DOI:10.1051/ps:2007038 www.esaim-ps.org
MINIMAX AND BAYES ESTIMATION IN DECONVOLUTION PROBLEM
∗Mikhail Ermakov
1Abstract. We consider a deconvolution problem of estimating a signal blurred with a random noise.
The noise is assumed to be a stationary Gaussian process multiplied by a weight function functionh where h ∈ L2(R1) and is a small parameter. The underlying solution is assumed to be infinitely differentiable. For this model we find asymptotically minimax and Bayes estimators. In the case of solutions having finite number of derivatives similar results were obtained in [5].
Mathematics Subject Classification. 62G05, 65R30, 65R32.
Received September 12, 2007.
1. Introduction and main results 1.1. Introduction
We study the deconvolution problem in the following setting. We wish to estimate x from the observed random process
Y(t) = ∞
−∞a(t−s)x(s)ds+η(t), t∈R1. (1) Here the kernelais known andη(t) is a random noise with a small parameter >0. Such a setting arises in many applications (seee.g. [17,20,21,29] and references therein).
The noiseηis usually considered as a stationary process. This allows to study the problem in terms of the Fourier transform and to derive the estimators in a simple analytic form. For this setting the statistical prop- erties of Tikhonov regularizing algorithm, the procedure of Kolmogorov-Wiener filtration, robust and minimax estimators were analyzed in numerous publications and textbook (see, for example, [10,21,29,31] and references therein). Recently interesting adaptive procedures (see [2,3,14,30]) and wavelet based estimators (see [7,23,25]) were proposed.
If the noise is a stationary process, the power of the noise2∞
−∞Eη2(t)dton the whole real line is infinite.
Consistent estimator of solutionx(t) exists only if the ratio of the power of the noise to the power of the solution is finite. Thus, if we want to consider the estimation problem on real line, we need to suppose that the power of the solution is also infinite or tends to infinity. The assumptions of such a type became standard in the study of the Bayes and minimax estimators (see [10,21,31]). In practice the power of the solution is usually finite. Thus
Keywords and phrases. Deconvolution, minimax estimation, Bayes estimation, Wiener filtration.
∗This paper was partially supported by RFFI Grants 02-01-00262, 4422.2006.1.
1Mechanical Engineering Problems Institute, Mechanical Engineering Problems Institute, Russian Academy of Sciences, Bolshoy pr. VO 61, 199178 St.Petersburg, Russia.
Article published by EDP Sciences c EDP Sciences, SMAI 2008
it seems reasonable to consider modifications of the models with the finite power of solution on the whole real line. The study of such a modification is the goal of the paper.
The deconvolution problem is often considered in another setting as deconvolution problem on a circle (see [8,18–20]). In this model the powers of the noise and the solution are finite. However this setting requires the additional analysis of errors caused by such a model reduction.
In this paper we deal with the following noise model
η(t) =h(t)ζ(t) (2)
whereζ(t) is a stationary Gaussian processζ(t),Eζ(t) = 0, E[ζ(t)ζ(0)] =r(t), t∈R1andhis a weight function h∈L2(R1). The assumptionh∈L2(R1) implies that the noise power2∞
−∞Eη2(t)dt =2||h||2r(0) is finite.
Thus we can assume that the power of solution is also finite.
From viewpoint of asymptotic statistics the model (1), (2) is closely related with another setting called often the problem of deconvolving a density (see [2,3,6,12,13,25]). In this setting we observe independent sample Z1 = X1+Y1, . . . , Zn = Xn+Yn. A density function a of i.i.d.r.v.’s Yi,1 ≤ i ≤ n is known and a density functionxof i.i.d.r.v.’sX1, . . . , Xn is unknown. We wish to estimate the densityx.
Thus the densityxsatisfies the convolution equation f(t) = (a∗x)(t) =
∞
−∞a(t−s)x(s) ds (3) wheref(t) is the density ofZ1, . . . , Zn.
As it is well-known (see [24]) the problem of estimation of density f is locally asymptotically equivalent to the estimation of function f in the following white noise model
dY(t) =f(t)dt+f01/2(t)dw(t). (4) Here dw(t) stands for the Gaussian white noise and it is supposed that f lies in the small vicinity of known functionf0 (see [24] for details).
We can rewrite (4) in the following form dY(t) =
∞
−∞a(t−s)x(s) ds+ ∞
−∞a(t−s)x0(s)ds 1/2
dw(t). (5)
Here we suppose thatxlies in the small vicinity of known functionx0(t),f0=a∗x0. If we puth= (a∗x0)1/2, we get the paper model. The local asymptotic equivalence of the model () and the model of deconvolving a density has not been studied. However one can show that a wide class of linear estimators in the problem of deconvolving a density have the same asymptotic behaviour as in the model (5).
The model of estimation in the weighted Gaussian white noise arises also in the problems of asymptotic estimation of regression function (see [1]). As first publication on asymptotically minimax estimation in this model one can mention [9].
The goal of our paper is to derive asymptotically minimax and asymptotically Bayes estimators for the model (1,2). We already considered this problem (see [11]) for the functions xwith a finite number of derivatives. In [11] we showed that asymptotically minimax and Bayes estimators have the same form as in the model with the Gaussian stationary noise (see [10,31]). This paper extend these results to the case of supersmooth functions.
The models with supersmooth functions naturally arise in statistical applications (see [4,5,28,30]).
In [4,5,30] the spectrum of operator corresponding to the kernel a is discrete. In our model the operator spectrum is continuous. The operator corresponding to the function h does not commute with the operators corresponding to the kernelaand the correlation functionr. These differences lead to implementation of serious analytical technique in the proof of the results.
The model of deconvolving a density with supersmooth densityxis rather natural. The asymptotic behaviour of kernel estimators having minimax rates of convergence was analyzed in [12]. The adaptive minimax rates of convergence were studied in [3,4]. As the first publications on the estimation of supersmooth functions with the sharp minimax asymptotic one should mention [10,15,16,27].
This paper can be considered as natural complement of [11]. The results of these papers cover the cases of the both smooth and supersmooth behaviour of kernel and solution. It turns out that, if the kernel or the solution is supersmooth, the asymptotically minimax and the asymptotically Bayes estimators are simple projection estimators. Only if both the solution and the kernel have finite smoothness, we are forced to define the estimators with the more complicated structure (see [11]). In this case the estimators based on the Kolmogorov- Wiener filters are asymptotically Bayes (see [21,31]) and the standard minimax estimators are asymptotically minimax (see [8,10,26]). The standard minimax estimators remain asymptotically minimax in the case of supersmooth kernel or solution. However in these cases we can define essentially more simple estimators with the same property.
For any function z∈L2(R1) denote by Z(ω) =
exp{2πiωt}z(t)dt the Fourier transform ofz and for anyz∈L2(R1) denote by
||z||=
z2(t)dt 1/2
theL2-norm ofz. Hereafter the limits of integration are omitted if the integration domain is the real lineR1. We suppose that the kernela satisfies A0,A1-A3 if the kernela has a finite number of derivatives and A0, A3, A4 in the case of supersmooth kernel.
A0. There holds A(ω) =A(−ω)>0 for allω∈R1. A1. There holds
t→∞lim t
0 A−2(ω)R(ω)dω=∞. (6)
If the left-hand side of (6) is finite, the rate of convergence of estimators equals. A2. There existsγ≥0 such that for allC >0
ω→∞lim A(Cω)
A(ω) =C−γ. A3. There existsC >0 such that for allω, ω1∈R1
|A(ω)−A(ω1)|< C|ω−ω1|.
A4. There existsγ >0 such that for allC >0
ω→∞lim
logA(Cω) logA(ω) =Cγ.
If A2 holds, the kernelahas a finite number of derivatives. If A4 holds, the kernelais infinitely differentiable.
The correlation functionrsatisfies the following R.There existsα >0 such that for anyC >0
ω→∞lim R(Cω)/R(ω) =C−α.
If A2, R hold with γ= 0, α= 0 andA(ω)≡1, R(ω)≡1, we get the standard setting estimation of signal in the weighted Gaussian white noise
dy(t) =x(t)dt+h(t)dw(t). (7)
1.2. Main results. Minimax estimation
The assumption about the solution is rather standard (see [8,10,26]). We suppose that a priori information is given
x∈Q=
x:
B2(ω)|X(ω)|2dω <1, x∈L2(R1)
(8) with the functionB satisfying the following.
B.The functionB(ω) is even, positive and there existsβ >0 such that for allC >0
ω→∞lim
lnB(Cω) lnB(ω) =Cβ.
Thus we have a priori information that the solutionxbelongs to an ellipsoid inL2(R1) and B implies that the solutionx(t) is infinitely differentiable.
The risk of any estimatorx∗(t) equals
ρ(x∗) = sup
x∈Q
E(x∗(t)−x(t))2dt.
We search for asymptotically minimax estimatorx∗∗ such that ρ=ρ(x∗∗ ) = inf
x∗ ρ(x∗)(1 +o(1)), →0. Here the infimum over all estimatorsx∗.
We suppose that the function h is smoother than the realizations of random processζ(t) (see H2 below).
Thus all information on the noise smoothness is contained inζ(t).
H1. The functionH(ω) is even,H(ω)∈L2(R1)∩L1(R1) andh(t)>0 for allt∈R1. H2. There existsδ >0 such that
ω→∞lim R−1(ω)H2(ω)ω1+δ= 0. H3.
|th(t)|dt <∞.
Define the functions
Ψ(µ) =2||h||2
A−2(ω)(1−µB(ω))+R(ω) dω and
Ψ¯(θ) =2||h||2 θ
−θA−2(ω)R(ω)dω.
Hereafter (u)+ = max{u,0} for all u ∈R1. We put ω1 = sup{ω : ¯Ψ(ω)≤ B−2(ω)} and ω = ω1(1 +δ) where δ >0 is such that B(ω1) = o(B(ω)), A−1(ω1) = A−1(ω)(1 +o(1)) and δ → 0 as → 0. Denote µ= sup{µ:B(ω)> µ}.
Define the kernels
Kµ(ω) =A−1(ω)(1−µB(ω))+ (9) and
Kω(ω) =A−1(ω)χ(|ω|< ω). (10)
Hereafter χ(U) denotes the indicator of event U. The kernel kµ is the kernel of standard asymptotically minimax estimator x∗∗µ(t) = (kµ ∗y)(t) (see [8–11,26]). The kernel kω is the kernel of simple projection estimatorx∗∗ω(t, y) = (kω∗y)(t).
Define also the kernelskαof Tikhonov regularizing algorithm Kα(ω) = A(ω)
A2(ω) +αM(ω) (11)
where the function M(ω) satisfies the following.
M1. The functionM(ω) is even, nonnegative and increasing inR1+. M2. The functionM(ω)|B−1(ω)A−2(ω)|is nondecreasing inR1+.
M2 implies that the functionM(ω) has the exponential growth. The exponential growth of functionM(ω) in Tikhonov regularizing algorithm is natural assumption in the case of a priori information on supersmooth solution (see [20]).
Define the parameters of regularizationα=A2(ω)M−1(ω).
If the solution x is smoother than the kernel a (A2 or A4 with β > γ holds) the asymptotic of minimax risks is attained for all estimators mentioned above: for the standard minimax estimatorkµ (see (9)), for the simple projection estimator kω (see (10)) and the estimator of Tikhonov regularizing algorithm (see (11)). If the kernelais smoother then the solutionx(A4,B holds withγ > β), then the lower bound of asymptotic of minimax risks is attained only in the cases of simple projection estimator and estimator of Tikhonov regularizing algorithm. These statements are given in Theorems 1 and 2 respectively. In both theorems we assume thath is smoother than the realizations of random processζ(t) (see H2 above).
Theorem 1.1. Assume A0–A3, B, H1–H3, R or A0, A3, A4, B, H1–H3, R with β > γ. Then the estimators x∗∗µ(t) = (kµ∗y)(t)andx∗∗ω(t, y) = (kω∗y)(t)are asymptotically minimax. The asymptotically minimax risks equal
ρ(x∗∗µ) =ρ(x∗∗ω)(1 +o(1)) = Ψ(µ)(1 +o(1)) = ¯Ψ(ω)(1 +o(1)) = ¯Ψ(ω1)(1 +o(1)). (12) If M1,M2 hold also, the Tikhonov regularizing algorithm x∗∗α=kα∗y is asymptotically minimax.
The presence of||h||2 in the asymptotic Ψ(µ) of asymptotically minimax riskρand in the definition (9) of asymptotically minimax estimator x∗∗µ is the unique difference from the standard definitions of asymptotically minimax risks and asymptotically minimax estimator (see [8–11,26]).
Remark 1. Denote Λ the set of functionshsatisfying H1,H3 and such that the convergence in H2 is uniform w.r.t. allh∈Λ. Suppose also that there exist functionsh01, h0 ∈Λ such thath01≥h(t)≥h0(t)>0, t∈R1 for all h∈Λ. Then the asymptotics of minimax risksρh(x∗∗µ) =ρh(x∗∗ω)(1 +o(1)) are uniform with respect to h∈Λ. Moreover all information on hin the estimatorsx∗∗ω, x∗∗µ is contained in||h||. Thus we can consider has unknown in the model supposing onlyh∈Λ. Naturally, since||h||is unknown, the problems of the choice of regularization parametersµ andω arise. However, since other parameters of models are usually unknown as well (for example R(ω), B(ω)), these problems arise also on other reasons. Note that similar remark on the uniform risks convergence hold for the other theorems of this paper and for theorems of [11] as well.
If the kernelais smoother than the solutionxandγ > β(see A3, B), the projection estimatorx∗∗ω=kω∗y and the Tikhonov regularizing algorithmx∗∗α =kα∗y remain asymptotically minimax. However the formula for asymptotic of minimax risks is defined in another form.
Letω1 satisfy the equation
B−2(ω1) =2||h||2 ω1
−ω1
A−2(ω)R(ω)dω. (13) Defineδ>0 such thatδ=o(ω−β1 ) andω1−γ =o(δ). Denoteω= (1−δ)ω1and α=A2(ω)M−1(ω).
Theorem 1.2. Assume A0, A4, B, H1, H2 and R. Let γ > β. Then the family of projection estimators x∗∗ω(t, y) = (kω∗y)(t)is asymptotically minimax. There holds
ρ(x∗∗ω) =B−2(ω)(1 +o(1)) =B−2(ω1)(1 +o(1)). (14) If M1 holds, the Tikhonov regularizing algorithm x∗∗α=kα∗y is also asymptotically minimax.
Example 1. Let
A(ω) = A1(ω)|ω|−γ, (15)
B(ω) = B2exp{B1(ω)|ω|β}, (16)
R(ω) = R1(ω)|ω|−α (17)
and let the assumptions of Theorem 1.1 hold.
Letω1 be defined the equation
B1(ω1)ωβ1=|ln|. (18)
Then one can putω=ω1(1 +δ) whereδ>0, δ|ln|1−κ→ ∞, δ→0 as→0 with 1> κ >0 and ρ= 2||h||2
2γ−α+ 12ω2γ−α+11 A−21 (ω1)R1(ω1)(1 +o(1)). (19) IfA1(ω) =A1, B1(ω) =B1, R1(ω) =R1 are constants, we get
ρ=A−21 R1B
α−2γ−1
1 β
2||h||2
2γ−α+ 12|ln|2γ−α+1β (1 +o(1)). (20) In the case of standard setting signal detection in the Gaussian white noise (A(ω)≡1, R(ω)≡1)
ρ= 2B−
1β
1 ||h||22|ln|1β(1 +o(1)). Example 2. Let
A(ω) =A2exp{−A1(ω)|ω|γ} (21)
and (16), (17) hold with β > 2γ. Then we can put ω = ω1(1 +δ) where ω1 is defined (18) and δ >
0, δ|ln|1−γ/β−κ→ ∞, δ→0 as→0 with 1−γ/β > κ >0. The asymptotically minimax risk equals and ρ= 2
γA−22 ||h||22ω1−γ−α1 R1(ω1) exp{2A1(ω1)|ω1|γ}(1 +o(1)). (22) IfA1(ω) =A1, B1(ω) =B1, R1(ω) =R1 are constants, we get
ρ= 2 γA−22 B
α+γ−1
1 β R1||h||22|ln|1−γ−αβ exp{2A1B−
γβ
1 |ln)|γβ}(1 +o(1)). (23) Example 3. Let (16), (17), (21) hold with 2γ ≥ β > 32γ and let A1(ω) = A1, B1(ω) = B1, R1(ω) = R1 be constants. Then
ω1=B1−1/β|ln|1/β
1−1
βA1B−γ/β1 |ln|γβ−1
and
ρ = γ−1A−11 A−22 B
α+γ−1
1 β R1||h||22|ln|1−γ−αβ
×exp
2A1B−
γβ
1 |ln|γβ
1−γ βA1B−
γβ
1 |ln|γβ−1
(1 +o(1)). (24)
Example 4. Let (16), (17), (21) hold withγ >2β. Defineω1the equation A1(ω1)ωγ1=|ln|
and put ω=ω1(1−δ) whereδ>0, δ→0, δ|ln|1−β/γ−κ→ ∞as→0 with 0< κ <1−β/γ. Then ρ=B22exp
−2B1(ω1)ω1β
(1 +o(1)). (25)
IfA1(ω) =A1, B1(ω) =B1are constants, we get ρ=B22exp
−2B1A−
βγ
1 |ln|βγ
(1 +o(1)). (26)
Example 5. Let (16), (17), (21) hold with 2β ≥γ > 3/2β and letA1(ω) =A1, B1(ω) =B1, R1(ω) =R1 be constants. Then
ω1=A−1/γ1 |ln|1/γ
1− 1
γB1A−β/γ1 |ln|βγ−1
(27) and
ρ=B22exp
−2A−
βγ
1 B1|ln|βγ
1−β γA−
βγ
1 B1|ln|βγ−1
(1 +o(1)). (28)
1.3. Main results. Bayes approach
In the Bayes setting we suppose that the solutionxis a realization of the random process x(t) =h1(t)ξ(t)
whereh1(t)∈L2(R1) andξis Gaussian stationary random process,Eξ(t) = 0, E[ξ(t)ξ(0)] =v(t), t∈R1. As follows from assumption V given bellow we suppose that the realizations of random process ξ(t) are infinitely differentiable.
V.There existsβ >0 such that for allC >0 there holds
ω→∞lim
lnV(Cω) lnV(ω) =Cβ. The functionh1(t) satisfies the following assumptions.
H4. The functionh1(t) is even, bounded andh1(t)> ch(t)>0 for allt∈R1 with constantc >0.
H5. There holds
ω→∞lim
lnH1(ω) lnV(ω) =∞.
H5 implies that the main information on the smoothness ofxis contained in the random processξ(t).
For any estimatorx∗ define the Bayes risk
ρˆ(x∗) =EξEζ||x∗−h1ξ||2. We say that the estimator ¯x∗ is asymptotically Bayes if
ρˆ= ˆρ(¯x∗) = inf
x∗ ρˆ(x∗)(1 +o(1)), →0. Here the infimum is over all estimatorx∗.
In this setting we could not prove that the Wiener filters
K(ω) =||h1||2A(ω)V(ω)(||h1||2A2(ω)V(ω) +2||h||2R(ω))−1
are asymptotically Bayes. At the same time we show that the simple projection estimatorx∗∗ωare asymptotically Bayes. The value ofω is defined by the equationω=ω1(1 +δ) whereω1 satisfies the equation
A2(ω1)||h1||2V(ω1) =2||h||2R(ω1), (29) andδ>0, δ→0 as→0, is such that
V(ω) =o(V(ω1)), H12 1
2δω1
=o(2A−2(ω)R(ω)). (30) Theorem 1.3. Assume A0, A1, A2, H1, H2, H4, H5, R, V. Then the family of estimatorsx∗∗ω = (kω∗y)(t) is asymptotically Bayes. There holds
ρˆ(x∗∗ω) =2||h||2 ω1
−ω1
A−2(ω)R(ω)dω(1 +o(1)). (31) Note thatω1 in (31) can be replaced byω.
Example 6. Let
V(ω) = exp{−V1|ω|β} and (15), (17) hold. Then
ω1=V11/β|ln|1/β
andω=ω1(1 +δ) whereδ>0, δ→0, δ|ln|1−κ→ ∞as→0 with 0< κ <1.
We have
ρˆ= 2||h||2 2γ−α+ 1V
2γ−α+1
1 β 2|ln|2γ−α+1β A−21 (ω)R1(ω)(1 +o(1)). For the model of signal detection in Gaussian white noiseA(ω)≡1, R(ω)≡1 we get
ρˆ= 2||h||2V
1β
1 2|ln|β1(1 +o(1)).
Remark 2. In applications the kernelais often unknown and the information about the kernel can be obtained only from statistical experiment (see [21,29]). As a consequence, the arising estimator ˆaof kernela is known with a random error. Usually it is supposed that the error admits the Gaussian approximation. If we study the quality of estimation of solutionxone needs to find the influence of this random error on the risk function.
We consider the following model of kernel estimator
ˆa(t) =a(t)dt+κ¯h(t)dw(t) with the function ¯h∈L2(R1) satisfying the following
t2¯h2(t)dt <∞, (32)
ω→∞lim H¯2(ω)|ω|1+δ= 0 (33) whereδ >0.
The Fourier transform of ˆacan be written in the following form Aˆ(ω) =A(ω) +κˆτ(ω)
where
τˆ(ω) =
H¯(ω−ω1)dw(ω1) is Gaussian stationary process.
Straightforward calculations show that, if M = sup{|ˆ. τ(ω),|ω| ≤ ω} = oP(A(ω)/κ), the random error κˆτ(ω) does not influence on the asymptotic of minimax and Bayes risks and the estimators x∗∗ω remains asymptotically minimax and asymptotically Bayes. The reasoning is the following.
If (32),(33) holds, by Theorem 12.3.5 in [22],
→0limP ||h||−1(2 ln(2ω))−1/2|M− ||h||(2 ln(2ω))1/2|> u
= exp{−Cexp{−u}}.
Hence, if κ = o
A(ω)|lnω|1/2
, the influence of noise κ¯h(t)dw(t) on the kernel a is negligible and the asymptotic of risks for the estimator x∗∗ω are the same both in the minimax setting of Theorems 1.1,1.2 and the Bayes setting of Theorem 1.3. A similar statements hold also for the asymptotically minimax settings Theorems 1.1 and 1.2 in [11]. Note that =o
A(ω)(lnω)1/2
. Thus we can estimate the solution xin the case of random kernel with the same risk asymptotic even if the kernel estimator ˆahave a larger error than.
2. Proofs of theorems 2.1. Proof of lower bound in Theorem 1.1
The proof of lower bounds in minimax setting is based traditionally on the fact that the Bayes risk does not exceed the minimax one. We define such Bayes a priori distributions that the powers of realizations ω+∆ω
ω X2(u)du on each interval (ω, ω+ ∆ω) ⊂ Ω = (−ω1(1−δ), ω1(1−δ)) have the large order then corresponding power of noise2||h||2ω+∆ω
ω A−2(u)R(u)du. Hence the bias of Bayes estimator does not influence on the asymptotic of Bayes risks. Thus we get that the asymptotic of Bayes risks is defined the random noise and the contribution of each interval (ω, ω+∆ω)⊂Ωin the Bayes risk equals2||h||2ω+∆ω
ω A−2(u)R(u)du(1+
o(1)). This allows to choose A−1(ω)Y(ω) as asymptotically Bayes estimatorsX(ω) in Ω and to get the lower bound (12).
We putω2= (1−δ)ω1 whereδ>0 is such thatA(ω2) =A(ω)(1 +o(1)),
B−1(ω1) =o(B−1(ω2)), 2A−2(ω)(ω1+γ+βR−1(ω) +ω3) =o(B−2(ω2)) (34) andδ→0 as→0.
Define the valuesD and ∆ such that
D=o(B−1(ω2)ω(1−β)/2), 2A−2(ω)(ω1+γ+β R−1(ω) +ω3) =o(D2) (35) and
∆ = ∆=o(2D−2R(ω)ω−γ+ω−3/2−γR(ω) +ω1−β ). (36) The Bayes a priori measuresλ=λ∆are defined as the conditional probability measures of Gaussian random processes ξ = ξ∆ under the condition ξ ∈ Q where Q is the set of all solution x (see (8)). The random processξ=ξ∆ is defined via the Fourier transform
ξˆ(ω) =ξ∆=D
l i=−l
ηiχ(ω∈Ii) (37)
where l =l = [ω2/∆], Ii = ((i−1/2)∆,(i+ 1/2)∆) andηi are independent Gaussian random variables, Eηi = 0, Eη2i = 1,−l≤i≤l. Hereafter [t] denotes the integer part oft∈R1.
On the base of the partition on intervals Ii,−l≤i ≤l, we reduce the problem to the discrete version and show that the discrete version of the problem has a simple solution. The choice of D satisfying (35) allows to get2E
Iiηˆ2(u)du=o(E
Iiξˆ2(u)du) on all intervalsIi,|i|> δl,0< δ <1. The variation of functionR(ω) on intervalIi is neglected, that isR(ω) =R(i∆)(1 +o(1)) if ∆ satisfies (36). At the same time
Ii
h2(ω−u) du=||h||2(1 +o(1))
if |ω|<(1−δ1)∆,0 < δ1 <1. This allows to show that asymptotically minimax risks ρ and asymptotically minimax estimatorsx∗∗µ, x∗∗ω depend only on||h||2.
Denoteνthe probability measure ofξ. Lemma 2.1. There holds
→0limP(ξ∈Q) = 1. (38)
Proof. By straightforward calculations, we get E
B2(ω) ˆξ2(ω)dω
=O
D2 ω2
−ω2
B2(ω)dω
=O(D2B2(ω2)ωβ−1) =o(1), Var
B2(ω) ˆξ2(ω)dω
=O
D4 ω2
−ω2
B4(ω)dω
=O(D4B4(ω2)ω1−β) =o(D4B4(ω2)ω2−2β) as →0. Hence, by Chebyshev inequality, using (35), we get (38). This completes the proof of Lemma2.1.
For any estimatorx∗ define the Bayes risks ρˆλ(x∗) =
Qdλ(x)E||x∗−x||2, ρˆν(x∗) =
Qdν(x)E||x∗ −x||2.
Denote ¯x∗ and ¯x the Bayes estimators corresponding to a priori measuresλ andν respectively.
Lemma 2.2. There holds
ρˆλ(¯x∗)≥ρˆν(¯x)(1 +o(1)), →0. The proof of Lemma2.2is akin to that of Lemma 2.3 in [11] and is omitted.
By Lemmas2.1and2.2, the proof of lower bound is reduced to the problem of obtaining asymptotic of Bayes risks ˆρν(¯x). Since a priori measuresν are Gaussian the analytic formula for the Bayes risks ˆρν(¯x) can be written straightforwardly (see (47) below). However this formula is rather cumbersome and the proof that the asymptotic of ˆρν(¯x) coincides with ¯Ψ(ω) requires special analytic technique.
Since the random processξ is Gaussian the Bayes estimator is linear x¯=
k(t, s)y(s)ds with the kernelksatisfying the equation (see (2.16) in [11])
Eτ[¯x(t)f(s)]−Eτ
k(t, u)f(u)duf(s)
−2
k(t, u)h(s)h(u)r(u−s)du= 0 (39)
for allt, s∈R1. Hereafter f =a∗x. Note that (39) is obtained as the Gateaux derivative of the risk function EξEζ
k(t, u)y(u)du−ξ
2.
In terms of Fourier transform (39) can be written as follows V∆(ω, ω0)A(ω0) =
K(ω, ω1)A(ω1)V∆(ω1, ω0)dω1A(ω0) +2
K(ω, ω2)
H(ω2−ω1)R(ω1)H(ω1−ω0)dω1dω2, ω, ω0∈R1. (40) HereafterV∆=V∆(ω1, ω0) =E[ξ(ω1)ξ(ω0)] =D2i=l
i=−lχ(ω1∈Ii, ω0∈Ii) withω0, ω1∈R1.
In what follows, we shall make use of operator symbolic in the problem of risk minimization. DenoteA, Rthe multiplication operators with the kernelsA(ω), R(ω). DenoteK, V∆, Hthe integral operatorsL2(R1)→L2(R1) with the kernelsK(ω, ω0),V∆(ω, ω0), H(ω−ω0) respectively and denoteI the unit operator. In this notation the Bayes estimator minimizes the Bayes risk
ρ(K) = Sp
(KTA−I)V∆(AK−I) +2KTHRHK
(41) among all estimators x∗ =k∗y. Here Sp[U] denotes the trace of operatorU and KT denotes the conjugate operator ofK.
DenoteR(ω) =R(ω)χ(|ω|< ω2) and R¯(ω, ω0) =
H(ω−ω1)R(ω1)H(ω1−ω0)dω1χ(ω∈(−ω2, ω2), ω0∈(−ω2, ω2)). SinceR≤R, we get
ρ¯(K) = Sp[(KTA−I)V∆(AK−I) +2KTR¯K]
≤ Sp[(KTA−I)V∆(AK−I) +2KTHRHK]≤ρ(K). (42) Therefore the problem of obtaining lower bound is reduced to that of minimization of ¯ρ(K).Define the operator K¯ such that ¯ρ( ¯K) = inf ¯ρ(K) where the infimum is over all operatorsK such that ¯ρ(K)<∞.
If we write a version of equation (40) for the operator ¯K, we get easily
K¯=V∆A(AV∆A+2R¯)−1. (43) It follows from the definition of ¯ρ(K) that in the problem of minimization of ¯ρ(K) it suffices to consider the operatorsA, V∆,R¯ as operators L2((−ω2, ω2)) →L2((−ω2, ω2)). Thus, in what follows, we suppose that such a contraction of operators takes place.
For any linear operatorG:L2(−ω2, ω2)→L2(−ω2, ω2) denote
||G||= sup{xGx: ||x||= 1, x∈L2(−ω2, ω2)} wherexy stands for the inner product ofx, y∈L2(−ω2, ω2).
In this notation there holds
||K¯|| ≤ ||A−1|| ||AK|| ≤ ||A−1|| ≤A−1(ω). (44)
We shall use also the following inequality. For any operatorGand positive operatorM it holds
Sp[GM]≤ ||G||Sp[M] (45)
if Sp[M]<∞and||G||<∞.
It is known also that, ifG(s, t) is the kernel of linear operatorG, then
||G||2≤ ω2
−ω2
ω2
−ω2
G2(s, t) dsdt. (46) Substituting (43) in the left-hand side of (42), we get
ρ¯( ¯K) = 4Sp[(AV∆A+2R¯)−1R¯V∆R¯(AV∆A+2R¯)−1]
+2Sp[(AV∆A+2R¯)−1AV∆R¯V∆A(AV∆A+2R¯)−1]. (47) Now we define some discrete version ¯ρ∆( ¯K) of ¯ρ( ¯K) and show that |¯ρ∆( ¯K)−ρ¯(K)| = o( ¯ρ∆( ¯K)) is negligible. After that we show that ¯Ψ(ω) is the lower bound of ¯ρ∆( ¯K). The proofs of these statements have technical character and are based on inequalities (44)–(46).
For eachi,|i| ≤lwe fix ωi∈Ii and define the functions A∆(ω) =
l i=−l
A(ωi)χ(ω∈Ii),
H∆(ω, ω1) = l i=−l
H(ωi−ω1)χ(ω∈Ii). R∆(ω0, ω1) = (H∆TRH∆)(ω0, ω1). Denote
ρ¯∆( ¯K) = Sp
( ¯KTA∆−I)V∆(A∆K¯−I) +2K¯TR∆K¯
. (48)
We have
ρ¯( ¯K)−ρ¯∆( ¯K) = Sp[ ¯KT(A−A∆)V∆(AK¯−I)]
+Sp[ ¯KTA∆V∆(A−A∆) ¯K] +2Sp[ ¯KT( ¯R−R∆) ¯K]=. J1+J2+2J3. (49) Since||AK¯−I|| ≤1 and||K¯A|| ≤1, by A4, (44), we get
|J1| ≤A−1(ω)||AK¯|| ||A−A∆||Sp[V∆]||AK¯−I|| ≤CA−1(ω)∆Sp[V∆]. (50) HereafterC stands for positive constants.
Arguing similarly, we get
|J2| ≤A−1(ω)||AK¯|| ||A∆||Sp[V∆]||A−A∆||A−1(ω)||AK¯|| ≤C∆A−2(ω)Sp[V∆]. (51) We have
J3= Sp[ ¯KK¯T(H−H∆)RH] + Sp[ ¯KK¯TH∆R(H−H∆)]=. J31+J32. (52) By H3, the functionH(ω) satisfies the Lipschitz conditions. Hence, using (46), we get
||H−H∆|| ≤
⎛
⎝ ω
−ω
dω0
ω
−ω
dω
H(ω0−ω) l i=−l
H(ωi−ω)χ(ω0∈Ii)
2⎞
⎠
1/2
≤C∆ω (53)
