Clément Marteau
email: clement.marteau@math.univ-toulouse.fr

Theofanis Sapatinas

Minimax Goodness-of-Fit Testing in Ill-Posed Inverse Problems with Partially Unknown Operators

Keywords: AMS 2000 subject classifications: 62G05, 62K20 Ellipsoids, compact operators, Gaussian sequence model, Gaussian white noise model, ill-posed inverse problems, minimax goodness-of-fit testing, minimax signal detection, singular value decomposition

We consider a Gaussian sequence model that contains ill-posed inverse problems as special cases. We assume that the associated operator is partially unknown in the sense that its singular functions are known and the corresponding singular values are unknown but observed with Gaussian noise. For the considered model, we study the minimax goodness-of-fit testing problem. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we obtain lower and upper bounds for the minimax separation radius in the non-asymptotic framework, i.e., for fixed values of the involved noise levels. Examples of mildly and severely ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth sequences are examined in detail and minimax rates of goodness-of-fit testing are obtained for illustrative purposes.

Introduction

We consider the following Gaussian sequence model (GSM),

Y j = b j θ j + ε ξ j , j ∈ N, X j = b j + σ η j , j ∈ N, (1.1)
where N = {1, 2, . . .} is the set of natural numbers, b = {b j } j∈N > 0 is an unknown sequence, θ = {θ j } j∈N ∈ l 2 (N) is the unknown signal of interest, ξ = {ξ j } j∈N and η = {η j } j∈N are sequences of independent standard Gaussian random variables (and independent of each other), and ε, σ > 0 are known parameters (the noise levels). The observations are given by the sequence (Y, X) = {(Y j , X j )} j∈N from the GSM (1.1) and their joint law is denoted by P θ,b . Here, l 2 (N) denotes the space of squared-summable sequence of real numbers, i.e., l 2 (N) = {θ ∈ R N : θ 2 := j∈N θ 2 j < +∞}.

The GSM (1.1) arises in the case of ill-posed inverse problems with noisy operators. Indeed, consider the Gaussian white noise model (GWNM)

dY ε (t) = Af (t)dt + ε dW (t), t ∈ V, (1.2)
where A is a linear bounded operator acting on a Hilbert space H 1 with values on another Hilbert space H 2 , f (•) ∈ H 1 is the unknown response function that one wants to detect or estimate, W (•) is a standard Wiener process on V ⊆ R, and ε > 0 is a known parameter (the noise level).

For the sake of simplicity, we only consider the case when A is injective (meaning that A has a trivial nullspace) and assume that V = [0, 1], H 1 = L 2 ([0, 1]), U ⊆ R and H 2 = L 2 (U ). In most cases of interest, A is a compact operator (see, e.g., Chapter 2 of [START_REF] Engl | Regularization of inverse problems[END_REF]). In particular, it admits a singular value decomposition (SVD) (b j , ψ j , ϕ j ) j∈N , in the sense that

Aϕ j = b j ψ j , A ⋆ ψ j = b j ϕ j , j ∈ N, (1.3)
where A ⋆ denotes the adjoint operator of A -here (b 2 j ) j∈N and (ϕ j ) j∈N are, respectively, the eigenvalues and the eigenfunctions of A ⋆ A. Thus, the (first equation in) GSM (1.1) arises where for all j ∈ N Y j = 1 0 ψ j (t)dY ε (t), θ j = 1 0 ϕ j (t)f (t)dt, ξ j = 1 0 ψ j (t)dW (t), j ∈ N, and b 2 j > 0 (since A is injective). In this case, the GWNM (1.2) corresponds to a so-called ill-posed inverse problem since the inversion of A * A is not bounded. Possible examples of such decompositions arise with, e.g., convolution or Radon-transform operators, see, e.g., [START_REF] Engl | Regularization of inverse problems[END_REF]. The effect of the ill-posedness of the model is clearly seen in the decay of the singular values b j as j → +∞. As j → +∞, b j θ j gets weaker and is then more difficult to perform inference on the sequence θ = {θ j } j∈N .

In the early literature, the compact operator A (and, hence, its sequence b = {b j } j∈N of singular values) was supposed to be fully known. (Note that, in this case, the second equation in the GSM (1.1) does not appear.) We refer, e.g., to [START_REF] Cavalier | Inverse problems in statistics[END_REF], [START_REF] Cavalier | Risk hull method and regularization by projections of ill-posed inverse problems[END_REF], [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF], [START_REF] Cavalier | Sharp adaptation for inverse problems with random noise[END_REF], [START_REF] Cavalier | Penalized blockwise stein's method, monotone oracles and sharp adaptative estimation[END_REF] (minimax estimation) and to [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF], [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF] (minimax signal detection/minimax goodness-of-fit testing). Therein, minimax rates/oracle inequalities (estimation) and minimax separation radius/minimax separation rates (signal detection or goodness-of-fit testing) were established, amongst other investigations, for ill-posed inverse problems with smoothness conditions on the sequence of interest.

The case of an unknown compact operator A that is observed with Gaussian noise has also been recently treated in the estimation literature, especially the situation where A is partially unknown, see, e.g., [START_REF] Cavalier | Adaptive estimation for inverse problems with noisy operators[END_REF], [START_REF] Delattre | Blockwise SVD with error in the operator and application to blind deconvolution[END_REF], [START_REF] Johannes | Adaptive Gaussian inverse regression with partially unknown operator[END_REF]. In these contributions, it is assumed for the corresponding SVD (1.3) that

• the sequence of singular functions (ψ, ϕ) = (ψ j , ϕ j ) j∈N is known,

• the sequence of singular values b = {b j } j∈N is unknown but observed with some Gaussian noise.

In other words, the following sequence model is considered

X j = b j + σ η j , j ∈ N,
where η = {η j } j∈N is a sequence of independent standard Gaussian random variables (and independent of the standard Gaussian sequence ξ = {ξ j } j∈N ), and σ > 0 is a known parameter (the noise level). Therefore, the second equation in the GSM (1.1) is also readily available.

To practically motivate the GSM (1.1), consider the following deconvolution model (see also [START_REF] Cavalier | Adaptive estimation for inverse problems with noisy operators[END_REF] for a complete discussion on this subject)

dY ε (t) = g ⋆ f (t) + ε dW (t), t ∈ [0, 1], (1.4)
where

g ⋆ f (t) = 1 0 g(t -x)f (x)dx, t ∈ [0, 1],
is the convolution between g(•) and f (•), g(•) is an unknown 1-periodic (convolution) kernel in L 2 ([0, 1]), f (•) is an unknown 1-periodic signal in L 2 ([0, 1]), dY ε (•) is observed, W (•) is a standard Wiener process, and ε > 0 is the noise level. Let φ j (•), j ∈ N, be the usual real trigonometric basis on V . The model (1.4) is equivalent to the (first equation in the) GSM (1.1) by a projection on the trigonometric basis φ j (•), j ∈ N. In the case where the kernel g(•) is unknown (i.e., the sequence (b k ) k∈N = ( g, φ k ) k∈N is unknown), suppose that we can pass the trigonometric basis φ j (•), j ∈ N, through the convolution kernel, i.e., to send each φ j (•), j ∈ N, as an input function f (•) and observe the corresponding dY ε,j (•), j ∈ N. In other words,we are able to obtain training data for the estimation of the unknown convolution kernel g(•) in this setting. In particular, we obtain exactly the two sequences of observations Y j and X j , j ∈ N, in the GSM (1.1). In this case, the corresponding noise levels coincide, i.e., ε = σ.

To the best of our knowledge, there is no research work on minimax goodness-of-fit testing in ill-posed inverse problems with partially unknown operators. Our aim is to fill this gap. In particular, considering model (1.1) and working with certain ellipsoids in the space of squaredsummable sequences of real numbers, with a ball of positive radius removed, we obtain lower and upper bounds for the minimax separation radius in the non-asymptotic framework, i.e., for fixed values of ε and σ. Examples of mildly and severely ill-posed inverse problems with ellipsoids of ordinary-smooth and super-smooth sequences are examined in detail and minimax rates of goodness-of-fit testing are obtained for illustrative purposes.

The paper is organized as follows. Section 2 presents the considered statistical setting and a brief overview of the main results. Section 3 is devoted to the construction of the suggested testing procedure. A general upper bound on the maximal second kind error is then displayed and special benchmark examples are presented for illustrative purposes. The corresponding lower bounds are proposed in Section 4. Some concluding remarks and open questions are discussed in Section 5. Finally, all proofs and technical arguments are gathered in Section 6.

Throughout the paper we set the following notations. For all x, y ∈ R, δ x (y) = 1 if x = y and δ x (y) = 0 if x = y. Also, x ∧ y := min{x, y} and x ∨ y := max{x, y}. Given two sequences (c j ) j∈N and (d j ) j∈N of real numbers, c j ∼ d j means that there exists 0 < κ 0 ≤ κ 1 < ∞ such that κ 0 ≤ c j /d j ≤ κ 1 for all j ∈ N. Let ν be either ε or σ or (ε, σ), and let V be either R + := (0, +∞) or R + × R + . Given two collections (c ν ) ν∈V and (d ν ) ν∈V of positive real numbers, c ν d ν means that there exists 0 < κ 0 < +∞ such that c ν ≥ κ 0 d ν for all ν ∈ V. Similarly, c ν d ν means that there exists 0 < κ 1 < +∞ such that c ν ≤ κ 1 d ν for all ν ∈ V.

2 Minimax Goodness-of-Fit Testing

The Statistical Setting

Given observations (Y, X) = {(Y j , X j )} j∈N from the GSM (1.1), the aim is to compare the underlying (unknown) signal θ ∈ l 2 (N) to a (known) benchmark signal θ 0 , i.e., to test

H 0 : θ = θ 0 versus H 1 : θ -θ 0 ∈ F, (2.1)
for some given θ 0 and a given subspace F. The statistical setting (2.1) is known as goodness-of-fit testing when θ 0 = 0 or signal detection when θ 0 = 0.

Remark 2.1 Given observations from the GWNM (1.2), the testing problem (2.1) is equivalent to

H 0 : f = f 0 versus H 1 : f -f 0 ∈ F ,
for a given benchmark function f 0 and a given subspace F . In most cases, F contains functions f ∈ L 2 ([0, 1]) that admit a Fourier series expansion with Fourier coefficients θ belonging to F (see, e.g., [START_REF] Yu | Nonparametric Goodness-of-Fit Testing Under Gaussian Models[END_REF], Section 3.2.).

The choice of the set F is important. Indeed, it should be rich enough in order to contain the true θ. At the same time, if it is too rich, it will not be possible to control the performances of a given test due to the complexity of the problem. The common approach for such problems is to impose both a regularity condition (which characterizes the smoothness of the underlying signal) and an energy condition (which measures the amount of the underlying signal).

Concerning the regularity condition, we will work with certain ellipsoids in l 2 (N). In particular, we assume that θ ∈ E a (R), the set E a (R) being defined as

E a (R) =    θ ∈ l 2 (N), j∈N a 2 j θ 2 j ≤ R    , (2.2)
where a = (a j ) j∈N denotes a non-decreasing sequence of positive real numbers with a j → +∞ as j → +∞, and R > 0 is a constant. The set E a (R) can be seen as a condition on the decay of θ. The cases where a increases very fast correspond to θ with a small amount of non-zero coefficients. In such a case, the corresponding signal can be considered as being 'smooth'. Without loss of generality, in what follows, we set R = 1, and write E a instead of E a (1).

Regarding the energy condition, it will be measured in the l 2 (N)-norm. In particular, given r ε,σ > 0 (called the radius), which is allowed to depend on the noise levels ε, σ > 0, we will consider θ ∈ E a such that θ > r ε,σ . Given a smoothness sequence a and a radius r ε,σ > 0, the set F can thus be defined as

F := Θ a (r ε,σ ) = {θ ∈ E a , θ ≥ r ε,σ } . (2.3)
In other words, the set F is an ellipsoid in l 2 (N) with a ball of radius r ε,σ > 0 removed. In many cases of interest, the set F provides constraints on the Fourier coefficients of f ∈ L 2 ([0, 1]) in the model (1.2) (see, e.g., [START_REF] Yu | Nonparametric Goodness-of-Fit Testing Under Gaussian Models[END_REF], Section 3.2).

We consider below the hypothesis testing setting (2.1) with θ 0 = 0 (i.e., goodness-of-fit testing). Formally, given observations from the GSM (1.1), for any given θ 0 = 0, we will be dealing with the following goodness-of-fit testing problem

H 0 : θ = θ 0 versus H 1 : θ 0 ∈ E a , θ -θ 0 ∈ Θ a (r ε,σ ), (2.4)
where Θ a (r ε,σ ) is defined in (2.3). The sequence a being fixed, the main issue for the problem (2.4) is then to characterize the values of r ε,σ > 0 for which both hypotheses H 0 (called the null hypothesis) and H 1 (called the alternative hypothesis) are 'separable' (in a sense which will be made precise later on).

Remark 2.2 We would like to stress that in the standard GSM (i.e., (1.1) with σ = 0), signal detection (i.e., θ 0 = 0) and goodness-of-fit testing (i.e., θ 0 = 0) problems are equivalent as soon as the involved operator is injective. Indeed, without loss of generality, we can still replace the observed sequence (Y j ) j∈N by ( Ỹj ) j∈N := (Y j -b j θ 0,j ) j∈N . This is no more the case in the GSM (1.1) since the sequence (b j ) j∈N is unknown. Signal detection and goodness-of-fit problems should therefore be treated in a different manner. In this work, we only address the goodness-of-fit testing problem (2.4).

In the following, a (non-randomized) test Ψ := Ψ(Y, X) will be defined as a measurable function of the observation (Y, X) = (Y j , X j ) j∈N from GSM (1.1) having values in the set {0, 1}. By convention, H 0 is rejected if Ψ = 1 and H 0 is not rejected if Ψ = 0. Then, given a test Ψ, we can investigate

• the first kind error probability defined as

α ε,σ (Ψ) := P θ 0 ,b (Ψ = 1), (2.5)
which measures the probability to reject H 0 when H 0 is true (i.e., θ = θ 0 ); it is often constrained as being bounded by a prescribed level α ∈]0, 1[, and

• the maximal second kind error probability defined as

β ε,σ (Θ a (r ε,σ ), Ψ) := sup θ 0 ∈Ea, θ-θ 0 ∈Θa(rε,σ) P θ,b (Ψ = 0), (2.6)
which measures the worst possible probability not to reject H 0 when H 0 is not true (i.e., when θ 0 ∈ E a and θ -θ 0 ∈ Θ a (r ε,σ )); one would like to ensure that it is bounded by a prescribed level β ∈]0, 1[.

For simplicity in our exposition, we will restrict ourselves to α-level tests, i.e., tests Ψ α satisfying α ε,σ (Ψ α ) ≤ α, for any fixed value α ∈]0, 1[. Let α, β ∈]0, 1[ be given, and let Ψ α be an α-level test.

Definition 2.1

The separation radius of the α-level test Ψ α over the class E a is defined as

r ε,σ (E a , Ψ α , β) := inf r ε,σ > 0 : β ε,σ (Θ a (r ε,σ ), Ψ α ) ≤ β ,
where the maximal second kind error probability

β ε,σ (Θ a (r ε,σ ), Ψ α ) is defined in (2.6).
In some sense, the separation radius r ε,σ (E a , Ψ α , β) corresponds to the smallest possible value of the available signal θ -θ 0 for which H 0 and H 1 can be 'separated' by the α-level test Ψ α with maximal second kind error probability, bounded by a prescribed level β ∈]0, 1[.

Definition 2.2

The minimax separation radius rε,σ := rε,σ (E a , α, β) > 0 over the class E a is defined as rε,σ := inf

Ψα: αε,σ( Ψα)≤α r ε,σ (E a , Ψα , β). (2.7)
The minimax separation radius rε,σ corresponds to the smallest radius r ε,σ > 0 such that there exists some α-level test Ψα for which the maximal second kind error probability β ε,σ (Θ a (r ε,σ ), Ψα ) is not greater than β.

Summary of the Results

Our aim is to establish 'optimal' separation conditions for the goodness-of-fit testing problem (2.4). This task requires, in particular, precise (non-asymptotic) controls of the first kind error probability α ε,σ (Ψ α ) and the maximal second kind error probability β ε,σ (Θ a (r ε,σ ), Ψ α ) (of a specific test Ψ α that will be made precise in Section 3) by prescribed levels α, β ∈]0, 1[, respectively. Such controls allow us to derive both upper and lower bounds on the minimax separation radius rε,σ , as summarized in the following theorem.

Theorem 2.1 Let α, β ∈]0, 1[ be fixed, such that α ≤ β. Consider the goodness-of-fit testing problem (2.4). Then, there exist explicit positive constants1 C(α, β) > 0, C α,β > 0, c α,β > 0 and σ 0 ∈]0, 1[ such that, for all 0 < σ ≤ σ 0 and for each ε > 0,

(i) r2 ε,σ ≤ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + 7 + 4 ln(2/α) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   ,
and, for all ε, σ > 0,

(ii) r2 ε,σ ≥ C 2 α,β 16 σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∨    sup D∈N   c α,β ε 2 D j=1 b -4 j ∧ a -2 D      ,
where the bandwidths M 0 , M 1 and M 2 depend 1 on both (b j ) j∈N and σ.

Theorem 2.1 provides a precise description on the behavior of the minimax separation radius rǫ,σ in terms of the sequences (a j ) j∈N and (b j ) j∈N and of the noise levels ǫ and σ. It is worth pointing out that this control is non-asymptotic. There is indeed a technical constraint on the value of σ (0 < σ ≤ σ 0 , σ 0 ∈]0, 1[), but we do not assume its convergence towards 0, i.e., we work with fixed values of the noise levels ε and σ.

Then, we apply the above result on specific problems. Namely, we consider various behaviors for both sequences (a j ) j∈N and (b j ) j∈N , and discuss the properties of the associated minimax separation radii rε,σ . Concerning the eigenvalues (b 2 j ) j∈N of the operator A * A, we will alternatively consider situations where b j ∼ j -t or b j ∼ exp(-jt), ∀j ∈ N, for some t > 0.

The first case corresponds to the so-called mildly ill-posed problems while the second one corresponds to severely ill-posed problems. Concerning the ellipsoids E a , i.e., the sequence (a j ) j∈N , two different kinds of smoothness will be investigated, namely, a j ∼ j s or a j ∼ exp(js), ∀j ∈ N, for some s > 0, the so-called ordinary-smooth and super-smooth cases, respectively. In the above scenarios, we apply Theorem 2.1 and describe the associated upper and lower bounds on the minimax separation radius rε,σ . They are, respectively, displayed in Table 2.1 and Table 2.2.

Looking at these tables, both lower and upper bounds coincide in every considered case, up to a logarithm term that depends on the noise level σ. Hence, Theorem 2.1 provides a sharp Goodness-of-Fit ordinary-smooth super-smooth Testing Problem a j ∼ j s a j ∼ exp{js} mildly ill-posed

ε 4s/(2s+2t+1/2) ∨ [σ ln 3/4 (1/σ)] 2[(s/t)∧1] ε 2 (ln(1/ε)) 2t+1/2 ∨ σ 2 ln 3/2 (1/σ) b j ∼ j -t severely ill-posed (ln(1/ε)) -2s ∨ [ln(1/σ ln -1/2 (1/σ))] -2s ε 2s/(s+t) ∨ [σ ln 1/2 (1/σ)] 2[(s/t)∧1]
b j ∼ exp{-jt} Table 2.1: Minimax goodness-of-fit testing with unknown singular values: upper bounds on the minimax separation radius r2 ε,σ for 0 < ε ≤ ε 0 , ε 0 ∈]0, 1[, and 0 < σ ≤ σ 0 , σ 0 ∈]0, 1[, for all t, s > 0.

Goodness-of-Fit ordinary smooth super smooth Testing Problem a j ∼ j s a j ∼ exp{js} mildly ill-posed

ε 4s/(2s+2t+1/2) ∨ σ 2[(s/t)∧1] ε 2 (ln ε -1 ) 2t+1/2 ∨ σ 2 b j ∼ j -t severely ill-posed (ln(1/ε)) -2s ∨ (ln(1/σ)) -2s ε 2s/(s+t) ∨ σ 2[(s/t)∧1]
b j ∼ exp{-jt} Table 2.2: Minimax goodness-of-fit testing with unknown singular values: lower bounds on the minimax separation radius r2 ε,σ for 0 < ε ≤ ε 0 , ε 0 ∈]0, 1[, and 0 < σ ≤ σ 0 , σ 0 ∈]0, 1[, for all t, s > 0.

control on the minimax separation radius rε,σ in various settings. The interesting property of such minimax separation radii is that they have the same structure whatever the considered situation: a maximum between two terms depending, respectively, on the noise levels ǫ and σ. It is also worth pointing out that the first term depending on ǫ corresponds to the minimax separation radius in the case where the operator is known (i.e., σ = 0), as displayed in Table 2.3.

Goodness-of-Fit ordinary-smooth super-smooth

Testing Problem a j ∼ j s a j ∼ exp{js} mildly ill-posed ε 4s/(2s+2t+1/2) ε 2 (ln ε -1 ) 2t+1/2 b j ∼ j -t severely ill-posed (ln ε -1 ) -2s ε 2s/(s+t) b j ∼ exp{-jt} Table 2.3: Minimax goodness-of-fit testing with known singular values: the separation rates r2

ε for 0 < ε ≤ ε 0 , ε 0 ∈]0, 1[, for all t, s > 0.
The results displayed in Theorem 2.1 and Tables 2.1, 2.2 can also be understood as follows. Two problems are at hand: detection of the underlying signal (with a minimax separation radius that only depends on ǫ) and detection of the 'frequencies' j for which the terms b j can be replaced by observations X j without loss of precision (with a minimax separation radius that depends only on σ). The final minimax separation radius is then the maximum of these two terms, i.e., the signal detection hardness is related to the most difficult underlying problem. We stress that such phenomenon has already been discussed in the minimax estimation framework, see e.g., [START_REF] Delattre | Blockwise SVD with error in the operator and application to blind deconvolution[END_REF], [START_REF] Johannes | Adaptive Gaussian inverse regression with partially unknown operator[END_REF].

Upper Bound on the Minimax Separation Radius

In this section, we first propose an α-level testing procedure. Then, we investigate its maximal second kind error probability and establish a non-asymptotic upper bound on the minimax separation radius (which corresponds to item (i) of Theorem 2.1). Finally, in Section 3.3, we provide a control of the upper bounds for minimax separation radii for the specific cases displayed in Table 2.1.

The Spectral Cut-Off Test

For a given θ 0 = 0, the aim of the goodness-of-fit testing problem (2.4) is to determine whether or not θ = θ 0 . In particular, for any given j ∈ N, one would like to infer the corresponding value θ j from the observation (Y, X) = (Y j , X j ) j∈N from GSM (1.1). Typically, for any given j ∈ N, one may use the 'naive' estimate θj of θ j , defined by θj :=

Y j X j = b j X j θ j + ε 1 X j ξ j , j ∈ N.
In order to ensure a 'good' approximation of θ j by θj (in a sense which will be made precise later on), a precise control of the ratio b j /X j is required. To this end, we want to avoid coefficients for which X j σ, namely for which the observation X j is of the order of the corresponding noise level σ, that does not have 'discriminatory' power . Therefore, we will restrict ourselves to coefficients X j with indices 1 ≤ j ≤ M , where the bandwidth M is defined by

M := inf{j ∈ N : |X j | ≤ σh j } -1, (3.1)
where, for all j ∈ N,

h j = 16 ln κj 2 α + 2 ln 10 α , (3.2)
for some κ > exp(1).

Remark 3.1 The value of κ is, in some sense, related to the value of the first kind error probability of the suggested testing procedure. We will see below that the value κ = 5(3π 2 +12)/6 is convenient to our purpose. We stress that κ is not a regularization parameter : an 'optimal' value of κ only allows to get 'optimal' constants in the final results but will not change the order of the corresponding minimax separation rates. Finding optimal constants is outside the scope of this work.

The bandwidth M is a random variable but can be controlled in the sense that M ∈ [M 0 , M 1 [ with high probability (see Lemma 6.1 for precise computations and Figure 3.1 for a graphical illustration), where the bandwidths M 0 and M 1 are defined by

M 0 := inf{j ∈ N : b j ≤ σh 0,j } -1, M 1 := inf{j ∈ N : b j ≤ σh 1,j }, (3.3)
and the sequences

h 0 = (h 0,j ) j∈N , h 1 = (h 1,j ) j∈N satisfy h 0,j = 18 ln κj 2 α + 2 ln 10 α , (3.4
) The decreasing solid curve corresponds to the values of the sequence b = (b j ) j∈N with respect to the index j ∈ N, while the oscillating curve demonstrates one realization of the random sequence X = (X j ) j∈N according to the GSM (1.1). The increasing dashed curve draws the behavior of the sequence σh j . For the corresponding random 'bandwidth' M defined in (3.3)

h 1,j = 16 ln κj 2 α , (3.5)
j 0 b j M 0 σh 0,M 0 M 1 σh 1,M 1
, Lemma 6.1 shows that M ∈ [M 0 , M 1 [ with high probability.
for all j ∈ N. The sequences h = (h j ) j∈N , h 0 = (h 0,j ) j∈N and h 1 = (h 1,j ) j∈N in the definition of M 0 , M 1 and M allow a 'uniform' control of the standard Gaussian sequence η = (η j ) j∈N (associated with X = (X j ) j∈N ), for all 1 ≤ j ≤ M 1 (see Lemmas 6.1, 6.2 and 6.3 in Section 6).

We are now in the position to construct a (spectral cut-off) testing procedure. According to the methodology proposed earlier in the literature (see e.g. [START_REF] Baraud | Non-asymptotic minimax rates of testing in signal detection[END_REF], [START_REF] Yu | Nonparametric Goodness-of-Fit Testing Under Gaussian Models[END_REF] or [START_REF] Marteau | A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection[END_REF]), our test will be based on an estimation of θ -θ 0 2 . For any fixed D ∈ N, consider the test statistic

T D,M := D∧M j=1 Y j X j -θ j,0 2 . ( 3.6)
Given a prescribed level α ∈]0, 1[ for the kind error probability, the associated spectral cut-off test is then defined as

Ψ D,M := 1{T D,M > t 1-α,D (X)}, (3.7)
where

t 1-α,D (X) := ε 2 D∧M j=1 X -2 j + C(α)ε 2 D∧M j=1 X -4 j + (1 + x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M , (3.8)
and

C(α) = 3 x α/2 + 2x α/2 > 0, x γ := ln(1/γ) ∀γ ∈]0, 1[. (3.9)
In other words, if the 'estimator' T D,M of θ -θ 0 2 is greater than the fixed threshold t 1-α,D (X), θ and θ 0 are very unlikely to be close to each other, and we will reject H 0 . Remark 3.2 Under H 0 , Y j = b j θ j,0 + εξ j , j ∈ N, and, hence,

T D,M = D∧M j=1 b j X j -1 θ j,0 + εX -1 j ξ j 2 .
Therefore, the law of T D,M is not available and, thus, its corresponding (1 -α)-quantile is not computable in practice, since the sequence b = (b j ) j∈N is unknown. However, Proposition 3.1 below ensures that the threshold t 1-α,D (X) defined in (3.8) provides a computable upper bound on this quantile.

First, we focus on the first kind error probability. The following proposition states that the spectral cut-off test Ψ D,M defined in (3.7)- (3.8), is an α-level test.

Proposition 3.1 Let α ∈]0, 1[ be fixed. Consider the goodness-of-fit testing problem (2.4). Then, setting κ = 5(3π 2 + 12)/6, there exists σ 0 ∈]0, 1[ such that, for all 0 < σ ≤ σ 0 and for each ε > 0, the spectral cut-off test Ψ D,M , defined in (3.7)- (3.8), is an α-level test, i.e.,

α ε,σ (Ψ D,M ) ≤ α. (3.10)
The proof is postponed to Section 6.2.1.

Remark 3.3

In order to shed light on the term σ 0 , we provide bellow a heuristic argument. Note that, under H 0 , thanks to a (rough) Taylor expansion,

T D,M ≃ D∧M j=1 εb -1 j ξ j + σb -1 j θ j,0 η j 2 .
Compared to the 'noise-free' case (i.e., σ = 0), we have in some sense to deal with the additional term σb -1 j θ j,0 η j . Two scenarios are at hand

• If sup j b -1 j a -1 j ≤ C 0 , the expected amount of additional signal is σ 2 D∧M j=1 b -2 j θ 2 j ≤ σ 2 C 0 θ 2 ,
which is of the order of the classical parametric rate σ 2 . However, since C 0 is unknown, we use a rough standard deviation control on this additional term, which requires a logarithmic term (i.e., ln 3/2 (1/σ)) in the right hand side of (3.8). We stress that this logarithmic term can be removed if the knowledge of C 0 is assumed.

• On the other hand, we can prove that σb -1 j η j (see Lemma 6.4) is bounded with controlled probability, according to the construction of the bandwidth M given in (3.1). In such case, the additional term can be controlled by the 'bias' a -2 D∧M .

Due to the additional logarithmic term mentioned above, the first kind error probability can be controlled as soon as σ is small enough (i.e., 0 < σ ≤ σ 0 for some σ 0 ∈]0, 1[). Unsurprisingly, it is impossible to retrieve any kind of information on the observations if the noise level σ is too large.

A Non-Asymptotic Upper Bound

We now turn our attention to the the maximal second error probability. The following proposition provides, for each noise level ε > 0 and for noise level σ small enough, an upper bound for the separation radius

r ε,σ (E a , Ψ D,M , β) of the spectral cut-off test Ψ D,M defined in (3.6)-(3.8). Proposition 3.2 Let α, β ∈]0, 1[ be fixed, such that α ≤ β.
Consider the goodness-of-fit testing problem (2.4). Let Ψ D,M be the spectral cut-off test, defined in (3.7)-(3.8). Then, there exists σ 0 ∈]0, 1[ such that, for all 0 < σ ≤ σ 0 and for each ε > 0,

r 2 ε,σ (E a , Ψ D,M , β) ≤ C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0 , (3.11)
where

C(α, β) = 16(C(α) + 3 x β/2 ). (3.12)
The proof is postponed to Section 6.2.2.

Remark 3.4 According to Proposition 3.2, given a radius r ε,σ > 0, then

r 2 ε,σ ≥ C(α, β)ε 2 D∧M 1 j=1 b -4 j +(7+4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0 ⇒ β ε,σ (Θ a (r ε,σ ), Ψ D,M ) ≤ β,
and, hence,

r2 ε,σ ≤ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   .
Note that the upper bound on the separation radius

r 2 ε,σ (E a , Ψ D,M , β) given in (3.15) depends on two antagonistic terms, namely, ε 2 D∧M 1 j=1 b -4 j and σ 2 ln 3/2 (1/σ) ∨ a -2
D∧M 0 . Ideally, one would like to make this upper bound as small as possible, i.e., to obtain the weakest possible condition on θ -θ 0 such that, for any fixed

β ∈]0, 1[, β ε,σ (Θ a (r ε,σ ), Ψ D,M ) ≤ β. Therefore, one would like to select D := D ⋆ such that D ⋆ := arg min D∈N    C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0    ,
where C(α, β) is defined in (3.12). However, this 'optimal' bandwith D ⋆ is not available in practice since the sequence b = (b j ) j∈N is not assumed to be known. To this end, we use instead the bandwidth D := D † defined as

D † := arg min D∈N    C(α, β)ε 2 D∧M j=1 X -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    , (3.13)
The following theorem illustrates the performances of the corresponding spectral cut-off test Ψ D † ,M , defined in (3.7), with D := D † , defined in (3.13).

Theorem 3.1 Let α, β ∈]0, 1[ be fixed, such that α ≤ β. Consider the goodness-of-fit testing problem (2.4). Let Ψ D † ,M be the spectral cut-off test, defined in (3.7) with D := D † , defined in (3.13). Then, there exists σ 0 ∈]0, 1[ such that, for all 0 < σ ≤ σ 0 and for each ε > 0,

α ε,σ (Ψ D † ,M ) ≤ α (3.14)
and

r 2 ε,σ (E a , Ψ D † ,M , β) ≤ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   , (3.15
) where the constant C(α, β) has been introduced in (3.12).

The proof of Theorem 3.1 is postponed to Section 6.2.3. Remark 3.5 According to Theorem 3.1, given a radius r ε,σ > 0, then

r 2 ε,σ ≥ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   ⇒ β ε,σ (Θ a (r ε,σ ), Ψ D † ,M ) ≤ β,
and, hence,

r2 ε,σ ≤ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   . (3.16)
This upper bound corresponds to item (i) of Theorem 2.1.

Upper Bounds: Specific Cases

Our aim in this section is to determine an explicit value (in terms of the noise levels ε and σ) for the upper bounds on the minimax separation radius rε,σ obtained in Theorem 3.1 above.

To this end, we will consider well-known specific cases regarding the behavior of both sequences (a j ) j∈N and (b j ) j∈N . According to the existing literature, we will essentially deal with mildly and severely ill-posed problems with ellipsoids of ordinary-smooth and super-smooth functions (see also Section 2.2 for formal definitions).

Theorem 3.2 Consider the goodness-of-fit testing problem (2.4) when observations are given by (1.1), and the signal of interest has smoothness governed by (2.2). Then, (i) If b j ∼ j -t , t > 0, and a j ∼ j s , s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 4s 2s+2t+1/2 ∨ σ ln 3/4 (1/σ) 2( s t ∧1)
.

(ii) If b j ∼ j -t , t > 0, and a j ∼ exp{js}, s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 2 [ln (1/ε)] (2t+ 1 2 ) ∨ σ 2 ln 3 2 (1/σ) .
(iii) If b j ∼ exp{-jt}, t > 0, and a j ∼ j s , s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ [ln (1/ε)] -2s ∨ ln 1 σ ln 1/2 (1/σ) -2s
.

(iv) If b j ∼ exp{-jt}, t > 0, and a j ∼ exp{js}, s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 2s s+t ∨ [σ ln 1/2 (1/σ)] 2( s t ∧1) .
The proof is postponed to Section 6.3. The main task is to compute the asymptotic tradeoff between both antagonistic terms ε 2

D∧M 1 j=1
b -4 j and σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0 in the upper bounds on the minimax separation radius rε,σ displayed in (3.16).

Lower Bounds on the Minimax Separation Radius

We establish a non-asymptotic lower bound on the minimax separation radius (which corresponds to item (ii) of Theorem 2.1). In order to do this, we consider two special cases of the GSM (1.1), namely the situations where (a) ε = 0: the signal is observed without noise but the eigenvalues of the operator at hand are still noisy, and (b) σ = 0: the 'classical' model (see, e.g., [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF] or [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF]) where the eigenvalues of the operator at hand are known.

Both models (a) and (b) correspond to some 'extreme' situations but provide, in some sense, a benchmark for the problem at hand. We first establish a lower bound for the case (a) in Section 4.1 and recall the lower bound for the case (b) (that has already been discussed in, e.g., [START_REF] Baraud | Non-asymptotic minimax rates of testing in signal detection[END_REF], [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF] or [START_REF] Marteau | A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection[END_REF]) in Section 4.2. Then, we establish in Section 4.3 that the minimax separation radius associated to goodness-of-fit testing problem (2.4) is always greater than the maximum of the minimax separation radii associated to the cases (a) and (b). Finally, in Section 4.4, we provide a control of the lower bounds for minimax separation radii for the specific cases displayed in Table 2.2.

Lower Bounds for a GSM with ε = 0

We consider the GSM (1.1) with b = b and ε = 0, i.e.,

Y j = bj θ j , j ∈ N, X j = bj + σ η j , j ∈ N. (4.1)
For a given sequence b = (b) j∈N , define

B(b) = {ν ∈ l 2 (N) : C 0 |b j | ≤ |ν j | ≤ C 1 |b j |, j ∈ N, 0 < C 0 ≤ 1 ≤ C 1 < +∞}.
Given observations from the GSM (4.1), for any given θ 0 = 0 and b ∈ B(b), we consider the following goodness-of-fit testing problem

H 0 : θ = θ 0 versus H 1 : θ 0 ∈ E a , θ -θ 0 ∈ Θ a (r σ ), b ∈ B(b), (4.2)
where Θ a (r σ ) = {µ ∈ E a , µ ≥ r σ }.

Our aim below is to provide a lower bound on the minimax separation radius r0,σ , defined as r0,σ := inf

Ψα: α 0,σ ( Ψα)≤α r 0,σ (E a , Ψα , β),
where r 0,σ (E a , Ψ α , β) is the separation radius of any given α-level test Ψ α , defined as

r 0,σ (E a , Ψ α , β) := inf r σ > 0 : β 0,σ,b (Θ a (r σ ), B(b), Ψ α ) ≤ β ,
and β 0,σ,b (Θ(r σ ), B(b), Ψ α ) is the associated maximal second kind error probability, defined as

β 0,σ,b (Θ(r σ ), B(b), Ψ α ) := sup θ 0 ∈Ea, θ-θ 0 ∈Θa(rσ) b∈B(b) P θ, b(Ψ α = 0).
The following proposition states a lower bound for the minimax separation radius r0,σ of the goodness-of-fit testing problem (4.2). Proposition 4.1 Assume that (Y, X) = (Y j , X j ) j∈N are observations from the GSM 4.1) and consider the goodness-of-fit testing problem (4.2). Let α ∈]0, 1[ and β ∈ ]0, 1 -α[ be given. Then, for every σ > 0, the minimax separation radius r0,σ is lower bounded by

r0,σ ≥ C α,β 4 σ max 1≤D≤M 2 b -1 D a -1 D , (4.3)
where

M 2 := sup D ∈ N : C α,β σb -1 D ≤ 2 and G D (C 0 , C1) ≥ 1 1 + 4(1 -α -β) 2 (4.4) with C α,β = ln(1 + 4(1 -α -β) 2 ) > 0 and G D (C 0 , C 1 ) = 1 σ √ 2π C 1 b D C 0 b D exp - 1 2σ 2 (t -b D ) 2 dt, (4.5) for some constants 0 < C 0 ≤ 1 ≤ C 1 < +∞.
The proof is postponed to Section 6.4.1.

Remark 4.1 Note that G D (C 0 , C1) = Φ (C 1 -1) b D σ -Φ (C 0 -1) b D σ ,
where Φ(•) is the cumulative distribution function of the standard Gaussian distribution. Hence,

G D (C 0 , C1) ≥ 1 1 + 4(1 -α -β) 2 ⇔ b D ≥ σK,
where K := K(C 0 , C 1 , α, β) > 0. Then M 2 in (4.4) can be re-expressed as

M 2 := sup {D ∈ N : b D ≥ σ[K ∨ C α,β /2]} . (4.6)
This expression M 2 in (4.6) can be compared to the respective expressions of M 0 and M 1 defined in (3.3). In particular, we point-out that there is no logarithmic term involved in M 2 .

Lower Bounds for the GSM when σ = 0

We consider the GSM (1.1) with σ = 0, i.e.,

Y j = b j θ j + εξ j , j ∈ N, X j = b j , j ∈ N. (4.7)
Note that, in this case, the above model can be re-expressed as

Y j = b j θ j + εξ j , j ∈ N, (4.8)
where b = (b j ) j∈N is a known positive sequence.

The following proposition states a lower bound for the minimax separation radius rε,0 , defined in (2.2) with σ = 0, of the following goodness-of-fit testing problem

H 0 : θ = θ 0 versus H 1 : θ 0 ∈ E a , θ -θ 0 ∈ Θ a (r ε,0 ), (4.9)
where Θ a (r ε,0 ) is defined in (2.3) with σ = 0.

Proposition 4.2 Assume that Y = (Y j ) j∈N are observations from the GSM 4.8) and consider the goodness-of-fit testing problem (4.9). Let α ∈]0, 1[ and β ∈ ]0, 1 -α[ be given. Then, for every ε > 0, the minimax separation radius rε,0 is lower bounded by

r2 ε,0 ≥ sup D∈N   c α,β ε 2 D j=1 b -4 j ∧ a -2 D   , (4.10)
where

c α,β = (2 ln(1 + 4(1 -α -β) 2 )) 1/4 > 0. (4.11)
The proof of the Proposition 4.2 with detailed arguments are related discussion can be found in e.g., [START_REF] Baraud | Non-asymptotic minimax rates of testing in signal detection[END_REF], [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF] and [START_REF] Marteau | A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection[END_REF].

A Combined Lower Bound

The following result provides a lower bound on the minimax separation radius rε,σ for the goodness-of-fit testing problem (2.4). This lower bound corresponds to item (ii) of Theorem 2.1.

In particular,

r2 ε,σ ≥ C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∨    sup D∈N   c α,β ε 2 D j=1 b -4 j ∧ a -2 D      , (4.13)
where C α,β is given in (4.5), M 2 is given in (4.4) and c α,β is given in (4.11).

The proof of Theorem 4.1 is postponed to Section 6.4.2.

Remark 4.2 At a first sight, the upper and lower bounds respectively displayed in (i) and (ii) of Theorem 2.1 do not exactly match up. However, a closer look at the involved formulas indicates that both quantities contain terms that have similar behaviors. This is, in some sense, confirmed in Section 4.4 below where specific sequences (a j ) j∈N and (b j ) j∈N are treated.

Lower Bounds: Specific Cases

Our aim in this section is to determine an explicit value (in terms of the noise levels ε and σ) for the lower bounds on the minimax separation radius rε,σ obtained in Theorem 4.1 above for the specific sequences (a j ) j∈N and (b j ) j∈N considered in Section 3.3.

Theorem 4.2 Consider the goodness-of-fit testing problem (2.4) when observations are given by (1.1), and the signal of interest has smoothness governed by (2.2). Then, (i) If b j ∼ j -t , t > 0, and a j ∼ j s , s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 4s 2s+2t+1/2 ∨ σ 2( s t ∧1) .
(ii) If b j ∼ j -t , t > 0, and a j ∼ exp{js}, s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 2 [ln (1/ε)] (2t+ 1 2 ) ∨ σ 2 .
(iii) If b j ∼ exp{-jt}, t > 0, and a j ∼ j s , s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ [ln (1/ε)] -2s ∨ [ln (1/σ)] -2s .
(iv) If b j ∼ exp{-jt}, t > 0, and a j ∼ exp{js}, s > 0, for all j ∈ N, then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 2s s+t ∨ σ 2( s t ∧1) .
The proof is postponed to Section 6.5. As in the case of the upper bound, the main task is to compute the trade-off between both different antagonistic terms involved in the lower bound on the minimax separation radius rε,σ displayed in (4.13).

Concluding Remarks

The main conclusion of this work is that goodness-of-fit testing in an inverse problem setting is 'feasible', even in the specific situation where some uncertainty is observed on the operator at hand in the model (1.1). We have established 'optimal' separation conditions for the goodnessof-fit testing problem (2.4) via a sharp control of the associated minimax separation radius.

We stress that several outcomes and open questions are still of interest. We can mention, among others,

• Adaptivity: As proved in Theorem 3.1, the test Ψ D † ,M introduced in (3.6)-(3.9) with D † defined in (3.13) is powerful in the sense that its separation radius is equal (up to constant) to the minimax one. However, this test strongly depends on the sequence a = (a j ) j∈N that characterizes the smoothness of the signal of interest. In practice, this sequence is unknown and adaptive procedures are necessary (see, e.g., [START_REF] Yu | Nonparametric Goodness-of-Fit Testing Under Gaussian Models[END_REF] or [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF]).

• Signal detection: We have already mentioned in Remark 2.2 that signal detection is different from goodness-of-fit testing (2.4) when the GSM (1.1) is at hand. In this work, we were concerned with the case where θ 0 = 0 (goodness-of-fit testing). However, some attention should also be paid in the future to the case where θ 0 = 0 (signal detection). In particular, testing methodologies and related minimax separation radii are quite different from those presented above.

• Errors-in-variables model: Density model with measurement errors have been at the core of several statistical studies in the past decades (see, e.g., [START_REF] Meister | Deconvolution Problems in Nonparametric Statistics[END_REF] for an overview). Formally, given a sample of independent and identical distributed random variables (Y i ) i=1,2,...,n satisfying

Y i = X i + ǫ i i = 1, 2, . . . , n,
the aim is to produce some inference on the unknown density of the X i denoted by f , the ǫ i corresponding to some error, with known density η. This appears to be an inverse (deconvolution) problem since the Y i are associated to the convolved density f * η. In a goodness-of-fit testing task, this model has been discussed in [START_REF] Butucea | Goodness-of-fit testing and quadratic functional estimation from indirect observations[END_REF] and minimax separation rates (in the asymptotic minimax testing framework) have been established in various settings. In the spirit of our contribution, it could be interesting to propose methods taking into account some possible uncertainty on the density η at hand.

All these topics require special attention that is beyond the scope of this paper. Nevertheless, they provide an avenue for future research.

6 Appendix: Proofs

Useful Lemmas

The constant C > 0 and 0 < τ < 1 below will vary from place to place.

The following lemma is inspired by Lemma 6.1 of [START_REF] Cavalier | Adaptive estimation for inverse problems with noisy operators[END_REF]. Then, for any σ ∈]0, 1[,

P(M c ) ≤ α 10 + απ 2 6κ . (6.2)
Proof of Lemma 6.1. It is easily seen that

P(M ≥ M 1 ) = P   M 1 j=1 {|X j | > σh j }   ≤ P (|X M 1 | > σh M 1 ) , ≤ P (|b M 1 | + σ|η M 1 | > σh M 1 ) , ≤ P (|η M 1 | > h M 1 -h 1,M 1 ) , = P |η M 1 | > 2 ln 10 α ,
where the sequences (h j ) j∈N and (h 1,j ) j∈N are defined in (3.2) and (3.5) respectively. Using the bound 1 2π

+∞ x e -x 2 2 dx ≤ 1 x e -x 2 2 √ 2π ∀x > 0, (6.3)
we get

P(M ≥ M 1 ) ≤ 2 √ 2π α 10 1 2 ln(10/α) ≤ α 10 , (6.4)
since 2 ln(10/α) > 1 for all α ∈]0, 1[. In the same spirit,

P(M < M 0 ) = P   M 0 j=1 {|X j | ≤ σh j }   ≤ M 0 j=1 P (|X j | ≤ σh j ) , ≤ M 0 j=1 P (|b j | -σ|η j | ≤ σh j ) , ≤ M 0 j=1 P (σ|η j | ≥ |b j | -σh j )) , ≤ M 0 j=1 P (|η j | ≥ h 0,j -h j ) .
According to the respective definition of (h j ) j∈N , (h 0,j ) j∈N (see (3.2) and (3.4)), and using again inequality (6.3), we obtain

P(M ≥ M 1 ) ≤ 2 √ 2π M 0 j=1 1 h 0,j -h j exp - 1 2 (h 0,j -h j ) 2 , ≤ 2 √ 2π M 0 j=1 1 2 ln κj 2 α α κj 2 , ≤ α κ j∈N 1 j 2 = απ 2 6κ , (6.5)
on noting that j∈N 1

j 2 = π 2 /6. Since P(M c ) ≤ P(M < M 0 ) + P(M ≥ M 1 ),
the lemma follows, thanks to (6.4) and (6.5). Then, for any σ ∈]0, 1[

P(B c ) ≤ α 10 + απ 2 3κ . (6.7)
Proof of Lemma 6.2 Using the definitions of M and M 1 , simple calculations give

P(B c ) = P(B c ∩ M) + P(B c ∩ M c ), ≤ P   M 1 -1 j=1 σ|η j | > b j 2   + P(M c ), ≤ M 1 -1 j=1 P |η j | > 1 2 h 1,j + P(M c ).
Using (3.5), Lemma 6.1 and (6.3), we obtain

P(B c ) ≤ 2 √ 2π M 1 j=1 1 8 2 ln κj 2 α α κj 2 + α 10 + απ 2 6κ ≤ α 10 + απ 2 3κ . (6.8)
Hence, the lemma holds true.

j X j -1 2 θ 2 j ≥ σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M   ≤ α 5 + α κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)},
for some C > 0 and 0 < τ < 1.

Proof of Lemma 6.3. Using Lemma 6.1, Lemma 6.2 and a Taylor expansion as in Lemma 6.6 of [START_REF] Cavalier | Adaptive estimation for inverse problems with noisy operators[END_REF], we get, for all j ≤ M ,

b j X j = 1 1 + σb -1 j η j = 1 -σb -1 j η j + σ 2 ζ -2 j η 2 j ,
where ζ -1 j ≤ 8b -1 j on the even B defined in (6.6). Hence

P   D∧M j=1 b j X j -1 2 θ 2 j ≥ σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M   = P   D∧M j=1 (-σb -1 j η j + σ 2 ζ -2 j η 2 j ) 2 θ 2 j ≥ σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M   , ≤ P   2σ 2 D∧M j=1 b -2 j θ 2 j η 2 j + 2σ 4 D∧M j=1 ζ -4 j θ 2 j η 4 j ≥ σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M   . Therefore P   D∧M j=1 b j X j -1 2 θ 2 j ≥ σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M   ≤ P      2σ 2 D∧M j=1 b -2 j θ 2 j η 2 j ≥ 1 2 σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ∩ (B ∩ M)   +P      2σ 4 D∧M j=1 ζ -4 j θ 2 j η 4 j ≥ 1 2 σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ∩ (B ∩ M)   + P ((B ∩ M) c ) , := T 1 + T 2 + P ((B ∩ M) c ) . (6.9)
We concentrate bellow our attention on the term T 1 defined as

P      2σ 2 D∧M j=1 b -2 j θ 2 j η 2 j ≥ 1 2 σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ∩ (B ∩ M)   .
We consider the two following possible scenarios: (i) a -1 j b -1 j ≤ C 0 as j → +∞, for some C 0 > 0, and (ii) a -1 j b -1 j → +∞ as j → +∞.

Consider first scenario (i). Then, using again (6.3)

T 1 ≤ P 2σ 2 max 1≤j≤D∧M (b -2 j a -2 j η 2 j ) ≥ 1 2 σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M ∩ (B ∩ M) ≤ P 2C 2 0 σ 2 max 1≤j≤D∧M (η 2 j ) ≥ 1 2 σ 2 ln 3/2 (1/σ) ∩ (B ∩ M) ≤ M 1 -1 j=1 P η j ≥ 1 2C 0 ln 3/4 (1/σ) , ≤ 2M 1 √ 2π 2C 0 ln 3/4 (1/σ) exp - C ln 3/2 (1/σ) 8C 2 0 ≤ C exp{-ln 1+τ (1/σ)}. (6.10)
for some constants C, τ ∈ R + . A similar bound occurs for the term T 2 for this scenario.

Consider now the second scenario (ii). Then

T 1 ≤ P 2b -2 D∧M a -2 D∧M σ 2 max 1≤j≤D∧M (η 2 j ) ≥ 1 2 a -2 D∧M ∩ (B ∩ M) ≤ P σ 2 max 1≤j≤D∧M (η 2 j ) ≥ 1 4 b 2 D∧M ∩ (B ∩ M) ≤ M 1 -1 j=1 P σ 2 η 2 j ≥ 1 4 b 2 M 1 -1 ,
since the sequence (b j ) j∈N is non-increasing. Using (6.3), we get

T 1 ≤ M 1 -1 j=1 P η j ≥ 1 2 h 1,M 1 -1 , ≤ 2M 1 √ 2π 2 h 1,M 1 -1 exp - h 2 1,M 1 -1 8 , ≤ M 1 exp -ln κM 2 1 α ≤ M 1 × α κM 2 1 ≤ α κ . (6.11)
By similar computations, we get

T 2 := P      2σ 4 D∧M j=1 ζ -4 j θ 2 j η 4 j ≥ 1 2 σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ∩ (B ∩ M)   , ≤ P      2 × 8 4 σ 4 D∧M j=1 b -4 j θ 2 j η 4 j ≥ 1 2 a -2 D∧M    ∩ (B ∩ M)   , ≤ P 2 × 8 4 σ 4 max j=1..D∧M η 4 j ≥ 1 2 b 4 D∧M ∩ (B ∩ M) , ≤ M 1 -1 j=1 P |η j | ≥ 1 8 √ 2 h 1,M 1 -1 , ≤ 2 √ 2π 8 √ 2M 1 h 1,M 1 exp - 1 4 × 8 2 4 × 8 2 ln κM 2 1 α ≤ α κ . (6.12)
Hence, the lemma follows from Lemmas 6.1, 6.2 and (6.9)-(6.12).

Lemma 6.4 Let

Z j = ν j + v j ω j , j ∈ N,
where ω = (ω j ) j∈N is a sequence of independent standard Gaussian random variables. For all D ∈ N, define

T = D j=1 Z 2 j and Σ = D j=1 v 4 j + 2 D j=1 v 2 j ν 2 j .
Then, for all x > 0,

P T -E(T ) > 2 √ Σx + 2x sup 1≤j≤D (v 2 j ) ≤ exp(-x) (6.13) P T -E(T ) < -2 √ Σx ≤ exp(-x). (6.14)
Proof of Lemma 6. [START_REF] Cavalier | Oracle inequalities for inverse problems[END_REF] The proof is given in Lemma 2 of [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF]. Conditionally to the sequence X = (X j ) j∈N , for each 1 ≤ j ≤ D ∧ M , the random variable X -1 j Y j -θ j,0 is Gaussian with mean ν j = (b j /X j -1)θ j,0 and standard deviation v j = εX -1 j . In particular, for all

Non-Asymptotic Upper bounds

D ∈ N E θ 0 ,b (T D,M | X) := E θ 0 ,b   D∧M j=1 Y j X j -θ j,0 2 | X   = D∧M j=1 b j X j -1 2 θ 2 j,0 + ε 2 D∧M j=1 X -2 j . (6.15) For all D ∈ N, define Σ D,M := ε 4 D∧M j=1 X -4 j + ε 2 D∧M j=1 X -2 j b j X j -1 2 θ 2 j,0 .
Applying Lemma 6.4 with T = T D,M , Σ = Σ D,M and x = x α/2 := ln(2/α), we get

P θ 0 ,b T D,M -E θ 0 (T D,M | X) > 2 Σ D,M x α/2 + 2ε 2 x α/2 max 1≤j≤D∧M (X -2 j ) | X ≤ α 2 . (6.16) Using the inequalities √ a + b ≤ √ a + √ b and ab ≤ a 2 /2 + b 2 /2 for a, b > 0, it is easily seen that Σ D,M ≤ ε 2 D∧M j=1 X -4 j + ε 2 D∧M j=1 X -2 j b j X j -1 2 θ 2 j,0 ≤ ε 2 D∧M j=1 X -4 j + ε 2 max 1≤j≤D∧M X -2 j D∧M j=1 b j X j -1 2 θ 2 j,0 ≤ ε 2 D∧M j=1 X -4 j + 1 2 ε 2 max 1≤j≤D∧M X -2 j + 1 2 D∧M j=1 b j X j -1 2 θ 2 j,0 . (6.17)
According to (6.15)-( 6.17), we obtain the following bound

P θ 0 ,b   T D,M > (1 + x α/2 ) D∧M j=1 b j X j -1 2 θ 2 j,0 + ε 2 D∧M j=1 X -2 j + C(α)ε 2 D∧M j=1 X -4 j | X   ≤ α 2 ,
where the constant C(α) is defined in (3.9). Since E[E(V | W )] = E(V ) for any random variables V and W , the previous inequality leads to

P θ 0 ,b   T D,M > (1 + x α/2 ) D∧M j=1 b j X j -1 2 θ 2 j,0 + ε 2 D∧M j=1 X -2 j + C(α)ε 2 D∧M j=1 X -4 j   ≤ α 2 .
Then, by defining

A =    D∧M j=1 b j X j -1 2 θ 2 j,0 < σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ,
and applying Lemma 6.3, we immediately get

α ε,σ (Ψ D,M ) ≤ P θ 0 ,b ({T D,M > t 1-α,D (X)} ∩ A) + P(A c ) ≤ α 2 + α 5 + α 6κ (3π 2 + 12) + C exp{-ln 1+τ (1/σ), = 7α 10 + α κ π 2 2 + 2 ) + C exp{-ln 1+τ (1/σ),
for some C > 0 and 0 < τ < 1. In particular, setting

κ = 5 π 2 2 + 2 ,
there exists σ 0 ∈]0, 1[ such that, for all σ ≤ σ 0 and for each ε > 0,

α ε,σ (Ψ D,M ) ≤ α.
This concludes the proof of the proposition.

Proof of Proposition 3.2

Let θ, θ 0 ∈ E a and θ -θ 0 ∈ Θ a (r ε,σ ). Then

P θ,b (Ψ D,M = 0) = P θ,b ({Ψ D,M = 0} ∩ (B ∩ M)) + P θ,b ({Ψ D,M = 0} ∩ (B ∩ M) c ) := T 1 + T 2 . (6.18)
Control of T 2 : Using Lemma 6.1, Lemma 6.2 and elementary probabilistic arguments, we get

T 2 := P θ,b ({Ψ D,M = 0} ∩ (B ∩ M) c ) ≤ P((B ∩ M) c ) ≤ P(B c ) + P(M c ) ≤ α 5 + α κ π 2 2 + 2 ≤ β 5 + β κ π 2 2 + 2 , (6.19) since β > α.
Control of T 1 : Define t β/2,D (θ, X) to be the β/2-quantile of T D,M , conditionally on X, i.e.,

P θ,b (T D,M ≤ t β/2,D (θ, X) | X) ≤ β 2 .
Then, by elementary probabilistic arguments, we get

T 1 := P θ,b ({Ψ D,M = 0} ∩ {B ∩ M}), = E P θ,b ({Ψ D,M = 0} | X) 1{B ∩ M} , = E P θ,b (T D,M ≤ t 1-α,D (X) | X) 1{B ∩ M} , ≤ β 2 E 1{t 1-α,D (X) ≤ t β/2,D (θ, X)}1{B ∩ M} + E 1{t 1-α,D (X) > t β/2,D (θ, X)}1{B ∩ M} , ≤ β 2 + E 1{t 1-α,D (X) > t β/2,D (θ, X)}1{B ∩ M} , ≤ β 2 + P θ,b {t 1-α,D (X) > t β/2,D (θ, X)} ∩ {B ∩ M} . (6.20)
Our next task is to provide a lower bound for t β/2,D (θ, X). Under H 1 , conditionally to the sequence X = (X j ) j∈N , for each 1 ≤ j ≤ D ∧ M , the random variable X -1 j Y j -θ j,0 is Gaussian with mean ν j and standard deviation v j defined as ν j = b j X j -1 θ j + (θ j -θ j,0 ) and v j = εX -1 j .

In particular,

E θ,b (T D,M | X) = D∧M j=1 b j X j -1 θ j + (θ j -θ j,0 ) 2 + ε 2 D∧M j=1 X -2 j = D∧M j=1 ν 2 j + ε 2 D∧M j=1 X -2 j . (6.21) Let ΣD,M := ε 4 D∧M j=1 X -4 j + ε 2 D∧M j=1 X -2 j b j X j -1 θ j + (θ j -θ j,0 ) 2 = ε 4 D∧M j=1 X -4 j + ε 2 D∧M j=1 X -2 j ν 2 j . (6.22)
Using Lemma 6.4 with T = T D,M , Σ = ΣD,M and x = x β/2 := ln(2/β), we obtain

P θ,b   T D,M < D∧M j=1 ν 2 j + ε 2 D∧M j=1 X -2 j -2 ΣD,M x β/2 | X   ≤ β 2 ⇒ t β/2,D (θ, X) ≥ D∧M j=1 ν 2 j + ε 2 D∧M j=1 X -2 j -2 ΣD,M x β/2 . (6.23)
Therefore, using (3.8) and (6.23), we get

P θ,b {t 1-α,D (X) > t β/2,D (θ, X)} ∩ {B ∩ M} ≤ P θ,b      D∧M j=1 ν 2 j < C(α) + 2 x β/2 ε 2 D∧M j=1 X -4 j + (1 + x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M + 2 x β/2 ε 2 D∧M j=1 X -2 j ν 2 j    ∩ {B ∩ M}   ≤ P θ,b      1 2 D∧M j=1 ν 2 j < C(α, β)ε 2 D∧M j=1 X -4 j + (1 + x α/2 )[σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M ]    ∩ {B ∩ M}   ,
where C(α, β) := C(α) + 3 x β/2 , (6.24)

and C(α) is defined in (3.9). Note that, for any a, b ∈ R, using the Young inequality 2ab ≤ γa 2 + γ -1 b 2 for γ = 1/2 we get (a + b) 2 ≥ a 2 /2 -b 2 . Applying the latter inequality with

a = θ j -θ j,0 , b = b j X j -1 θ j , j = 1, . . . , D ∧ M,
and using Lemma 6.3, we arrive at

P θ,b {t 1-α,D (X) > t β/2,D (θ, X)} ∩ {B ∩ M} ≤ P θ,b      D∧M j=1 (θ j -θ j,0 ) 2 < 4C(α, β)ε 2 D∧M j=1 X -4 j + 4(1 + x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M + 2 D∧M j=1 b j X j -1 2 θ 2 j    ∩ {B ∩ M}   ≤ P θ,b      D∧M j=1 (θ j -θ j,0 ) 2 < 4C(α, β)ε 2 D∧M j=1 X -4 j + (6 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ∩ {B ∩ M}   + α 5 + α κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)}.
Using the fact that θ ∈ E a , we get

P θ,b {t 1-α,D (X) > t β/2,D (θ, X)} ∩ {B ∩ M} ≤ P θ,b      θ -θ 0 2 < 4C(α, β)ε 2 D∧M j=1 X -4 j + (6 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M + j>D∧M (θ j -θ j,0 ) 2    ∩ {B ∩ M}   + α 5 + α κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)} ≤ P θ,b      θ -θ 0 2 < 4C(α, β)ε 2 D∧M j=1 X -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M    ∩ {B ∩ M}   + α 5 + α κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)}. (6.25)
To conclude the proof, note that on the event {B ∩ M}, we have

M 0 ≤ M < M 1 and b j X j ∈ 2 3 , 2 ∀ j = 1, . . . , M. (6.26)
Hence, using (6.25) and (6.26)

P θ,b {t 1-α,D (X) > t β/2,D (θ, X)} ∩ {B ∩ M} ≤ P θ,b   θ -θ 0 2 < 16C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   + α 5 + α κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)}, = α 5 + α κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)},
as soon as

θ -θ 0 2 ≥ C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0 , (6.27)
where C(α, β) = 16C(α, β) is defined in (3.12). Therefore, for any fixed β ∈]α, 1[, (6.27) implies that, there exists σ 0 ∈]0, 1[ such that, for all 0 < σ < σ 0 and for each ε > 0,

P θ,b (Ψ D,M = 0) ≤ 7β 10 + β κ π 2 2 + 2 + C exp{-ln 1+τ (1/σ)} ≤ β,
for some C > 0 and 0 < τ < 1, which, in turn, implies that (3.15) holds true. The last part of the theorem is a direct consequence of (2.6) and (3.15). This completes the proof of the proposition.

Proof of Theorem 3.1

The validity of (3.14) can be immediately derived from Proposition 3.1 taking into account that Lemma 6.3 is still valid with D := D † (that depends on the sequence X = (X j ) j∈N ). For the proof of (3.15), note first that (6.18), (6.19) and (6.20) still holds true with D := D † . In the same spirit, is is easy to see that Lemma 6.3 is still valid when the bandwidth D is measurable with respect to the sequence (X k ) k∈N . Hence, the same inequality than (6.25) can be obtained with D := D † , namely

P θ,b {t 1-α,D † (X) > t β/2,D † (θ, X)} ∩ {B ∩ M} ≤ P θ,b      θ -θ 0 2 < 4C(α, β)ε 2 D † ∧M j=1 X -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D † ∧M    ∩ {B ∩ M}   ≤ P θ,b   θ -θ 0 2 < inf D∈N   16C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0     = 0,
as soon as

θ -θ 0 2 ≥ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   ,
where C(α, β) is defined in (3.12). Therefore, we immediately get that (3.15) holds true.Finally, the validity of (3.16) follows immediately on noting that r2 ε,σ := inf

Ψα: αε,σ( Ψα)≤α r 2 ε,σ (E a , Ψα , β) ≤ r 2 ε,σ (E a , Ψ D † ,M , β) ≤ inf D∈N   C(α, β)ε 2 D∧M 1 j=1 b -4 j + (7 + 4 x α/2 ) σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   .
This completes the proof of the theorem.

Upper Bounds: Specific Cases

For the sake of convenience, we give the proof of each item (i)-(iv) in Theorem 3.2 in different sections. ε,σ (see (3.16), where the bandwidths M 0 and M 1 are defined in (3.3). The bandwidth J ⋆ corresponds to the value J ∈ N where the two dashed lines cross, i.e.,

D 0 ε 2 D∧M 1 j=1 b -4 j a -2 D∧M 0 J ⋆ M0 M1 D 0 ε 2 D∧M 1 j=1 b -4 j a -2 D∧M 0 J ⋆ M0 M1 D 0 ε 2 D∧M 1 j=1 b -4 j a -2 D∧M 0 J ⋆ M1 M0
J ∈ N : ε 2 J j=1 b -4 j = a -2 J .
The computation of the separation radius rε,σ , for

0 < ε ≤ ε 0 , ε 0 ∈]0, 1[ and 0 < σ ≤ σ 0 , σ 0 ∈]0, 1[, leads to three different scenarios: J ⋆ M 0 M 1 (left figure), M 0 J ⋆ M 1 (center figure) and M 0 M 1 J ⋆ (right figure). D 0 ε 2 D∧M 1 j=1 b -4 j a -2 D∧M ⋆ J ⋆ M ⋆ M1 D 0 ε 2 D∧M 1 j=1 b -4 j a -2 D∧M ⋆ J ⋆ M ⋆ M1 D 0 ε 2 D∧M 1 j=1 b -4 j a -2 D∧M ⋆ J ⋆ M1 M ⋆ Figure 6.2: [Case II: a -2 D∧M 0 ∼ σ 2 ln 3/2 (1/σ)]
An illustration of the two resulting two terms (red color), namely ε 2

D∧M 1 j=1 b -4 j and a -2
D∧M ⋆ , for each D ∈ N, involved in the (upper bound of the) minimax separation radius r2 ε,σ (see (3.16), where the bandwidth M 1 is defined in (3.3) and the bandwidth M ⋆ is the value of M ∈ N such that the two terms a -2 M and σ 2 ln 3/2 (1/σ) are of the same order, i.e., M ∈ N : a -2 M ∼ σ 2 ln 3/2 (1/σ). The bandwidth J ⋆ corresponds to the value J ∈ N where the two dashed lines cross, i.e., J ∈ N :

ε 2 J j=1 b -4 j = a -2 J .
The computation of the separation radius rε,σ , for 0 < ε ≤ ε 0 , ε 0 ∈]0, 1[ and 0 < σ ≤ σ 0 , σ 0 ∈]0, 1[, leads to three different scenarios:

J ⋆ M ⋆ M 1 (left figure), M ⋆ J ⋆ M 1 (center figure) and M ⋆ M 1 J ⋆ (right figure).

Case (i): Mildly ill-posed problems with ordinary smooth functions

Recall that b j ∼ j -t , t > 0, and a j ∼ j s , s > 0, j ∈ N. (6.28) Proposition 6.1 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.28).

Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 4s 2s+2t+1/2 ∨ σ ln 3/4 (1/σ) 2( s t ∧1) . (6.29)
Proof of Proposition 6.1 In a first time, we determine the order of the bandwidths M 0 and M 1 . Setting

M1 := σ 1 t ln (1/σ) -1/t and M1 := σ 1 2t ln (1/σ) -1/t we get σh 1, M1 ∼ σ ln( M1 ) = σ 1 t ln (1/σ) - 1 2t ln 1 t ln (1/σ) ≤ σ 1 t ln (1/σ) ∼ b M1 , which implies that M 1 M1 . At the same time σh 1, M1 ∼ σ ln( M1 ) = σ 1 t ln (1/σ) - 1 2t ln 1 2t ln (1/σ) , = σ 1 2t ln (1/σ) + 1 2t ln (1/σ) - 1 2t ln 1 2t ln (1/σ) , σ 1 2t (1/σ) ∼ b M1 ,
which implies that M 1 M1 . Hence, we can conclude that

M 1 ∼ σ ln (1/σ) -1/t .
Similarly, we get that

M 0 ∼ σ ln (1/σ) -1/t .
In order to control the terms involved in the upper bound on the minimax separation radius, we consider the cases s < t and s ≥ t separately.

Consider first the case s < t. In this case, for all D ∈ N,

a -2 D∧M 0 a -2 M 0 ∼ M -2s 0 ∼ σ ln 1/2 (1/σ) 2s/t σ 2 ln 3/2 (1/σ). Hence, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0   .
Define now the value of J ∈ N that satisfies the following equation

ε 2 J j=1 b -4 j ∼ a -2 J ⇔ ε 2 J 2t+1/2 ∼ J 2s ⇔ J := J ⋆ ∼ ε -2 2s+2t+1/2 .
We now consider the following situations (see Figure 6.1 for a graphical illustration):

• (J ⋆ M 0 ) In this case, r2 ε,σ a -2 J ⋆ ε 4s 2s+2t+1/2 . • (J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N a -2 D∧M 0 a -2 M 0 ∼ σ ln 1/2 (1/σ) 2s t . • (M 0 J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0   . =    inf D≤M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0      ∧    inf D>M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0      a -2 M 0 ∧    inf D>M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 M 0      a -2 M 0 ∧    ε 2 M 0 j=1 b -4 j + a -2 M 0    a -2 M 0 ∼ σ ln 1/2 (1/σ) 2s t .
Combining the above terms, we immediately get

r2 ε,σ ε 4s 2s+2t+1/2 ∨ σ ln 1/2 (1/σ) 2s t . (6.30)
Consider now the case s ≥ t. Define the value of M ∈ N that satisfies the following equation

a -2 M ∼ σ 2 ln 3/2 (1/σ) ⇔ M := M ⋆ ∼ σ ln 3/4 (1/σ) -1 s . Hence, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M ⋆   .
Working along the lines of the case s ≤ t, by replacing M 0 by M ⋆ (see Figure 6.2), we get r2

ε,σ ε 4s 2s+2t+1/2 ∨ σ ln 3/4 (1/σ) 2 . (6.31)
Hence, (6.29) follows thanks to (6.30) and (6.31). This completes the proof of the proposition.

6.3.2

Case (ii): Mildly ill-posed problems with super smooth functions

Recall that b j ∼ j -t , t > 0, and a j ∼ exp{js}, s > 0, j ∈ N. (6.32) Proposition 6.2 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.32). Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 2 [ln (1/ε)] (2t+ 1 2 ) ∨ σ 2 ln 3 2 (1/σ) . (6.33)
Proof of Proposition 6.2 According to Section 6.3.1, we obtain again

M 1 ∼ σ ln (1/σ) -1/t and M 0 ∼ σ ln (1/σ) -1/t .
Then, for all D ∈ N,

a -2 D∧M 0 a -2 M 0 ∼ exp{-2M 0 s} ∼ exp -2s σ ln 1/2 (1/σ) -1/t σ 2 ln 3/2 (1/σ).
Define as in the previous case the value M ∈ N that satisfies the following equation

a -2 M ∼ σ 2 ln 3/2 (1/σ) ⇔ M =: M ⋆ ∼ 1 s ln 1 σ ln 3/4 (1/σ) . Hence, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M ⋆   .
Define now the value of J ∈ N that satisfies the following equation

ε 2 J j=1 b -4 j ∼ a -2 J ⇔ ε 2 J 2t+1/2 ∼ exp{-2Js} ⇔ J := J ⋆ ∼ 1 σ ln (1/ε) -ln 1 σ ln (1/ε) 2t+ 1 2
.

We now consider the following situations:

• (J ⋆ M ⋆ ) In this case, r2 ε,σ a -2 J ⋆ ε 2 [ln (1/ε)] (2t+ 1 2 ) . • (J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N a -2 D∧M ⋆ a -2 M ⋆ ∼ σ 2 ln • (M ⋆ J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M ⋆   . =    inf D≤M ⋆   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M ⋆      ∧    inf D>M ⋆   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M ⋆      a -2 M ⋆ ∧    inf D>M ⋆   ε 2 D∧M 1 j=1 b -4 j + a -2 M ⋆      a -2 M ⋆ ∧    ε 2 M ⋆ j=1 b -4 j + a -2 M ⋆    a -2 M ⋆ ∼ σ 2 ln 3 2 (1/σ) .
Combining the above terms, we immediately get (6.33). This completes the proof of the proposition.

Case (iii): Severely ill-posed problems with ordinary smooth functions

Recall that b j ∼ exp{-jt}, t > 0, and a j ∼ j s , s > 0, j ∈ N. (6.34) Proposition 6.3 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.34).

Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

σ ln 1/2 (1/σ) e -M1 t = σ ln(1/σ) ∼ b M1 ,
which implies that M 1 ≥ M1 for σ small enough. At the same time

σh 1, M1 ∼ σ ln( M1 ) = σ ln 1 t ln (1/σ) b M1 ∼ e -M1 t = σ,
which implies that M 1 ≤ M1 for σ small enough. Hence, we can conclude that 1 t ln 1

σ ln 1/2 (1/σ) ≤ M 1 ≤ 1 t ln (1/σ) ,
for σ small enough. Similarly, we get that

1 t ln 1 σ ln 1/2 (1/σ) ≤ M 0 ≤ 1 t ln (1/σ) ,
for σ small enough. Now, we turn our attention to the proof of (6.35). For all D ∈ N,

a -2 D∧M 0 a -2 M 0 M -2s 0 ∼ ln 1 σ ln 1/2 (1/σ) -2s σ 2 ln 3/2 (1/σ). Hence, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0   .
Define now the value of J ∈ N that satisfies the following equation

ε 2 J j=1 b -4 j ∼ a -2 J ⇔ ε 2 exp{2tJ} ∼ J -2s ⇔ J := J ⋆ ∼ 1 t ln (1/ε) -ln 1 t ln (1/ε) 2s .
We now consider the following situations:

• (J ⋆ M 0 ) In this case, r2 ε,σ a -2 J ⋆ [ln (1/ε)] -2s . • (J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N a -2 D∧M 0 a -2 M 0 ∼ ln 1 σ ln 1/2 (1/σ) -2s
.

• (M 0 J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0   . =    inf D≤M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0      ∧    inf D>M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0      a -2 M 0 ∧    inf D>M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 M 0      a -2 M 0 ∧    ε 2 M 0 j=1 b -4 j + a -2 M 0    a -2 M 0 ln 1 σ ln 1/2 (1/σ) -2s
.

Combining the above terms, we immediately get (6.35). This completes the proof of the proposition.

6.3.4 Case (iv): Severely ill-posed problems with super smooth functions

Recall that b j ∼ exp{-jt}, t > 0, and a j ∼ exp{js}, s > 0, j ∈ N. (6.36) Proposition 6.4 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.36).

Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies

r2 ε,σ ε 2s s+t ∨ [σ ln 1/2 (1/σ)]
σ ln 1/2 (1/σ) ≤ M 0 ≤ 1 t ln (1/σ) ,
for σ small enough. Now, we consider the cases s < t and s ≥ t separately.

Consider first the case s < t. In this case, for all D ∈ N,

a -2 D∧M 0 a -2 M 0 ∼ exp{-2sM 0 } σ ln 1/2 (1/σ) 2s/t σ 2 ln 3/2 (1/σ). Hence, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0   .
Define now the value of J ∈ N that satisfies the following equation

ε 2 J j=1 b -4 j ∼ a -2 J ⇔ ε 2 exp{2tJ} ∼ exp{2sJ} ⇔ J := J ⋆ ∼ 1 s + t ln (1/ε) .
We now consider the following situations:

• (J ⋆ M 0 ) In this case, r2 ε,σ a -2 J ⋆ ε 2s s+t . • (J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N a -2 D∧M 0 a -2 M 0 σ ln 1/2 (1/σ) 2s t . • (M 0 J ⋆ M 1 ) In this case, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0   . =    inf D≤M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0      ∧    inf D>M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 D∧M 0      a -2 M 0 ∧    inf D>M 0   ε 2 D∧M 1 j=1 b -4 j + a -2 M 0      a -2 M 0 ∧    ε 2 M 0 j=1 b -4 j + a -2 M 0    a -2 M 0 σ ln 1/2 (1/σ) 2s t .
Combining the above terms, we immediately get

r2 ε,σ ε 2s s+t ∨ σ ln 1/2 (1/σ) 2s t . (6.38)
Consider now the case s ≥ t. Define the value M ∈ N that satisfies the following equation

a -2 M ∼ σ 2 ln 3/2 (1/σ) ⇔ M := M ⋆ ∼ 1 s ln 1 σ ln 3/4 (1/σ) . Hence, r2 ε,σ inf D∈N   ε 2 D∧M 1 j=1 b -4 j + σ 2 ln 3/2 (1/σ) ∨ a -2 D∧M 0   inf D∈N   ε 2 D∧M 1 j=1 b -4 j + a -2 M ⋆   .
Working along the lines of the case s < t by replacing M 0 by M ⋆ , we get r2

ε,σ ε 2s s+t ∨ σ ln 1/2 (1/σ) 2 . (6.39)
Hence, (6.37) follows thanks to (6.38) and (6.39). This completes the proof of the proposition.

6.4 Non-Asymptotic Lower Bounds Let θ 0 ∈ E a be given sequence (to be made precise below). Given a (prior) probability measure π on the set associated with H 1 , i.e., a probability measure π on Θa,θ 0 (r σ , b) := Θ a,θ 0 (r σ ) × B(b), where Θ a,θ 0 (r σ ) = θ 0 + Θ a (r σ ), by standard Bayesian arguments (see, e.g., Section 3.1 of [START_REF] Marteau | A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection[END_REF]), we arrive at

β 0,σ,b (Θ(r σ ), B(b)) = inf Ψα: α 0,σ ( Ψα)≤α sup θ 0 ∈Ea, θ-θ 0 ∈Θa(rσ) b∈B(b) P θ, b( Ψα = 0) ≥ 1 -α - 1 2 (E 0 [L 2 π (Y, X)] -1) 1/2 , (6.40)
where L π (Y, X) denotes the likelihood ratio between the two measures P π and P 0 , E 0 denotes the expectation with respect to P 0 , with P 0 = Θa,θ 0 (rσ,b) P θ 0 , b dπ(θ, b) and P π = Θa,θ 0 (rσ,b) P θ, b dπ(θ, b), and the last inequality is obtained by standard calculations (see, e.g., Section 3.1 of [START_REF] Marteau | A unified treatment for non-asymptotic and asymptotic approaches to minimax signal detection[END_REF]).

The probability measure π on Θa,θ 0 (r σ , b) is selected as product probability measure, i.e., π = j∈N π j , π j = π j,1 × π j,2 , j ∈ N.

Then, given the sequence θ and the bandwidth D ∈ N (to be made precise below), we set π j,1 = δ θ j,0 and π j,2 = δ b j , j = D, and

π D,1 = G -1 D (C 0 , C 1 )δ θ D and dπ D,2 (t) = 1 σ √ 2π exp - 1 2σ 2 (t -b D ) 2 dt, where G D (C 0 , C 1 ) = (1/σ √ 2π) C 1 b D C 0 b D exp -(t -b D ) 2 /(2σ 2 ) dt.
In some sense, using the above product probability measure π, we deal with observations (Y, X) = (Y j , X j ) j∈N from the following Bayesian sequence model

Y j = b j θ j,0 , X j = b j + ση j , j ∈ N \ {D}, and Y D = B D θ D , X D = B D + ση D , (6.41)
where B D is Gaussian random variable with mean b D and variance σ 2 , that is independent of the standard Gaussian sequence {η j } j∈N . Note that

π( Θa,θ 0 (r σ , b)) := π(Θ a,θ 0 (r σ ) × B(b)) = π 1 (Θ a,θ 0 (r σ )) × π 2 (B(b)) = G -1 D (C 0 , C 1 ) 1 σ √ 2π C 1 b D C 0 b D exp - 1 2σ 2 (t -b D ) 2 dt (6.42) = 1. (6.43)
In view of the above, it is immediately seen that

L π (Y, X) = j∈N L π j (Y j , X j ) = L π D (Y D , X D ).
Hence, as before, we arrive at

β 0,σ,b (Θ(r σ ), B(b)) ≥ 1 -α - 1 2 (E 0 [L 2 π D (Y D , X D )] -1) 1/2 . (6.44)
Our task below is then to provide an upper bound on

E 0 [L 2 π D (Y D , X D )].
To this end, it is easily seen from model (6.41) that Z D = (X D , Y D ), D ∈ N, is Gaussian random vector with mean U θ,D and covariance matrix σ 2 Σ θ,D , where

U θ,D = b D b D θ D , Σ θ,D = 2 θ D θ D θ 2 D . Note that Σ -1 θ,D = 1 θ 2 D θ 2 D -θ D -θ D 2 ,
where

ρ D = 1 - θ D,0 θ D .
Using simple algebra, we get

E 0 L 2 π (Z D ) | B D = exp 4B 2 D σ 2 (ρ D -1)(ρ D -2) - 4B D b D σ 2 (1 -ρ D ) = exp 4 σ 2 (ρ D -1)[B 2 D (ρ D -2) + B D b D ] .
It is easily seen that

B 2 D (ρ D -2) + B D b D = (b D + σ ηD ) 2 (ρ D -2) + (b D + σ ηD )b D = b 2 D (ρ D -1) + 2σ ηD b D (ρ D -3/2) + σ 2 η2 D (ρ D -2),
where {η D } D∈N is a sequence of independent standard Gaussian random variables. Therefore,

E 0 L 2 π (Z D ) | B D = exp 4b 2 D σ 2 (1 -ρ D ) 2 × exp 8 σ b D ηD (ρ D -1) ρ D - 3 2 + 4η 2 D (ρ D -2) . Since ρ D ∈]1, 2[, then 4η 2 D (ρ D -2) < 0 and, hence, E 0 L 2 π (Z D ) | B D ≤ exp 4b 2 D σ 2 (1 -ρ D ) 2 + 8 σ b D ηD (ρ D -1) ρ D - 3 2 .
Using (6.45) with

λ 1 = 0, λ 2 = 8 σ b D (ρ D -1) ρ D - 3 2 ,
we get Choice of θ: The sequence θ = (θ j ) j∈N is chosen as follows

E 0 L 2 π (Z D ) = G -1 D (C 0 , C 1 ) E E 0 L 2 π (Z D ) | B D ≤ G -1 D (C 0 , C 1 ) E exp 4b 2 D σ 2 (1 -ρ D ) 2 + 8 σ b D ηD (ρ D -1) ρ D - 3 2 = G -1 D (C 0 , C 1 ) exp 4b 2 D σ 2 (1 -ρ D ) 2 1 + ρ D - 3 2 2 ≤ G -1 D (C 0 , C 1 ) exp 5b 2 D σ 2 (1 -ρ D ) 2 ≤ 1 + 4(1 -α -β) 2 , as soon as 5b 2 D σ 2 (1 -ρ D ) 2 ≤ ln(1 + 4(1 -α -β) 2 ) + ln(G(C 0 , C 1 )), or, equivalently, as soon as |θ D -θ D,0 | ≤ C α,β,D σ|θ D |b -1 D , where C α,β,D = ln(1 + 4(1 -α -β) 2 ) + ln(G D (C 0 , C 1 )) := C α,β + ln(G D (C 0 , C 1 )). ( 6
θ j = 0 if j = D, a -1 D /2 if j = D.
It can be easily seen that θ ∈ E a .

Choice of θ 0 : The sequence θ 0 = (θ j,0 ) j∈N is chosen as follows

θ j,0 = 0 if j = D, a -1 D /2 + C α,β,D σa -1 D b -1 D /2 if j = D. Note that θ 0 ∈ E a as soon as C α,β,D σb -1 D ≤ 1. (6.47)
Indeed, using the standard inequality (x + y) 2 ≤ 2(x 2 + y 2 ), for x, y ∈ R, we immediately get

j∈N a 2 j θ 2 j,0 = a 2 D θ 2 D,0 ≤ a 2 D 2 a -2 D 4 + 2C 2 α,β,D σ 2 b 2 D a -2 D 4 ≤ 1 2 + C 2 α,β,D σ 2 2b 2 D ≤ 1, (6.48)
as soon as (6.47) is satisfied. Furthermore, as soon as (6.47) is satisfied, it is easily seen that θ -θ 0 ∈ E a .

Moreover, for the specific choices of θ and θ 0 given above, it is immediately seen that

|θ D -θ D,0 | = C α,β,D σb -1 D |θ D | ⇔ θ -θ 0 = C α,β,D 2 σb -1 D a -1 D .
In other words, we have proved that for all D ∈ N satisfying (6.47) then

β 0,σ,b (Θ(r σ,D ), B(b)) > β, where r σ,D = C α,β,D 2 σb -1 D a -1 D ,
for any given β ∈]0, 1 -α[. This implies that, for every ρ > 0, β 0,σ,b (Θ(ρ), B(b)) > β as soon as

ρ ≤ C α,β,D 2 σb -1 D a -1 D for some D ∈ N : C α,β,D σb -1 D ≤ 1,
which holds, as soon as

ρ ≤ C α,β 4 σb -1 D a -1 D for some 1 ≤ D ≤ M 2 ,
on noting that

M 2 := sup D ∈ N : C α,β σ|b -1 D | ≤ 2 and G D (C 0 , C1) ≥ 1 1 + 4(1 -α -β) 2 ,
and that

C α,β,D ≥ C α,β 2 , 1 ≤ D ≤ M 2 .
In particular,

β 0,σ,b (Θ(ρ), B(b)) > β for all ρ ≤ C α,β 4 σ max 1≤D≤M 2 [b -1 D a -1 D ]. Hence, r0,σ ≥ C α,β 4 σ max 1≤D≤M 2 [b -1 D a -1 D ].
This completes the proof of the proposition. The proof is splitted in two parts. We first show that rε,σ ≥ rε,0 and then show that rε,σ ≥ r0,σ .

Consider observations Y = (Y j ) j∈N from the GSM (4.8). Introduce the following goodnessof-fit testing algorithm:

• Generate a sequence X = ( Xj ) j∈N according to the GSM Xj = b j + σ ηj , j ∈ N, (6.49) where η = (η j ) j∈N is a sequence of independent standard Gaussian random variables (that is also independent of the sequence ξ = (ξ j ) j∈N ). (Note that the GSM (6.49) is an independent copy of the second equation in the GSM (1.1).)

• Let Ψα := Ψα (Y, X) be a given (non-randomized) α-level goodness-of-fit testing procedure based on observations (Y, X) = (Y j , Xj ) j∈N from the GSMs (4.8) and (6.49).

• Define the randomized test Ψ α := Ψ α (Y ) 2 as

Ψ α (Y ) := E[ Ψα | Y ],
where E[•] refers to expectation with respect to the independent standard Gaussian sequence η.

In particular, for every ε > 0 and σ > 0, the randomized test Ψ α is an α-level test. Indeed, This implies that for any α-level goodness-of-fit testing procedure Ψα , based on observations (Y, X) from the GSMs (4.8)-(6.49), we can associate an α-level goodness-of-fit testing procedure Ψ α , based on observations Y from the GSM (4.8), such that the separation radius of Ψ α is smaller than the separation radius of Ψα , i.e., r ε,0 (E a , Ψ α , β) ≤ r ε,σ (E a , Ψα , β).

α ε (Ψ α ) = E θ 0 ,b [Ψ α ] = E θ 0 ,b [E[ Ψα | Y ]] = E θ
2 a measurable function of the observation Y = (Yj) j∈N from the GSM (4.8) with values in the interval [0, 1]: for any given radius ρ > 0, the null hypothesis is rejected with probability Ψα(Y ) and it is not rejected with probability 1 -Ψα(Y ). In this case, αε(Ψα) := E θ 0 ,b (Ψα(Y )) and β ε (Θa(ρ), Ψα) := sup θ 0 ∈Ea θ-θ 0 ∈Θa(ρ) E θ,b (1 -Ψα(Y ))).

Hence, it is immediately seen that, for any α-level goodness-of-fit testing procedure Ψα , based on observations (Y, X) from the GSMs (4.8) and (6.49), rε,0 := inf Ψα: α ε,0 ( Ψα)≤α r ε,0 (E a , Ψα , β)

≤ r ε,0 (E a , Ψ α , β)

≤ r ε,σ (E a , Ψα , β), (6.52) implying that rε,0 ≤ rε,σ .

The proof of the assertion r0,σ ≤ rε,σ .

follows similarly, along the lines of the proof of the previous assertion, and it is therefore omitted. This completes the proof of (4.12)

Finally, (4.13) follows immediately form (4.12), taking into account (4.3) and (4.10). This completes the proof of the theorem.

Lower Bounds: Specific Cases

For the sake of convenience, we give the proof of each item (i)-(iv) in Theorem 4.2 in different sections.

Case (i): Mildly ill-posed problems with ordinary smooth functions

We assume that (6.28) holds true, i.e., b j ∼ j -t , t > 0, and a j ∼ j s , s > 0, j ∈ N. Proposition 6.5 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.28). Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies r2 ε,σ ε 4s 2s+2t+1/2 ∨ σ 2( s t ∧1) . (6.53)

Proof of Proposition 6.5 For the second term in (4.13), it is known that (see [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF], [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF]), Consider now the first term in (4.13). If s > t, then the sequence {b -1 j a -1 j } j∈N is nonincreasing and, thus,

C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∼ σ 2 .
On the other hand, if s ≤ t, then the sequence {b -1 j a -1 j } j∈N is non-decreasing. Hence, thanks to (4.4), σ 2 ∼ b 2 M 2 ⇔ M 2 ∼ σ -1/t , and, thus,

C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∼ σ 2 b -2 M 2 a -2 M 2 ∼ a -2 M 2 ∼ σ 2s t .
Combining the above terms, we arrive at (6.53). This completes the proof of the proposition. We assume that (6.32) holds true, i.e., b j ∼ j -t , t > 0, and a j ∼ exp{js}, s > 0, j ∈ N.

Proposition 6.6 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.32). Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies r2 ε,σ ε 2 [ln (1/ε)] (2t+ 1 2 ) ∨ σ 2 . (6.54)

Proof of Proposition 6.6 For the second term in (4.13), it is known that (see [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF], [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF]), Consider now the first term in (4.13). Then, the sequence {b -1 j a -1 j } j∈N is non-increasing for each s, t > 0, and, thus,

C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∼ σ 2 .
Combining the above terms, we arrive at (6.54). This completes the proof of the proposition.

Case (iii): Severely ill-posed problems with ordinary smooth functions

We assume that (6.34) holds true, i.e., b j ∼ exp{-jt}, t > 0, and a j ∼ j s , s > 0, j ∈ N.

Proposition 6.7 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.34).

Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies r2 ε,σ

[ln (1/ε)] -2s ∨ [ln (1/σ)] -2s . (6.55)

Proof of Proposition 6.7 For the second term in (4.13), it is known that (see [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF], [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF]), Consider now the first term in (4.13). Then, the sequence {b -1 j a -1 j } j∈N is non-decreasing. Hence, thanks to (4.4),

σ 2 ∼ b 2 M 2 ⇔ M 2 ∼ 1 t ln (1/σ)
and, thus,

C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∼ σ 2 b -2 M 2 a -2 M 2 ∼ a -2 M 2 ∼ [ln (1/σ)] -2s .
Combining the above terms, we arrive at (6.55). This completes the proof of the proposition. We assume that (6.36) holds true, i.e., b j ∼ exp{-jt}, t > 0, and a j ∼ exp{js}, s > 0, j ∈ N.

Proposition 6.8 Assume that the sequences b = (b j ) j∈N and a = (a j ) j∈N are given by (6.36).

Then, there exists ε 0 , σ 0 ∈]0, 1[ such that, for all 0 < ε ≤ ε 0 and 0 < σ ≤ σ 0 , the minimax separation radius rε,σ satisfies r2 ε,σ ε 2s s+t ∨ σ 2( s t ∧1) . (6.56)

Proof of Proposition 6.8 For the second term in (4.13), it is known that (see [START_REF] Laurent | Non asymptotic minimax rates of testing in signal detection with heterogeneous variances[END_REF], [START_REF] Yu | Minimax signal detection in ill-posed inverse problems[END_REF]), Consider now the first term in (4.13). If s > t, then the sequence {b -1 j a -1 j } j∈N is nonincreasing and, thus,

C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∼ σ 2 .
On the other hand, if s ≤ t, then the sequence {b -1 j a -1 j } j∈N is non-decreasing. Hence, thanks to (4.4),

σ 2 ∼ b 2 M 2 ⇔ M 2 ∼ 1 t ln (1/σ) ,
and, thus,

C 2 α,β 16
σ 2 max 1≤D≤M 2 [b -2 D a -2 D ] ∼ σ 2 b -2 M 2 a -2 M 2 ∼ a -2 M 2 ∼ σ 2s t .
Combining the above terms, we arrive at (6.53). This completes the proof of the proposition.

Figure 3 . 1 :

 31 Figure 3.1: An illustration of the spatial positions of the bandwidths M 0 and M 1 , defined in (3.3).The decreasing solid curve corresponds to the values of the sequence b = (b j ) j∈N with respect to the index j ∈ N, while the oscillating curve demonstrates one realization of the random sequence X = (X j ) j∈N according to the GSM (1.1). The increasing dashed curve draws the behavior of the sequence σh j . For the corresponding random 'bandwidth' M defined in (3.3), Lemma 6.1 shows that M ∈ [M 0 , M 1 [ with high probability.

Theorem 4 . 1

 41 Consider the GSMs (1.1), (4.1) and (4.7). Denote by rε,σ , r0,σ and rε,0 the corresponding minimax separation radii. Then, for every ε > 0 and σ > 0, rε,σ ≥ r0,σ ∨ rε,0 .

Lemma 6 . 1

 61 Let M, M 0 , M 1 be defined as in (3.1)-(3.5) where α ∈]0, 1[ and κ ≥ exp(1) are fixed values. Define the event M = {M 0 ≤ M < M 1 }. (6.1)

Lemma 6 . 2

 62 Let M be defined as in (3.1) and (3.4) where α ∈]0, 1[ and κ ≥ exp(1) are fixed values. Define the event

Lemma 6 . 3

 63 Let θ ∈ E a be given. Let M be defined as in (3.1) and (3.4) where α ∈]0, 1[ and κ ≥ exp(1) are fixed values. Then, for any σ ∈]0, 1[ and for any D ∈ N,

6. 2 . 1 1

 211 Proof of Proposition 3.By definition, α ε,σ (Ψ D,M ) := P θ 0 ,b (Ψ D,M = 1) = P θ 0 ,b (T D,M > t 1-α,D (X)).

Figure 6 . 1 : 2 D∧M 0 σ 2 2 D∧M 1 j=1 b - 4 j

 6120224 Figure 6.1: [Case I: a -2 D∧M 0

1 σh 1 ,

 11 Proof of Proposition 6.3 In a first time, we determine the order of the bandwidths M 0 and M M1 ∼ σ ln( M1 ) = σ ln 1 t ln 1

6. 4 . 1

 41 Proof of Proposition 4.1

  .46) (Note that, according to (4.4), for all D ≤ M 2 , C α,β,D ≥ C α,β /2.)

6. 4 . 2

 42 Proof of Theorem 4.1

  0 ,b [ Ψα ] = P θ 0 ,b ( Ψα = 1) = α, (6.50) since Ψα is an α-level test.Let θ ∈ l 2 (N) and θ -θ 0 ∈ E a be fixed. Then, the associated second kind error probability satisfiesE θ,b (1 -Ψ α (Y )) = E θ,b (1 -E[ Ψα | Y ]) = E θ,b (1 -Ψα ) = P θ,b ( Ψα = 0) ≤ β,(6.51)as soon as θ -θ 0 ≥ r ǫ,σ (E a , Ψα , β).

6. 5 . 2

 52 Case (ii): Mildly ill-posed problems with super smooth functions

  ln (1/ε)] -2s .

6. 5 . 4

 54 Case (iv): Severely ill-posed problems with super smooth functions

For the sake of brevity, these quantities are made precise in the subsequent sections

(1/σ) .