Non asymptotic minimax rates of testing in signal detection with heterogeneous variances

The aim of this paper is to establish non-asymptotic minimax rates of testing for goodness-of-fit hypotheses in a heteroscedastic setting. More precisely, we deal with sequences $(Y_j)_{j\in J}$ of independent Gaussian random variables, having mean $(\theta_j)_{j\in J}$ and variance $(\sigma_j)_{j\in J}$. The set $J$ will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to $l_2$ and $l_{\infty}$ norms, without assumption on $(\sigma_j)_{j\in J}$ and on several functions spaces. Our point of view is completely non-asymptotic.


Introduction
We consider the following heteroscedastic statistical model : where θ = (θ j ) j∈J is unknown, σ = (σ j ) j∈J is assumed to be known, and the variables (ǫ j ) j∈J are i.i.d.standard normal variables.The set J is either {1, . . ., N } for some N ∈ N * (which corresponds to a Gaussian regression model) or N * (which corresponds to the Gaussian sequence model).The sequence θ has to be tested from the observations (Y j ) j∈J in order to decide whether "θ = 0" or not.The particular case σ j = σ for all j ∈ J corresponds to the classical statistical model where the variance of the observations is always the same.It has been widely considered in the literature, both for test and estimation approaches.In this paper, we consider a slightly different setting in the sense that the variance of the sequence is allowed to depend on j.
We point out that the model (1.1) can describe inverse problems.Indeed, for a linear operator T on an Hilbert space H with inner product (., .),consider an unknown function f indirectly observed in a Gaussian white noise model where ǫ(g) is a centered Gaussian variable with variance g 2 := (g, g).If T is assumed to be compact, it admits a singular value decomposition (SVD) (b j , ψ j , φ j ) j≥1 in the sense that T φ j = b j ψ j , T * ψ j = b j φ j , with T * the adjoint operator of T .Hence considering the observations Y (ψ j ), model (1.2) becomes Z j = b j θ j + σǫ j , j ∈ N ⋆ , ( with ǫ j = ǫ(ψ j ), (T f, ψ j ) = b j θ j and θ j = (f, φ j ).This model is often considered in the inverse problem literature, see eg [7].Setting Y j = b −1 j Z j and σ j = σb −1 j for all j ∈ N ⋆ , we obtain (1.1).Hence inference on observations from model (1.1) provides the same results for inverse problems.We stress that if estimation issues for inverse problem have been well studied over the past years (see for instance [19], [7] or [18] for a model selection approach), tests for inverse problems have either only been tackled, in the general case, by very few authors among them we refer to [5], [12], [16] or have been investigated only for the very specific case of the convolution problem, see in [6] and references therein.
Let us now define ρ q (F, α, β) as the infimum over all level α test Φ α of the quantity ρ q (Φ α , F, β).This quantity will be called the (α, β) minimax rate of testing over the class F. The aim of the paper is to determine this minimax rate of testing over various classes of alternatives F, for the test of null hypothesis "θ = 0" in Model (1.1) with respect to the l 2 and l ∞ norms.
The main reference for computing minimax rates of testing over non parametric alternatives is the series of paper due to Ingster [11], where various statistical models and a wide range of sets of alternatives are considered.Lepski and Spokoiny [17] obtained minimax rates of testing over Besov bodies B s,p,q (R) in the irregular case (when 0 < p < 2), see also Ingster and Suslina [13].Ermakov [10] determines a family of asymptotic minimax tests for testing that the signal belongs to a parametric set against nonparametric sets of alternatives in the heteroscedastic Gaussian white noise.In all these references, asymptotic minimax rates of testing are established.In Model (1.1), with σ j = σ for all j ∈ J, Baraud [2] consider a non asymptotic point of view, which means that the noise level σ is not assumed to converge towards 0. This is the point of view that we adopt in this paper.We give a precise expression of the dependency of the minimax rates of testing with respect to the sequence (σ j ) j∈J .The particular cases of interest correspond to polynomial and exponentially increasing sequences, which in the case of Model (1.3) leads to the so-called mildly and severely ill-posed inverse problems.We do not aim at providing the adaptive minimax rates of testing, which will be the core of a future work.
The paper is organised as follows.In Section 2, we provide lower bounds for the minimax separation rate over classes of vectors θ with a finite number of non-zero coefficients, which yet covers sparse signals.In Section 3, we determine upper bounds for those minimax rates.In Section 4, we compute minimax rates of testing over ellipsoids and l p balls.The proofs are gathered in Sections 5 and 6.
To end this introduction, let us define some notations.Let Y = (Y j ) j∈J obey to Model (1.1).We denote by θ the vector (or sequence) (θ j ) j∈J and by P θ the distribution of Y .All along the paper, we consider the test of null hypothesis "θ = 0".Let α ∈]0, 1[ be some prescribed level.A test function Φ α is a measurable function of the observation Y , with values in {0, 1}.The null hypothesis is accepted if Φ α = 0 and rejected if Φ α = 1.Finally, for all x ∈ R, we denote by ⌊x⌋ the greater integer smaller than x and we set ⌈x⌉ = ⌊x⌋ + 1.

Lower bounds
The bounds will be established for two classes of signals characterized by their non zeros coefficients.The first one deals with the elementary case where the coefficients are equal to zero after a certain rank.The second one concerns the so-called sparse signals which are defined by the amount of non zeroes coefficients which can be located at different scales.

Lower bounds in l 2 norm
In this section, we generalize the results obtained by Baraud [2] in an homoscedastic model to the heteroscedastic Model (1.1).
We first give a lower bound for the minimax rate of testing over the set S D , defined for all D ≥ 1 by S D = {θ ∈ l 2 (J), ∀j > D, θ j = 0} .
The following result holds: This implies that the minimax rate of signal detection over S D with respect to the l 2 norm satisfies ρ 2 (S D , α, β) ≥ ρ D .
This proposition can be understood as follows: whatever the α-level test chosen, for all ρ ≤ ρ D , there exists a signal θ ∈ S D with norm ρ such that the error of the second kind is greater than β.The results obtained in Proposition 1 coincide with the lower bound established by Baraud [2] in the homoscedastic model (σ j = σ ∀j ∈ J).
Let us now consider the problem of sparse signal detection.Let k, n ∈ N * with k ≤ n.When J = {1, . . ., N }, we assume that n ≤ N .We want to obtain lower bounds for the minimax separation rate of signal detection over the set S k,n defined by (2.4) , we define for all l ∈ {0, . . ., n − k}, (2.5) This implies that the minimax rate of signal detection over S k,n with respect to the l 2 norm satisfies ρ 2 (S k,n , α, β) ≥ ρ k,n .
Comments : Let us consider three cases governing the behaviour of the sequence (σ j ) j∈J .
1.In the homoscedastic case, σ j = σ for all j ∈ J.In this case, Σ 2 l,k = σ 2 k for all l and, taking l = 0, we obtain that which corresponds to the lower bound established by Baraud [2].
2. When k ≤ n/2 and Σ 2 ⌊n/2⌋,k ≥ CΣ 2 n−k,k for some absolute constant C (independent of k and n), we obtain that At the price of a factor 2 in the logarithm (n is replaced by n/2), the variance term appearing in the lower bound for ρ 2 k,n is Σ 2 n−k,k which corresponds to the largest possible variance for a set of cardinality k in {1, . . ., n}, indeed where M k,n denotes the set of all subsets of {1, . . ., n} with cardinality k.This situation occurs for example when (σ j ) j∈J grows at a polynomial rate : σ j = j 2γ for some γ > 0. In this case, 3. When (σ j ) j∈J grows at an exponential rate : σ j = exp(γj) for some γ > 0, we obtain that In this section, we provide lower bounds for the minimax rate of signal detection in Model (1.1) with respect to the l ∞ norm.These bounds can be derived from the Theorem 1. Indeed, let S k,n be defined by (2.4).For θ ∈ S 1,n , θ 2 = θ ∞ .Hence, denoting by ρ ∞ (F, α, β) the (α, β) minimax rate of testing over the class F with respect to the l ∞ norm, we obtain that By definition of the (α, β) minimax rate of testing it is obvious that if This leads to the following corollary of Theorem 1 : (2.8) The following result holds : The proof of the corollary follows directly from the arguments given above and therefore will be omitted.

Upper bounds
In this section, we give upper bounds for the (α, β) minimax rates of testing over the sets S D and S k,n that we compare with the lower bounds obtained in the previous section.
In order to show that the (α, β) minimax rate of testing with respect to the l q norm over a set F is bounded from above by ρ, it suffices to define a test statistic Φ α such that the power of the test at each point θ in F satisfying θ q ≥ ρ is greater than 1 − β.
Proposition 2 Assume that Y = (Y j ) j∈J obeys to Model (1.1).Let α, β ∈]0, 1[, and let t D,1−α (σ) denote the 1 − α quantile of D j=1 σ 2 j ǫ 2 j : Let Φ α be the test defined by Then, Φ α is a level-α test : Moreover, there exists an absolute constant C such that for all θ ∈ S D , setting Hence, we obtain that Note that this bound coincides to the upper bound found in Proposition 1.Hence, it proves that this lower bound is sharp.
Let us now propose a testing procedure for sparse signal detection.This procedure will be defined by a combination of two tests.The first one is based on a thresholding method, which was already used for detection of irregular alternatives in Baraud et al [3] and in Fromont et al [9].The second one is the test considered in Proposition 2 with D = n, which will be powerful when k is larger that √ n.
α be the test defined by α be the test defined by α/2 .Then, Φ α is a level-α test : There exists a constant C(α, β) such that for all θ ∈ S k,n satisfying ) Hence, we obtain that where Σ 2 l,k has been defined in (2.5).
Comments : Let us compare these results with the lower bounds obtained in Theorem 1.
1. We first assume that (σ j ) j∈J grows at a polynomial rate : ∀j ∈ J, σ j = σj γ for some γ ≥ 0 (this includes the homoscedastic case).In this case, when k ≤ n/2 there exists a constant A lower bound for the (α, β) minimax separation rate of signal detection over S k,n is given by (2.7).This lower bound has to be compared with the upper bound (3.11).
• When k = n l with l < 1/2, the upper and lower bounds coincide and are of order , which leads to a lower bound of order Cσ 2 n 2γ+1/2 .The upper bound is smaller that , which is smaller than σ 2 n 2γ+1/2 .Hence, the two bounds coincide.
and the upper bound is of order Σ 2 n−k,k ln(n).In this case, the upper and lower bound do not coincide, up to a logarithmic term.
Hence the two bounds coincide.Note that in this case, the test Φ (2) α based on thresholding is useless and one can simply consider that test Φ α = Φ (1)  α , which achieves the lower bound for the separation rate.
3. The result stated in (3.10) is more precise than the minimax upper bound given in (3.11).
If the set J 1 = {j, θ j = 0} corresponds to small values for the variances (σ j ) j∈J 1 , it is not required that θ 2 2 is greater than the right hand term in (3.11) for the test to be powerful for this value of θ.The minimax bound given in (3.11) corresponds to the worst situation, that is the case where the set J 1 corresponds to the largest values for the variances.
Let us now present an upper bound for the minimax separation rate with respect to the l ∞ norm.
This implies that for all k ∈ {1, . . ., n}, This upper bound coincides with the lower bound obtained in Corollary 1 when the sequence (σ j ) j∈J is constant or grows at a polynomial or at an exponential rate.

Minimax rates over ellipsoids and l p balls
In the previous sections, the only constraint on the signal was expressed through the number of non-zero coefficients.In several situations, one deals instead with infinite sequences having a finite number of significant coefficients, the reminder being considered as negligible (in a sense which will be precised later on).To this end, we consider in this section a slightly different framework.Our aim is to study the link between the decay of the θ k 's and the associated rate of testing.We consider in the following two different kinds of function spaces: ellipsoids and l p -bodies.

Minimax rates of testing over ellipsoids
In the following, we assume that the sequence θ = (θ j ) j∈J belongs to the ellipsoid E a,2 (R) defined as where a = (a k ) k∈J denotes a monotone non-decreasing sequence.For instance, if θ corresponds to the sequence of Fourier coefficients of a function f and a j is of order j s with s > 0, then assuming that θ ∈ E a,2 (R) is equivalent to impose conditions on the s-th derivative of f .The belonging to E a,2 (R) may be seen as a regularity assumption on our signal.The following result characterises the minimax rate of testing over E a,2 (R).
Proposition 3 Let α, β be fixed and denote by ρ 2 (E a,2 (R), α, β) the minimax rate of testing over E a,2 (R) with respect to the l 2 norm.Then where ρ 2 D has been introduced in Proposition 1.Moreover, for all D ∈ J, where C = C(α, β) is a positive constant depending only on α and β and Φ α denotes the test introduced in Proposition 2. Hence, Proposition 3 presents both an upper and a lower bound for the minimax rate of testing over E a,2 (R).Remark that the upper bound is attained by the test Φ α introduced in Proposition 2 where only signals with a finite number of non-zero coefficients were considered.We do not use the whole sequence (Y j ) j∈J in order to test the null hypothesis "θ = 0" but only the first D coefficients.The price to pay is to introduce some bias in the testing procedure.However this bias can be controlled by taking advantage of the constraint expressed on the decay of θ.
A good characterization of ρ 2 (E a,2 (R), α, β) can be obtained as soon as the lower and upper bounds are of the same order.As many statistical problems encountered in the literature, one has to find a trade-off between the bias R 2 a −2 D and some kind of variance term ρ 2 D .This tradeoff can be performed in several situations, hence leading to explicit rates of convergence: see Corollary 3 below.Proposition 3 presents the minimax rate of testing in a general setting.Several explicit rates can be obtained when introducing specific constraints on the sequences (a k ) k∈J and (b k ) k∈J .These rates are summarized in Corollary 3. Let (ν k ) k∈N ⋆ be a sequence real numbers.In the following, we write ν k ∼ k l if there exist positive constants c 1 and c 2 such that, for all k ∈ N ⋆ , Corollary 3 Let α, β be fixed.We assume that J = N ⋆ and (Z j ) j∈J obeys to Model (1.3).The table below presents the minimax rates of testing over the ellipsoids E a,2 (R) with respect to the l 2 norm.We consider various behaviours for the sequences (a k ) k∈N ⋆ and (b k ) k∈N ⋆ .For each case, we give f (σ) such that for all Here D denotes the integer part of the solution of ρ 2 D = R 2 a −2 D .These rates have already been presented in the literature.The case a k ∼ k s and b k ∼ k −t was first studied in [14].More recently, [12] deals with other cases.Similar rates are also available in [6] in the context of density estimation with errors in the variables.The aim of Corollary 3 is to show that our approach can lead to important minimax results.Our point of view in this paper is indeed entirely non-asymptotic and is not restricted to ill-posed inverse problems.
Concerning severely ill-posed problems with supersmooth functions (i.e.b k ∼ exp(−γk r ) and a k ∼ exp(νk s )), we do not handle the general case since we assume that s ≤ 1.When this assumption is violated, the upper and lower bounds in Proposition 3 do not coincide: our test does not attain the minimax rate of testing.This is certainly due to our approach, which in some sense is related to a rough regularization scheme.We mention [12] for a complete study of this case.
4.2 Minimax rates of testing over l p -bodies with 0 < p < 2 Ellipsoids contain essentially smooth functions.In the particular case where θ corresponds to the Fourier coefficients of a given function f , the constraints expressed through the belonging to one of the spaces introduced above may be incompatible with the presence of discontinuities.In order to extend the covered cases, we consider in this subsection sequences θ belonging to l p -bodies E a,p (R) defined as where a = (a k ) k∈J denotes a monotone non-decreasing sequence and 0 < p < 2. The following theorem proposes a lower bound for the minimax rate of testing over such spaces.
Theorem 3 Let (Y j ) j∈J obey to Model (1.1).Let α, β be fixed and denote by ρ 2 (E a,p (R), α, β) the minimax rate of testing over E a,p (R) with respect to the l 2 norm.For all D ∈ J and for all 0 ≤ l ≤ D − ⌈ √ D⌉, we set where Σ 2 l,⌈ √ D⌉ is given in (2.5).Then where To the end of this section, we assume that the sequence (b j ) j∈J is polynomially or exponentially increasing, which yet correspond to the main case of interest in inverse problems.The lower bounds in this particular setting are easier to handle, as proved in the following corollary.
Corollary 4 Let (Z j ) j∈N ⋆ obey to the Model (1.3) with b j = σσ −1 j for all j ∈ N ⋆ .Then, assuming that for all j ∈ N ⋆ , σ j = j γ or that for all j ∈ N ⋆ , σ j = exp(γj) for some γ ≥ 0, we obtain

6).
In order to attain the lower bound presented above, a test similar to the one introduced in Proposition 2 is not sufficient.On l p -bodies, the bias after a given rank D is indeed more difficult to control than for ellipsoids.Some significant coefficients (in a sense which will be precised in the proof) may be contained in the sequence θ after the rank D. Hence, we have to introduce specific tests in order to detect these coefficients.More precisely, for all j ∈ J and α ∈ (0, 1), introduce where q j,α denotes the 1 − α quantile of a Gaussian random variable with mean 0 and variance σ 2 j .Then define where Φ D † ,α/2 denotes the test constructed in Proposition 2 and By convention, D † = N if J = {1, . . ., N } and the set in (4.12) is empty.The following theorem emphasizes the performances of the test Φ † α .
Our test reaches the lower bound established in Proposition 3 up to a log term, it is not sharp.Nevertheless this drawback is not characteristic of the heteroscedastic model since a similar property occurs in the homoscedastic case: see [2] for more details.Hence, the lower bound established in Proposition 3 corresponds certainly to the minimax rate on E a,p (R).
For the sake of convenience, the upper bound is presented for J = {1, . . ., N } which, roughly speaking, corresponds to the regression setting.Nevertheless, our result can be easily extended to the case where J = N ⋆ .In such a situation, our test will be performed on {1, . . ., Ñ }, where Ñ is a trade-off between the bias after the rank Ñ on E a,p (R) and the growth of log(N ).A good candidate for Ñ is a power of σ −2 .
In order to conclude this discussion, we point out that we impose a condition on the sequence a −p j b . This condition is necessary in order to control the bias after the rank D † .It always hold when p = 2 since (a j ) j∈N is an increasing sequence.When p < 2, the considered function has to be sufficiently smooth with respect to the ill-posedness of the problem.A similar condition can be found for instance in [8].
The corollary below deals with the particular case of mildly ill-posed problems with polynomial l p -bodies, where the situation is easier to handle.
Corollary 5 Assume that a k ∼ k s and b k ∼ k −t for all k ∈ N * where s, t denote positive constants such that s > t(2/p − 1).Then where C 1 , C 2 denote positive constant independent of σ.
Remark that the sequence a −p j b −(2−p) j j∈N * is monotone non-increasing as soon as s > t(2/p − 1).Hence the conditions of Proposition 4 are satisfied.The proof follows the same argument as in Corollary 3.

Proof of the lower bounds
The proofs of the lower bounds use a Bayesian approach extending the methods developed in the papers by Ingster [11] and by Baraud [2].We use the following lemma : Lemma 1 Let F be some subset of l 2 (J).Let µ ρ be some probability measure on Assuming that P µρ is absolutely continuous with respect to P 0 , we define For all α > 0, This implies that ρ(F, α, β) ≥ ρ * .
For the proof of this lemma, we refer to Baraud [2], Section 7.1.

Proof of Theorem 1
Without loss of generality, we can assume that the sequence (σ j ) j∈J is non decreasing (if this is not the case, we can reorder the observations Y j ).We fix some l ∈ {0, 1, . . ., n − k}.Let M k,l,n denote the set of all subsets of {l + 1, . . ., n} with cardinality k.Let m be a random set of {l + 1, . . ., n}, which is uniformly distributed on M k,l,n .This means that for all m ∈ M k,l,n , We set Note that θ = (θ j ) j∈J ∈ S k,n and that, since (σ j ) j∈J is non decreasing, We use (5.13) together with E(cosh(λZ)) = exp(λ 2 /2) for a standard Gaussian variable Z. Since Y j /σ j is a standard normal variable, we obtain that .
Since, for all m, m ′ ∈ M k,l,n , we obtain The end of the proof is similar to the proof of Theorem 1 in Baraud [2], similar arguments are also given in Fromont et al. [9].Let us recall these arguments.
, where m, m′ are independent random subsets with uniform distribution on M k,l,n .For fixed m, | m ∩ m′ | is an hypergeometric variable with parameters (n − l, k, k/(n − l)).We know from Aldous [1] that there exists a binomial variable B with parameters (k, k/(n − l)) and a σ− algebra B such that .
Since B is a binomial variable with parameters (k, k/(n − l)), 2 , and A = n−l k 2 ln(c).Since the function cosh is increasing on R + , we obtain that if We finally obtain that By Lemma 1, this implies that Since this result holds for all l ∈ {0, n − k}, we get In order to prove that , we define, as in the proof of Proposition 1, where (ω j , n − k + 1 ≤ j ≤ n) are i.i.d.Rademacher random variables.Note that (θ j ) j∈J ∈ S k,n and that θ 2 2 = ρ 2 .We now conclude as in the proof of Proposition 1, using that C(α, β) = 2 ln(c) ≥ 1.

Proof of Proposition 2
In order to prove Proposition 2, we have to show that for all θ ∈ S D such that θ 2 2 ≥ C(α, β) We denote by t D,β (θ, σ) the β quantile of D j=1 Y 2 j , when Y = (Y j ) j∈J obeys to Model (1.1).In order to prove (5.15), it suffices to show that To prove this inequality, we will first give an upper bound for t D,1−α (σ) and then a lower bound for t D,β (θ, σ).Upper bound for t D,1−α (σ) : We use an exponential inequality for chi-square distributions due to Laurent and Massart [15] (see Lemma 1).It follows from this inequality that for all x ≥ 0, Setting x α = ln(1/α), we obtain that Lower bound for t D,β (θ, σ) : We prove the following lemma, which generalizes the results obtained by Birgé [4] to the heteroscedastic framework : where ǫ 1 , . . .ǫ D are i.i.d.Gaussian variables with mean 0 and variance 1.
The following inequalities hold for all x ≥ 0 : (5.17) The proof of this lemma is given in the Appendix.Inequality (5.18) provides a lower bound for t D,β (θ, σ).Indeed, setting x β = log(1/β), we obtain that Let us note that Hence, the following inequality implies (5.19) : This inequality holds if where C is an absolute constant (which can be taken equal to 8).Hence, we have proved that which concludes the proof of Proposition 2.

Proof of Theorem 2
The test Φ α is obviously of level α thanks to Bonferroni's inequality : α/2 = 1) + P 0 (Φ Let us now evaluate the power of the test.
We obtained in the proof of Theorem 2 that if (5.20) holds, then P θ (Φ α = 0) < β.Since q n,1−α ≤ 2 ln(n/α), we obtain that there exists a constant 5.3 Proof of minimax rates on ellipsoids and l p -bodies

Proof of Proposition 3
We first prove the lower bound.For all D ∈ J, introduce r 2 where the infinimum is taken over all possible level-α testing procedures.Since inequality (5.21) holds for all D ∈ J, we obtain ρ 2 (E a,2 (R), α, β) ≥ sup D∈J (ρ 2 D ∧ R 2 a −2 D ).Concerning the upper bound, we know from Proposition 2 that the test Φ α is powerful as soon as: where C(α, β) denotes a positive constant.Since θ ∈ E a,2 (R), we get where C = C(α, β).This concludes the proof since the previous result holds for all D ∈ J.

Proof of Corollary 3
First case: , we can remark that ρ 2 D and R 2 a −2 D are of the same order, hence leading to the desired rate.
Second case: a k ∼ e νk s and b k ∼ k −t .Set where C denotes a constant independent of σ.Concerning the lower bound, we set for some C > 0.
Third case: a k ∼ k s and b k ∼ e −γk r .Set for some C > 0. Concerning the lower bound, we set for some C > 0.
Fourth case: a k ∼ e νks and b k ∼ e −γk r .Denote by D the solution of the equation Remark that In order to conclude the proof, we have to prove that the lower and upper bounds coincide.To this end, remark that for some constant C as soon as s ≤ 1.

Proof of Theorem 3
The proof will use the one of Theorem 1.We assume that (σ j ) j∈J is non-decreasing.Let us first establish a relation between the l p ball E a,p (R) and the sets S k,n .For all D ∈ J, for all θ ∈ S √ R). Indeed, using Hölder's inequality We set k = ⌈ √ D⌉, n = D and for all l ∈ {0, 1, . . ., n − k}, we define θ = (θ j , j ∈ J) by (5.14).As pointed out in the proof of Theorem 1, θ ∈ S k,n and θ 2 2 ≥ ρ 2 .We also have Moreover, in the proof of Theorem 1, we proved that if This implies by Lemma 1 that ρ 2 2 (E a,p (R)) ≥ ρ 2 .We finally get Since the result holds for all l ∈ {0, 1, . . ., n − k}, we obtain that ρ 2 2 (E a,p (R)) ≥ ρ 1 (D).To obtain that ρ 2 2 (E a,p (R)) ≥ ρ 2 (D), we consider, as in the proof of Theorem 1, for k = ⌈ √ D⌉ and n = D for all D ∈ J, the result follows.

Proof of Corollary 4
For the sake of simplicity, we assume that √ D is an integer.We derive from Comment 2. of Theorem 1 that when σ j = j γ , then Moreover, When σ j = exp γj , we have from Theorem 3 that We conclude by noticing that, in this case, we also have that ρ

Proof of Proposition 4
It follows from Bonferonis's inequality that Φ † α is a level-α test.Introduce In a first time, we suppose that A is empty.From the definition of D † , we get D † = N and for some positive constant C. Since A is empty and using the comments following Theorem 1 Hence, our test is powerful as soon as θ 2 ≥ ρ 2 a,p,R .
From now on, we assume that the set A is not empty: D † ≤ N .For all j ∈ J, set Two different situations may occur: 1/ For all j > D † , b 2 j θ 2 j ≤ σ 2 µ 2 j for some sequence µ j , i.e. all the coefficients θ k have poor importance after the rank D † .
2/ There exists at least j ∈ {D † , . . ., N } such that b 2 j θ 2 j ≥ σ 2 µ 2 j , i.e. there exist significant coefficients after the rank D † .First consider the case 2/.Recall that in this case, the set A is not empty and there exists j In this particular setting, we have to use the threshold test in order to detect these coefficients.More precisely, Thanks to inequality (29) of [2], we know that this probability is smaller than β as soon as: This is exactly the assumption made in case 2/.Now, we consider point 1/.Let j > D † , Then, we get ) j∈N is assumed to be monotone non increasing, we can control the bias as follows In order to conclude the proof, we have to bound the right hand side of the above inequality.First assume that the problem is mildly ill-posed, i.e. (b k ) k∈N ⋆ ∼ (k −t ) k∈N for some t > 0. Then An inequality similar to (5.23) holds, which concludes the second point.

Appendix
Proof of Lemma 2 : We first compute the Laplace transform of T .Easy computations show that for t < 1/(2σ 2 j ), E exp(t(θ j + σ j ǫ j ) 2 ) = 1 This implies that for t < min 1≤j≤D 1/(2σ  We use the following inequality which holds for x < 1/2 : This inequality implies that for all t < min 1≤j≤D 1/2σ . We now use the following lemma which is proved in Birgé [4] (see Lemma 8. We now use (6.24) for x = tσ 2 j with t < 0. We obtain Moreover, E e tx( T −E( T )+2 We use the following inequality which holds for all u ≥ 0 : and we apply this inequality to u = −t x σ 2 j .We obtain that for all x ≥ 0, This concludes the proof of Lemma 2.