ( ) ( ) ( ),n n nn nn n n nJ J L x B x M xn nB e B eP P P Pn n,, ,σ−σγ − γ = γ − γ + + + ++ γ − γ − ρ γ + ρ γσ σ_ ___ _where_0x = γ−γ_, e = e(γ)_ _and 0 0 e = e(γ ) . Note that by (19)_| L(x) |≤| L(ѓБ) | + | L(ѓБ) |≤ 2ѓК._Therefore, by making use of the condition 1C ) and taking into account that the costfunction J attains its minimum at γ_, one comes to the inequality__102 ( )Err ( ) Err ( ) 4 2 ( )n nc nM xn n∗| ѓР−ѓР|ѓБ− ѓБ ≤ ѓК+ + +_2 _ _ 2 00 0( ) ( )2 () () ( ) 2 ( ) n nn n n nB e B eP P P Pn n, , + γ −ρ γ + ρ γ − γ .σ σ__ _(39)Applying the elementary inequality2 1 22 ab a b− | |≤ѓГ +ѓГ (40)32 V. Konev, S. Pergamenshchikovwith ε = ρ implies the estimate22 2 ( ( )) ( ( ))2 () () n nn nB e B eP Pn n, , | ѓБ | ѓБѓБ ≤ ѓП ѓБ+ .ѓР ѓРѓПWe recall that 0 < ρ < 1. Therefore, from here and (39), it follows that_ 2 102 2 ( )Err ( ) Err ( ) 2 ( ) nn nB c nM xn n∗ ∗, ѓБ ≤ ѓБ + + + +ѓРѓП__ ( ) 2 20 014 2 ( ) nPn+ |σ−σ| | γ | + | γ | + μ + ρ γ_,where 22 2sup ( ( ))n nB∗ B e, ҐгЎфҐГ ,= γ . In view of (19), one has2supѓБЃёѓЎ| ѓБ | ≤ ѓК.Thus, one gets_ 2 102 2 ( )Err ( ) Err ( ) 2 ( ) nn nB c nM xn n∗ ∗, ѓБ ≤ ѓБ + + + +ѓРѓП__062 ( ) n nPnμ+ | σ − σ | + ρ γ . (41)In view of Condition 2 C ) , one has22 2 2 E E ( ( )) ( )S n S nB∗ B e c∗ n, ,ҐгЎфҐГ≤Σ ѓБ ≤ѓЛ , (42)where ν = card(Γ) .Now we examine the first term in the right-hand side of (39). Substituting (13) in(37) and taking into account (3), one obtains that for any non-random sequence1 ( ( ))jx x j≥= with #(x) < Ѓ‡2 2 2 211 1E ( ) ( ) || ||S j xjM x x j Sn nЎД∗ ∗=≤ѓР Σ ѓЖ =ѓР , (43)where 1 ( ) x jjjS xjЃ‡==Σ θ φ . Let denote122( )supx || ||xnM xZS∗ЎфҐГ= ,where 1 0 Γ =Γ−γ . In view of (43), this quantity can be estimated as1 122( )|| ||EE SSx x xn M xZS∗ ∗ ∗ЎфҐГ ЎфҐГ≤ Σ ≤ Σ ѓР = ѓР ѓЛ. (44)Further, by making use of the inequality (40) with || || xε = ρ S , one gets22 ( ) || ||xZM x Sn∗| |≤ѓП + .ѓП(45)Non-parametric estimation in a semimartingale regression model 33Note that, for any 1xЃёѓЎ ,2 _ 2 2 2 _ 211|| || || || ( )( ) 2 ( ) x x j j njS S x j M xЃ‡,=− = ѓЖ − ≤ −ѓЖ Σ , (46)where2111( ) ( ) j jnjM x x jnЃ‡,== Σ θ ξ .Since | x( j) |≤ 1 for any 1xЃёѓЎ , one gets221|| ||E ( ) xSSM xn∗≤ ѓР .Denoting1211 2( )supx || ||xnM xZS∗ЎфҐГ= ,one has1ESZ∗ ∗≤ ѓР ѓЛ. (47)By the same argument as in (45), one derives2 112 () || ||xZM x Sn∗| |≤ ѓП + .ѓПFrom here and (46), one finds the upper bound for || || xS , i.e._ 22 1 || |||| ||1 (1 )xxS ZSn∗≤ + .−ѓП ѓП −ѓП(48)Using this bound in (45) gives_ 21 || ||2 ( )1 (1 )xS Z ZM xn∗ ∗ ѓП +≤ + .−ѓП ѓП −ѓПSetting_x = x in this inequality and taking into account that__ _ _02 20 || || || || 2(Err ( ) Err ( )) x n n S S S γ γ= _ − ≤ ѓБ + ѓБ ,_we obtain_0 12 (Err ( ) Err ( ))2 ( )1 (1 )n nZ ZM xn∗ ∗ѓП ѓБ+ ѓБ +≤ + .−ѓП ѓП −ѓП_From here and (41), it follows that_ 20 1101 2(1 )Err ( ) Err ( ) ( ) 31 3 (1 3 )2 (1 )( ).(1 3 ) 1 3nn nnBc nnZ ZPn∗, ∗∗ ∗+ ѓП − ѓП ⎛ ⎞ѓБ ≤ ѓБ + ⎜⎜ + + ѓК | ѓР − ѓР |⎟⎟ + − ѓП − ѓП ⎝ ѓРѓП ⎠+ ѓП −ѓП+ + ѓБѓП − ѓП − ѓП_34 V. Konev, S. PergamenshchikovTaking the expectation yields_ _ _02101 2(1 ) ( )( ) ( ) () 31 3 (1 3 )2 2(1 )( )(1 3 ) 1 3Snc nR S S R S S c nnPn∗∗∗ Ґг∗+ ѓП − ѓП ⎛ ѓЛ ⎞, ≤ , + ⎜ + + ѓК | ѓР − ѓР |⎟ +− ѓП − ѓП ⎝ ѓРѓП ⎠ѓР ѓЛ ѓП −ѓП+ + ѓБ .ѓП − ѓП − ѓПEUsing the upper bound for 0 ( ) nP γ in Lemma A.1, one obtains_ _02 1 3 2 1( ) ( ) ()1 3 nR S S R S S Bn∗∗ Ґг+ ѓП− ѓП, ≤ , + ѓП,− ѓПwhere ( ) nB∗ρ is defined in (27).Since this inequality holds for each ѓБ0 ЃёѓЎ , this completes the proof of Theorem 1.2 . P r o o f o f T h e o r e m 2We have to verify Conditions 1C ) and 2 C ) for the process (4).Condition 1C ) holds with 1c (n) 0∗= . This follows from Lemma A.2 if one putsj f = g = φ , j ≥ 1. Now we check Condition 2 C ) . By the Ito formula and Lemma A.2,one gets2 22 2 2 21 20( ) 2 ( ) ( ) ( ) ( )( ) t t t ss tdI f I f dI f f t dt f s z−≤ ≤= +_ +_ Σ Δand 2 20E ( ) ( )ttI f f t dt∗= σ Ѓз .Therefore, putting2 2 ( ) ( ) E ( )t t t I f I f I f = − ,_we obtain2 22 0 ( ) ( ) 2 ( ) ( ) ( ) 0 t t t td I f f t dm I f f t d I f −= + ξ , =_ __and 20( ) t ss tm z t≤ ≤= Σ Δ −λ . (49)Now we set1( ) ( ) t j t jjI x x IЃ‡==Σ ѓУ _ ,where 1 ( ) j jx x≥= with #(x) ≤ n and | x |≤ 1. This process obeys the equation22 0 ( ) 2 ( ) ( ) 0 t t t t td I x dm x d I x − = _ ѓі + ѓД ѓМ , = ,where 21 1( ) ( ) and ( ) ( ) ( ) t j j t j t j jj jx x t x x I t≥ ≥ѓі =Σ ѓУ ѓД =Σ ѓУ ѓУ .Now we show that0E ( ) ( ) 0nt tI x d I x −Ѓз = . (50)Non-parametric estimation in a semimartingale regression model 35Indeed, note that220 0101( ) ( ) ( ) ( )2 ( ) ( )n nt t j t j t tjnj t j t tjI x d I x x I x dmx I x d− −≥− −≥= φ Φ ++ φζ ξ .Ѓз ѓ° Ѓзѓ° Ѓз___Therefore, Lemma A.4 directly implies( )2 20 012 201( ) ( ) ( ) ()( ) () 0E EE En nt j t t l t j l tlnl t j l tlx dm x I t dmIx I t dm− −≥−≥φ Φ = φ φ− φ φ = .Ѓз ѓ° Ѓзѓ° Ѓз_Moreover, we note that0 01( ) ( ) ( ) ( ) ()n nt j t t l t j t l l tlI x d x I I t d − − − −≥Ѓз φ ζ ξ =ѓ° Ѓз φ φ φ ξ _ _and( )20 020( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ()n nt j t l l t t j t l l tnt j t l l tI I t d I I t dI I t d− − −−−φ φ φ ξ= φ φ φ ξ−− φ φ φ ξ .Ѓз ЃзЃз_EFrom Lemma A.5, it follows0E ( ) ( ) ( ) 0nt j t l l tI I t d −−Ѓз φ φ φ ξ = _and we come to (50). Furthermore, by the Ito formula one obtains2 2 2 4 2 41 2 { }0 012 2 2 3 2 32 { } 2 { }1 1( ) 2 ( ) ( ) 4 ( ) ( ) 14 () 1 4 ( ) ( ) 1k kk k k k kn nn t t t T k T nkT k T n T T k T nk kI x I x d I x x dt x Yx Y x x Y+Ѓ‡− ≤=+Ѓ‡ +Ѓ‡− ≤ − ≤= == + ζ + Φ ++ ζ + Φ ζ .Ѓз Ѓз ѓ°ѓ° ѓ°_ __ _By Lemma A.3 one has E ( ) 0k T kT− ζ | = . Therefore, taking into account (50), wecalculate2 2 2 4 4 21 2 1 1 2 20( ) 4 ( ) ( ) 4 ( )nn t n nI x x dt Y D x D x, ,E = _ EЃз ζ +_ E + _ , (51)where 2 21 { } 2 { }1 1( ) E ( )1 and ( ) E ( )1k k k k n T T n n T T nk kD x x D x xЃ‡ Ѓ‡, ≤ , − ≤= ==Σ Φ =Σ ζ .By applying Lemma A.2, one has20 0E ( ) ( ) ( )E ( ) ( )n nt ij i j t i t ji jx dt x x t t I I dt,Ѓз ѓД =ѓ° Ѓз ѓУ ѓУ ѓУ ѓУ =( ) 0 0( ) ( ) ( ) ( )n ti j i j i ji jx x t t s s ds dt ∗,= ѓР ѓ° Ѓз ѓУ ѓУ Ѓз ѓУ ѓУ =( )220( ) ( )2 2ni j i ji jx x t t dt n∗ ∗,ѓР ѓР= ѓ° Ѓз ѓУ ѓУ ≤ . (52)36 V. Konev, S. PergamenshchikovFurther it is easy to check that22 210 01( ) ( )n nn t j jjD, x dt x t dt≥⎛ ⎞= ѓЙ ѓі = ѓЙ ⎜⎜ ѓУ ⎟⎟ .⎝ ⎠Ѓз Ѓз ѓ°Therefore, taking into account that #(x) ≤ n and | x |≤ 1, we estimate 1 nD,byapplying the Causchy - Schwarts - Bounyakovskii inequality22114 4 ( ) 4 n jjD n x n#x n⎛ ⎞⎜ ⎟, ⎜ ⎟⎜ ≥ ⎟ ⎝ ⎠≤ ѓЙ ѓ° ≤ ѓЙ ≤ ѓЙ . (53)Finally, we write down the process ( ) tζ x as01( ) ( ) with ( ) ( ) ( )tt x s x j j jjx Q tsd Qts x s t≥ѓД = Ѓз , ѓМ , =ѓ° ѓУ ѓУ .By putting _122 { }2 1E 1 ( )kkn T n x k lk lD Q T TЃ‡ −, ≤= == Σ Σ ,and applying Lemma A.3 we obtain__2 2 22 1 0 { } 2 212 2 21 0 0 2 2( ) 1( )kkTn x k T n nkn tx nD QT s ds DQ t s dsdt DЃ‡, ≤ ,=,= , + == λ , + .ѓ° ЃзЃз Ѓз_ E __ _Moreover, one can rewrite the second term in the last equality as_( )( )22 { } { }1 12 20 02 20 01 ( )1( )( )El k n T n x k l T nl kln n sxn txD Q T TQ s z s dz dsQ t s ds dtЃ‡ Ѓ‡, ≤ ≤= =+−= , ==λ + , == λ , .ѓ° ѓ°Ѓз ЃзЃз ЃзThus, ( ) 2 2 2 22 1 20 0( ) ( )n nn xD Q t s ds dt,≤ ѓЙ_ + ѓЙ _ Ѓз Ѓз , =2 2 2 2 21 2 ( )n n∗= λ +λ =λσ . _ _ (54)The equation (51) and the inequalities (52), (53) imply the validity of condition 2 C )for the process (4). Hence Theorem 2.3 . P r o o f o f P r o p o s i t i o n 4Substituting (13) in (29) yields_ 2 2 n 2 n 1 nn j j j n j nj l j l j ln n, ,= = == θ + θ ξ + ξ .σ Σ Σ Σ (55)Further, denoting{ } { }11 and 1x j l j n x j l j nn≤ ≤ ≤ ≤ЃЊ = ЃЊЃЊ = ,we represent the last term in (55) asNon-parametric estimation in a semimartingale regression model 3721 21 1 1 1( ) ( )nj n n nj ln lB x B xn n n n, , ,=− +ΣѓМ = ЃЊ + ЃЊЃЊ + ѓР,where the functions 1 ( ) nB,⋅ and 2 ( ) nB,⋅ are defined in conditions 1C ) and 2 C ) .Combining these equations leads to the inequality_ 21 221 1 1( ) ( )nS n j S j j nj l j ln nnlB x B xn n n,≥ =, ,| ѓР − ѓР | ≤ ѓЖ + | ѓЖ ѓМ | +−+ | ЃЊ | + | ЃЊЃЊ | + ѓР.E Σ E ΣEBy Lemma A.6 and conditions 1C ) , 2 C ) , one gets_ 2 1 2 2 ( ) ( ) nS n j S j j nj l j lc n c nn n n n∗ ∗,ЎГ =ѓРE | ѓР − ѓР | ≤ΣѓЖ + E |ΣѓЖ ѓМ | + + + .In view of the inequality (3), the last term can be estimated as212En nS j j n jj l j lSl∗ ∗,= =|ΣѓЖ ѓМ |≤ ѓР ΣѓЖ ≤ ѓР | | . _Hence Proposition 4.4. AppendixA . 1 . P r o p e r t y o f t h e p e n a l t y t e rm ( 2 4 )Lemma A.1. Assume that the condition 1C ) holds with σ > 0 . Then for any n ≥1and ѓБЃёѓЎ ,1( )( ) E Err ( )n S nc nPn∗ѓБ ≤ ѓБ+ .Proof. By the definition of Err ( ) n ѓБ one has211Err ( ) ( ( ) 1) ( ) n j j njj jnЃ‡⎛ ⎞⎜ ⎟⎜ , ⎟= ⎝ ⎠γ =Σ γ − θ + γ ξ .In view of the condition 1C ) this leads to the desired result2 2 111 ( )E Err ( ) ( ) E ( )S n jn njc nj Pn nЎД ∗,=ѓБ ≥ ΣѓБ ѓМ = ѓБ − .A . 2 . P r o p e r t i e s o f t h e p r o c e s s ( 4 )Lemma A.2. Let f and g be any non-random functions from 2L [0,n] and0 ( ( )) t tI f≥ be the process defined by (4). Then, for any 0 ≤ t ≤ n ,0E ( ) ( ) ( ) ( )tt tI f I g f s g s ds∗= σ Ѓз ,where 2 21 2∗σ = _ + λ_ .This Lemma is a direct consequence of Itos formula as well as the following result.38 V. Konev, S. PergamenshchikovLemma A.3. Let Q be a bounded [0 ) R,Ѓ‡ Ѓ~ѓ¶ЃЁ function measurable with respectto [0 ) kB ,+Ѓ‡ ⊗G , where1 { } with some 2 k kG = ѓР T ,…,T k ≥ . (A.1)ThenE ( ( ) ) 0kT kI Q G−| =and ( )12 2 2 2 21 201E ( ) ( ) ( )kkkTT k llI Q G Q s ds Q T−−=| = _ Ѓз +_ ѓ° .Now we will study stochastic cadlag processes 0 ( )ѓЕ = ѓЕt ≤t≤n of the form{ 1}0( )1t l Tl t Tllt+Ѓ‡≤

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