Pierre Neuvial

Etienne Roquain

ON FALSE DISCOVERY RATE THRESHOLDING FOR CLASSIFICATION UNDER SPARSITY

Keywords: False discovery rate, sparsity, classification, multiple testing, Bayes' rule, adaptive procedure, oracle inequality

We study the properties of false discovery rate (FDR) thresholding, viewed as a classification procedure. The "0"-class (null) is assumed to have a known density while the "1"-class (alternative) is obtained from the "0"-class either by translation or by scaling. Furthermore, the "1"-class is assumed to have a small number of elements w.r.t. the "0"-class (sparsity). We focus on densities of the Subbotin family, including Gaussian and Laplace models. Nonasymptotic oracle inequalities are derived for the excess risk of FDR thresholding. These inequalities lead to explicit rates of convergence of the excess risk to zero, as the number m of items to be classified tends to infinity and in a regime where the power of the Bayes rule is away from 0 and 1. Moreover, these theoretical investigations suggest an explicit choice for the target level αm of FDR thresholding, as a function of m. Our oracle inequalities show theoretically that the resulting FDR thresholding adapts to the unknown sparsity regime contained in the data. This property is illustrated with numerical experiments.

Introduction.

1.1. Background. In many high-dimensional settings, such as microarray or neuro-imaging data analysis, we aim at detecting signal among several thousands of items (e.g., genes or voxels). For such problems, a standard error measure is the false discovery rate (FDR), which is defined as the expected proportion of errors among the items declared as significant.

Albeit motivated by pure testing considerations, the Benjamini-Hochberg FDR controlling procedure [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF] has recently been shown to enjoy remarkable properties as an estimation procedure [START_REF] Abramovich | Adapting to unknown sparsity by controlling the false discovery rate[END_REF][START_REF] Donoho | Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data[END_REF]. More specifically, it turns out to be adaptive to the amount of signal contained in the data, which has been referred to as "adaptation to unknown sparsity."

In a classification framework, while [START_REF] Genovese | Operating characteristics and extensions of the false discovery rate procedure[END_REF] contains what is to our knowledge the first analysis of FDR thresholding with respect to the mis-classification risk, an important theoretical breakthrough has recently been made by Bogdan et al. [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]; see also [START_REF] Bogdan | A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing[END_REF]. The major contribution of Bogdan et al. [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] is to create an asymptotic framework in which several multiple testing procedures can be compared in a sparse Gaussian scale mixture model. In particular, they proved that FDR thresholding is asymptotically optimal (as the number m of items goes to infinity) with respect to the mis-classification risk and thus adapts to unknown sparsity in that setting (for a suitable choice of the level parameter α m ). Also, they proposed an optimal choice for the rate of α m as m grows to infinity.

The present paper can be seen as an extension of [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]. First, we prove that the property of adaptation to unknown sparsity also holds nonasymptotically, by using finite sample oracle inequalities. This leads to a more accurate asymptotic analysis, for which explicit convergence rates can be provided. Second, we show that these theoretical properties are not specific to the Gaussian scale model, but carry over to Subbotin location/scale models. They can also be extended to (fairly general) log-concave densities (as shown in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]), but we choose to focus on Subbotin densities in the main manuscript for simplicity. Finally, we additionally supply an explicit, finite sample, choice of the level α m and provide an extensive numerical study that aims at illustrating graphically the property of adaptation to unknown sparsity. 1.2. Initial setting. Let us consider the following classification setting: let (X i , H i ) ∈ R × {0, 1}, 1 ≤ i ≤ m, be m i.i.d. variables. Assume that the sample X 1 , . . . , X m is observed without the labels H 1 , . . . , H m and that the distribution of X 1 conditionally on H 1 = 0 is known a priori. We consider the following general classification problem: build a (measurable) classification rule ĥm : R → {0, 1}, depending on X 1 , . . . , X m , such that the (integrated) misclassification risk R m ( ĥm ) is as small as possible. We consider two possible choices for the risk R m (•):

R T m ( ĥm ) = E m -1 m i=1 1{ ĥm (X i ) = H i } ; (1) R I m ( ĥm ) = P( ĥm (X m+1 ) = H m+1 ), (2)
where the expectation is taken with respect to (X i , H i ) 1≤i≤m in [START_REF] Abramovich | Adapting to unknown sparsity by controlling the false discovery rate[END_REF] and to (X i , H i ) 1≤i≤m+1 in [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF], for a new labeled data point (X m+1 , H m+1 ) ∼ (X 1 , H 1 ) independent of (X i , H i ) 1≤i≤m . The risks R T m ( ĥm ) and R I m ( ĥm ) are usually referred to as transductive and inductive risks, respectively; see Remark 1.1 for a short discussion on the choice of the risk. Note that these two risks can be different in general because X i appears "twice" in ĥm (X i ). However, they coincide for procedures of the form ĥm (•) = h m (•), where h m : R → {0, 1} is a deterministic function. The methodology investigated here can also be easily extended to a class of weighted mis-classification risks, as originally proposed by Bogdan et al. [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] (in the case of the transductive risk) and further discussed in Section 6.2.

The distribution of (X 1 , H 1 ) is assumed to belong to a specific parametric subset of distributions on R × {0, 1}, which is defined as follows:

(i) the distribution of H 1 is such that the (unknown) mixture parameter τ m = π 0,m /π 1,m satisfies τ m > 1, where π 0,m = P(H 1 = 0) and π 1,m = P(H 1 = 1) = 1 -π 0,m ; (ii) the distribution of X 1 conditionally on H 1 = 0 has a density d(•) w.r.t.

the Lebesgue measure on R that belongs to the family of so-called ζ-Subbotin densities, parametrized by ζ ≥ 1, and defined by

d(x) = (L ζ ) -1 e -|x| ζ /ζ (3) with L ζ = +∞ -∞ e -|x| ζ /ζ dx = 2Γ(1/ζ)ζ 1/ζ-1 ;
(iii) the distribution of X 1 conditionally on H 1 = 1 has a density d 1,m (•) w.r.t. the Lebesgue measure on R of either of the following two types:

-location: d 1,m (x) = d(x -µ m ), for an (unknown) location parameter µ m > 0; -scale: d 1,m (x)=d(x/σ m )/σ m , for an (unknown) scale parameter σ m >1.

The density d is hence of the form d(x) = e -φ (|x|) where φ(u) = u ζ /ζ + log(L ζ ) is convex on R + (log-concave density). This property is of primary interest when applying our methodology; see the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. The particular values ζ = 1, 2 give rise to the Laplace and Gaussian case, respectively. The classification problem under investigation is illustrated by Figure 1 (left panel), in the Gaussian location case. Moreover, let us note that we will exclude in our study the Laplace location model (i.e., the location model using ζ = 1). This particular model is not directly covered by our methodology and needs specific investigations; see Section 10.3 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF].

Our modeling is motivated by the following application: consider a microarray experiment for which measurements Z 1 , . . . , Z m for m genes are observed, each corresponding to a difference of expression levels between two experimental conditions (e.g., test versus reference sample). Let H 1 , . . . , H m be binary variables coded as 1 if the gene is differentially expressed and 0 for αm = 0.3 (dashed line). Right: illustration of the FDR algorithm for αm = 0.3; k ∈ {1, . . . , m} → Φ -1 (αmk/m) (solid line); X (k) 's (crosses); ŝFDR m (dashed horizontal line); k = 5 (dashed vertical line). Here, Φ(x) = P(X ≥ x) for X ∼ N (0, 1). m = 18; µm = 3; τm = 5. For this realization, 5 labels "1" and 13 labels "0."

if not. Assume that each Z i is N (δ i , σ 2
ε ) where δ i is the (unknown) effect for gene i while σ 2 ε quantifies the (known) measurement error. Next, assume the Bayesian paradigm that sets the following prior distribution for δ i : the distribution of δ i is N (0, σ 2 0 ) conditionally on H i = 0 and N (δ, σ 2 0 + τ 2 ) conditionally on H i = 1. Generally, σ 2 0 (≥0), the dispersion of the nondifferentially expressed genes, is assumed to be known while δ (≥0) and τ 2 (≥0), the shift and additional dispersion of the nondifferentially expressed genes, are unknown. Let X i = Z i /σ for σ 2 = σ 2 ε + σ 2 0 and consider the distribution unconditionally on the δ i 's. This corresponds to our model (in the Gaussian case) as follows:

-δ > 0 and τ 2 = 0: location model with µ m = δ/σ > 0; -δ = 0 and τ 2 > 0: scale model with σ 2 m = (σ 2 + τ 2 )/σ 2 > 1. The above convolution argument was originally proposed in [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] for a Gaussian scale model: it explains how we can obtain test statistics that have the same distribution under the alternative even if the effects of the measurements are not equal.

Going back to our general setting, an important point is that the parameters-(τ m , µ m ) in the location model, or (τ m , σ m ) in the scale model-are assumed to depend on sample size m. The parameter τ m , called the sparsity parameter, is assumed to tend to infinity as m tends to infinity, which means that the unlabeled sample only contains a small, vanishing proportion of label 1. This condition is denoted (Sp). As a counterpart, the other parameter-µ m in the location model, or σ m in the scale model-is assumed to tend to infinity fast enough to balance sparsity. This makes the problem "just solvable" under the sparsity constraint. More precisely, our setting corresponds to the case where the power of the Bayes procedure is bounded away from 0 and 1, and is denoted (BP). This is motivated by sparse highdimensional problems for which the signal is strong but only carried by a small part of the data. For instance, in the above-mentioned application to microarray data, the two experimental conditions compared can be so close that only a very small proportion of genes are truly differentially expressed (e.g., two groups of patients having the same type of cancer but a different response to a cancer treatment [START_REF] Sawyers | The cancer biomarker problem[END_REF]).

Remark 1.1. Our setting is close to the semi-supervised novelty detection (SSND) framework proposed in [START_REF] Blanchard | Semi-supervised novelty detection[END_REF], for which the knowledge of the distribution X 1 conditionally on H 1 = 0 is replaced by the observation of a finite i.i.d. sample with this distribution. In the latter work, the authors use the unlabeled data X 1 , . . . , X m to design a procedure ĥm that aims at classifying a new unlabeled data X m+1 . This approach is in accordance with the inductive risk defined by [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF]. However, in other situations closer to standard multiple testing situations, one wants to classify X 1 , . . . , X m meanwhile designing ĥm . This gives rise to the transductive risk defined by (1). 1.3. Thresholding procedures. Classically, the solution that minimizes the misclassification risks (1) and ( 2) is the so-called Bayes rule h B m that chooses the label 1 whenever d 1,m (x)/d(x) is larger than a specific threshold. We easily check that the likelihood ratio d 1,m (x)/d(x) is nondecreasing in x and |x| for the location and the scale model, respectively. As a consequence, we can only focus on classification rules ĥm (x) of the form 1{x ≥ ŝm }, ŝm ∈ R for the location model and 1{|x| ≥ ŝm }, ŝm ∈ R + for the scale model. Therefore, to minimize the mis-classification risks, thresholding procedures are classification rules of primary interest, and the main challenge consists of choosing the threshold ŝm in function of X 1 , . . . , X m .

The FDR controlling method proposed by Benjamini and Hochberg [2] (also called "Benjamini-Hochberg" thresholding) provides such a thresholding ŝm in a very simple way once we can compute the quantile function

D -1 (•), where D(u) = (L ζ ) -1 +∞ u e -|x| ζ /ζ
dx is the (known) upper-tail cumulative distribution function of X 1 conditionally on H 1 = 0. We recall below the algorithm for computing the FDR threshold in the location model (using test statistics rather than p-values).

Algorithm 1.2. (1) choose a nominal level α m ∈ (0, 1);

(2) consider the order statistics of the X k 's: X

(1) ≥ X (2) ≥ • • • ≥ X (m) ; (3) take the integer k = max{1 ≤ k ≤ m : X (k) ≥ D -1 (α m k/m)} when this set is nonempty and k = 1 otherwise; (4) use ĥFDR m (x) = 1{x ≥ ŝFDR m } for ŝFDR m = D -1 (α m k/m).
For the scale model, FDR thresholding has a similar form, ĥFDR

m (x) = 1{|x| ≥ ŝFDR m } for ŝFDR m = D -1 (α m k/(2m)), where k = max{1 ≤ k ≤ m : |X| (k) ≥ D -1 (α m k/(2m))} ( k = 1 if the set is empty) and |X| (1) ≥ |X| (2) ≥ • • • ≥ |X| (m) . Algorithm 1.2 is illustrated in Figure 1 (right panel), in a Gaussian location setting. Since ŝFDR m = D -1 (α m k/m) takes its values in the range [D -1 (α m ), D -1 (α m /m)],
it can be seen as an intermediate thresholding rule between the Bonferroni thresholding [D -1 (α m /m)] and the uncorrected thresholding [D -1 (α m )]. Finally, an important feature of the FDR procedure is that it depends on a pre-specified level α m ∈ (0, 1). In this work, the level α m is simply used as a tuning parameter, chosen to make the corresponding misclassification risk as small as possible. This contrasts with the standard philosophy of (multiple) testing for which α m is meant to be a bound on the error rate and thus is fixed in the overall setting.

1.4. Aim and scope of the paper. Let R m (•) be the risk defined either by [START_REF] Abramovich | Adapting to unknown sparsity by controlling the false discovery rate[END_REF] or [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF]. In this paper, we aim at studying the performance of FDR thresholding ĥm = ĥFDR m as a classification rule in terms of the excess risk R m ( ĥm ) -R m (h B m ) both in location and scale models. We investigate two types of theoretical results: (i) Nonasymptotic oracle inequalities: prove for each (or some) m ≥ 2, an inequality of the form

R m ( ĥm ) -R m (h B m ) ≤ b m , (4)
where b m is an upper-bound which we aim to be "as small as possible." Typically, b m depends on ζ, α m and on the model parameters.

(ii) Convergence rates: find a sequence (α m ) m for which there exists D > 0 such that for a large m,

R m ( ĥm ) -R m (h B m ) ≤ D × R m (h B m ) × ρ m (5)
for a given rate ρ m = o(1).

Inequality (4) is of interest in its own right, but is also used to derive inequalities of type [START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF], which are of asymptotic nature. Property (5) is called "asymptotic optimality at rate ρ m ." It implies that R m ( ĥm ) ∼ R m (h B m ); that is, ĥm is "asymptotically optimal," as defined in [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]. However, (5) is substantially more informative because it provides a rate of convergence.

It should be emphasized at this point that the trivial procedure ĥ0 m ≡ 0 (which always chooses the label "0") satisfies [START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF] with ρ m = O(1) [under our setting (BP)]. Therefore, proving [START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF] with ρ m = O(1) is not sufficient to get an interesting result, and our goal is to obtain a rate ρ m that tends to zero in [START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF]. The reason for which ĥ0 m is already "competitive" is that we consider a sparse model in which the label "0" is generated with high probability.

1.5. Overview of the paper. First, Section 2 presents a more general setting than the one of Section 1.2. Namely, the location and scale models are particular cases of a general "p-value model" after a standardization of the original X i 's into p-values p i 's. While the "test statistic" formulation is often considered as more natural than the p-value one for many statisticians, the p-value formulation will be very convenient to provide a general answer to our problem. The so-obtained p-values are uniformly distributed on (0, 1) under the label 0 while they follow a distribution with decreasing density f m under the label 1. Hence, procedures of primary interest (including the Bayes rule) are p-value thresholding procedures that choose label 1 for p-values smaller than some threshold tm . Throughout the paper, we focus on this type of procedures, and any procedure ĥm is identified by its corresponding threshold tm in the notation. Translated into this "p-value world," we describe in Section 2 the Bayes rule, the Bayes risk, condition (BP), BFDR and FDR thresholding.

The fundamental results are stated in Section 3 in the general p-value model. Following [START_REF] Abramovich | Adapting to unknown sparsity by controlling the false discovery rate[END_REF][START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF][START_REF] Donoho | Higher criticism for detecting sparse heterogeneous mixtures[END_REF][START_REF] Donoho | Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data[END_REF], as BFDR thresholding is much easier to study than FDR thresholding from a mathematical point of view, the approach advocated here is as follows: first, we state an oracle inequality for BFDR; see Theorem 3.1. Second, we use a concentration argument of the FDR threshold around the BFDR threshold to obtain an oracle inequality of the form (4); see Theorem 3.2. At this point, the bounds involve quantities that are not written in an explicit form, and that depend on the density f m of the p-values corresponding to the label 1.

The particular case where f m comes either from a location or a scale model is investigated in Section 4. An important property is that in these models, the upper-tail distribution function D(•) and the quantile function D -1 (•) can be suitably bounded; see Section 12 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. By using this property, we derive from Theorems 3.1 and 3.2 several inequalities of the form (4) and [START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF]. In particular, in the sparsity regime τ m = m β , 0 < β ≤ 1, we derive that the FDR threshold tFDR m at level α m is asymptotically optimal [under (BP) and (Sp)] in either of the following two cases:

-for the location model, ζ > 1, if α m → 0 and log α m = o((log m) 1-1/ζ ); -for the scale model, ζ ≥ 1, if α m → 0 and log α m = o(log m).
The latter is in accordance with the condition found in [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] in the Gaussian scale model. Furthermore, choosing

α m ∝ 1/(log m) 1-1/ζ (location) or α m ∝ 1/(log m) (scale) provides a convergence rate ρ m = 1/(log m) 1-1/ζ (location) or ρ m = 1/(log m) (scale), respectively.
At this point, one can argue that the latter convergence results are not fully satisfactory: first, these results do not provide an explicit choice for α m for a given finite value of m. Second, the rate of convergence ρ m being rather slow, we should check numerically that FDR thresholding has reasonably good performance for a moderately large m.

We investigate the choice of α m by carefully studying Bayes' thresholding and how it is related to BFDR thresholding; see Sections 2.4 and 4.4. Next, for this choice of α m , the performance of FDR thresholding is evaluated numerically in terms of (relative) excess risk, for several values of m; see Section 5. We show that the excess risk of FDR thresholding is small for a remarkably wide range of values for β, and increasingly so as m grows to infinity. This illustrates the adaptation of FDR thresholding to the unknown sparsity regime. Also, for comparison, we show that choosing α m fixed with m (say, α m ≡ 0.1) can lead to higher FDR thresholding excess risk.

2. General setting.

2.1. p-value model. Let (p i , H i ) ∈ [0, 1] × {0, 1}, 1 ≤ i ≤ m, be m i.i.d.
variables. The distribution of (p 1 , H 1 ) is assumed to belong to a specific subset of distributions on [0, 1] × {0, 1}, which is defined as follows:

(i) same as (i) in Section 1.2; (ii) the distribution of p 1 conditionally on H 1 = 0 is uniform on (0, 1); (iii) the distribution of p 1 conditionally on H 1 = 1 has a c.d.f. F m satisfying F m is continuously increasing on [0, 1] and differentiable on (0, 1),

f m = F m is continuously decreasing with f m (0 + ) > τ m > f m (1 -). (A(F m , τ m ))
This way, we obtain a family of i.i.d. p-values, where each p-value has a marginal distribution following the mixture model

p i ∼ π 0,m U (0, 1) + π 1,m F m . (6)
Model ( 6) is classical in the multiple testing literature and is usually called the "two-group mixture model." It has been widely used since its introduction by [START_REF] Efron | Empirical Bayes analysis of a microarray experiment[END_REF] [START_REF] Efron | Empirical Bayes analysis of a microarray experiment[END_REF]; see, for instance, [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF][START_REF] Efron | Microarrays, empirical Bayes and the two-groups model[END_REF][START_REF] Genovese | A stochastic process approach to false discovery control[END_REF][START_REF] Storey | The positive false discovery rate: A Bayesian interpretation and the q-value[END_REF].

The models presented in Section 1.2 are particular instances of this p-value model. In the scale model, we apply the standardization

p i = 2D(|X i |), which yields F m (t) = 2D(D -1 (t/2)/σ m ). In the location model, we let p i = D(X i ), which yields F m (t) = D(D -1 (t)-µ m ).
We can easily check that in both cases (A(F m , τ m )) is satisfied (additionally assuming ζ > 1 for the location model), with f m (0 + ) = +∞ and f m (1 -) < 1 (scale) and f m (1 -) = 0 (location), as proved in Section 9.1 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF].

2.2. Procedures, risks and the Bayes threshold. A classification procedure is identified with a threshold tm ∈ [0, 1], that is, a measurable function of the p-value family (p i , i ∈ {1, . . . , m}). The corresponding procedure chooses label 1 whenever the p-value is smaller than tm . In the p-value setting, the transductive and inductive misclassification risks of a threshold tm can be written as follows:

R T m ( tm ) = m -1 m i=1 P(p i ≤ tm , H i = 0) + m -1 m i=1 P(p i > tm , H i = 1), (7)
R I m ( tm ) = E(π 0,m tm + π 1,m (1 -F m ( tm ))). ( 8
)
In the particular case of a deterministic threshold t m ∈ [0, 1], these two risks coincide and are equal to

R m (t m ) = π 0,m t m + π 1,m (1 -F m (t m ))
. The following lemma identifies a solution minimizing both risks ( 7) and ( 8).

Lemma 2.1. Let R m (•) being either R T m (•) or R I m (•). Under assumption (A(F m , τ m )), the threshold t B m = f -1 m (τ m ) ∈ (0, 1) (9) minimizes R m (•), that is, satisfies R m (t B m ) = min t m {R m ( t m )}
, where the minimum is taken over all measurable functions from [0, 1] m to [0, 1] that take as input the p-value family (p i , i ∈ {1, . . . , m}).

The threshold t B

m is called the Bayes threshold, and R m (t B m ) is called the Bayes risk. The Bayes threshold is unknown because it depends on τ m and on the data distribution f m .

Notation. In this paper, all the statements hold for both risks. Hence, throughout the paper, R m (•) denotes either R T m (•) defined by [START_REF] Bogdan | A comparison of the Benjamini-Hochberg procedure with some Bayesian rules for multiple testing[END_REF] or R I m (•) defined by (8).

2.3. Assumptions on the power of the Bayes rule and sparsity. Under assumption (A(F m , τ m )), let us denote the power of the Bayes procedure by

C m = F m (t B m ) ∈ (0, 1). ( 10
)
In our setting, we will typically assume that the signal is sparse while the power C m of the Bayes procedure remains away from 0 or 1:

∃(C -, C + ) s.t. ∀m ≥ 2, 0 < C -≤ C m ≤ C + < 1; (BP) (τ m ) m is such that τ m → +∞ as m → +∞. (Sp)
First note that assumption (Sp) is very weak: it is required as soon as we assume some sparsity in the data. As a typical instance, τ m = m β satisfies (Sp), for any β > 0. Next, assumption (BP) means that the best procedure is able to detect a "moderate" amount of signal. In [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF], a slightly stronger assumption has been introduced, ∃C ∈ (0, 1) s.t. C m → C as m tends to infinity, (VD) Fig. 2. Left: plot of the family of curves {t → t 1/(2+j/2) }j=0,...,56 (thin solid curves). Right: choice (thick solid curve) within the family of curves {t → t 1/σ }σ>1 that fulfills ( 9) and ( 10) for Cm = 1/2 (given by the dashed horizontal line) and τm = 2 (slope of the dashed oblique line). This gives σm 4. The Bayes threshold t B m is given by the dotted vertical line.

which is referred to as "the verge of detectability." Condition (BP) encompasses (VD) and is more suitable to state explicit finite sample oracle inequalities; see, for example, Remark 4.6 further on.

In the location (resp., scale) model, while the original parameters are (µ m , τ m ) [resp., (σ m , τ m )], the model can be parametrized in function of (C m , τ m ) by using ( 9) and [START_REF] Donoho | Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data[END_REF]. This way, F m is uniquely determined from (C m , τ m ) as follows: among the family of curves {D(D -1 (•) -µ)} µ∈R (resp., {2D(D -1 (•/2)/σ)} σ>1 ), F m is the unique curve such that the pre-image of C m has a tangent of slope τ m , that is, f m (F -1 m (C m )) = τ m . This is illustrated in Figure 2 for the Laplace scale model. In this case, D(x) = d(x) = e -x /2 for x ≥ 0 and thus F m (t) = t 1/σm , so that the family of curves is simply {t → t 1/σ } σ>1 .

Remark 2.2. Condition (BP) constrains the model parameters to be located in a very specific region. For instance, in the Gaussian location model with τ m = m β , condition (BP) implies that µ m ∼ √ 2β log m (see Table 3 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]), which corresponds to choosing (µ m , β) on the "estimation boundary," as displayed in Figure 1 of [START_REF] Donoho | Higher criticism for detecting sparse heterogeneous mixtures[END_REF].

BFDR m (t) = P(H i = 0|p i ≤ t) = π 0,m t G m (t) = (1 + τ -1 m F m (t)/t) -1 (11)
for any t ∈ (0, 1) and where G m (t) = π 0,m t+π 1,m F m (t). As introduced by [START_REF] Efron | Empirical Bayes methods and false discovery rates for microarrays[END_REF], the quantity defined by ( 11) is called "Bayesian FDR." It is not to be confounded with "Bayes FDR" defined by [START_REF] Sarkar | A general decision theoretic formulation of procedures controlling FDR and FNR from a Bayesian perspective[END_REF]. Also, under a two-class mix-ture model, BFDR m (t) coincides with the so-called "positive false discovery rate," itself connected to the original false discovery rate of [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF]; see [START_REF] Storey | The positive false discovery rate: A Bayesian interpretation and the q-value[END_REF] and Section 4 of [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF].

Under assumption (A(F m , τ m )), the function Ψ m : t ∈ (0, 1) → F m (t)/t is decreasing from f m (0 + ) to 1, with f m (0 + ) ∈ (1, +∞]. Hence, the following result holds.

Lemma 2.3. Assume (A(F m , τ m )) and α m ∈ ((1 + f m (0 + )/τ m ) -1 , π 0,m ). Then, equation BFDR m (t) = α m has a unique solution t = t m (α m ) ∈ (0, 1), given by t m (α m ) = Ψ -1 m (q m τ m ) (12) for q m = α -1 m -1 > 0 and Ψ m (t) = F m (t)/t.
The threshold t m (α m ) is called the BFDR threshold at level α m . Typically, it is well defined for any α m ∈ (0, 1/2), because π 0,m > 1/2 and f m (0 + ) = +∞2 in the Subbotin location and scale models (additionally assuming ζ > 1 for the location model). Obviously, the BFDR threshold is unknown because it depends on τ m and on the distribution of the data. However, its interest lies in that it is close to the FDR threshold which is observable. When not ambiguous, t m (α m ) will be denoted by t m for short.

Next, a quantity of interest in Lemma 2.3 is q m = α -1 m -1 > 0, called the recovery parameter (associated to α m ). As α m = (1 + q m ) -1 , considering α m or q m is equivalent. Since we would like to have

t m = Ψ -1 m (q m τ m ) close to t B m = f -1
m (τ m ), the recovery parameter can be interpreted as a correction factor that cancels the difference between Ψ m (t) = F m (t)/t and f m (t) = F m (t). Clearly, the best choice for the recovery parameter is such that t m = t B m , that is,

q opt m = τ -1 m Ψ m (f -1 m (τ m )) = C m τ m t B m , (13)
which is an unknown quantity, called the optimal recovery parameter. Note that from the concavity of F m , we have Ψ m (t) ≥ f m (t) and thus q opt m ≥ 1. As an illustration, for the Laplace scale model, we have σ m f m (t) = Ψ m (t) and thus the optimal recovery parameter is q opt m = σ m . The fact that q opt m ≥ 1 suggests to always choose q m ≥ 1 (i.e., α m ≤ 1/2) into the BFDR threshold. A related result is that taking any sequence (α m ) m such that α m ≥ α -> 1/2 for all m ≥ 2 never leads to an asymptotically optimal BFDR procedure; see Section 13 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF].

2.5. FDR thresholding. The FDR threholding procedure was introduced by Benjamini and Hochberg (1995) by proving that it controls the FDR; see [START_REF] Benjamini | Controlling the false discovery rate: A practical and powerful approach to multiple testing[END_REF]. From an historical perspective, it is interesting to note that this Fig. 3. Left: illustration of the FDR threshold ( 15): e.c.d.f. of the p-value (solid line), line of slope 1/αm (dotted line), the FDR threshold at level αm (X-coordinate of the vertical dashed dotted line). Right: illustration of the FDR threshold as an empirical surrogate for the BFDR threshold; compared to the left picture, we added the c.d.f. of the p-values (thick solid line), the BFDR threshold at level αmπ0,m (dotted vertical line) and the Bayes threshold (dashed vertical line). In both panels, we consider the Laplace scale model with Cm = 0.5; m = 50; β = 0.2; τm = m β ; σm 4.2; αm = 0.4.

procedure has a prior occurrence in a series of papers by Eklund (1961)(1962)(1963); see [START_REF] Seeger | A note on a method for the analysis of significances en masse[END_REF]. As noted by many authors (see, e.g., [START_REF] Abramovich | Adapting to unknown sparsity by controlling the false discovery rate[END_REF][START_REF] Efron | Empirical Bayes methods and false discovery rates for microarrays[END_REF][START_REF] Genovese | Operating characteristics and extensions of the false discovery rate procedure[END_REF][START_REF] Sen | Some remarks on Simes-type multiple tests of significance[END_REF][START_REF] Storey | A direct approach to false discovery rates[END_REF]), this thresholding rule can be expressed as a function of the empirical c.d.f. G m of the p-values in the following way: for any α m ∈ (0, 1),

tBH m (α m ) = max{t ∈ [0, 1] : G m (t) ≥ t/α m }. ( 14
)
We simply denote tBH m (α m ) by tBH m when not ambiguous. Classically, this implies that t = tBH m solves the equation G m (t) = t/α m [this can be easily shown by using [START_REF] Ferreira | On the Benjamini-Hochberg method[END_REF] together with the fact that G m (•) is a nondecreasing function]. Hence, according to Lemma 2.3 and as already mentioned in the literature (see [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]), tBH m can be seen as an empirical counterpart of the BFDR threshold at level α m π 0,m , in which the theoretical c.d.f. G m (t) = π 0,m t + π 1,m F m (t) of the p-values has been replaced by the empirical c.d.f. G m of the p-values. Next, once α m has been chosen, [START_REF] Ferreira | On the Benjamini-Hochberg method[END_REF] only involves observable quantities, so that the threshold tBH m only depends on the data. This is further illustrated on the left panel of Figure 3. Also, as already observed in Section 5.2 of [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF], since the BH procedure is never more conservative than the Bonferroni procedure, the following modification of tBH m can be proposed:

Definition 2.4.
The FDR threshold at level α m is defined by

tFDR m (α m ) = tBH m (α m ) ∨ (α m /m), (15)
where tBH m (α m ) is defined by [START_REF] Ferreira | On the Benjamini-Hochberg method[END_REF].

We simply denote tFDR m (α m ) by tFDR m when not ambiguous. The threshold tFDR m is the one that we use throughout this paper. This modification does not change the risk

R T m (•), that is, R T m ( tBH m ) = R T m ( tFDR m ), but can affect the risk R I m (•), that is, R I m ( tBH m ) = R I m ( tFDR m )
, in general. Finally, while relation [START_REF] Finner | On the false discovery rate and an asymptotically optimal rejection curve[END_REF] uses p-values whereas the algorithms defined in Section 1.3 use test statistics, it is easy to check that the resulting procedures are the same.

Remark 2.5 (Adaptive FDR procedures under sparsity). To get a better FDR controlling procedure, one classical approach is to modify (15) by dividing α m by a (more or less explicit) estimator of π 0,m and by possibly using a step-up-down algorithm; see, for example, [START_REF] Benjamini | Adaptive linear stepup procedures that control the false discovery rate[END_REF][START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF][START_REF] Finner | On the false discovery rate and an asymptotically optimal rejection curve[END_REF][START_REF] Gavrilov | An adaptive step-down procedure with proven FDR control under independence[END_REF][START_REF] Sarkar | Some results on false discovery rate in stepwise multiple testing procedures[END_REF][START_REF] Sarkar | On methods controlling the false discovery rate[END_REF][START_REF] Tamhane | A generalized step-up-down multiple test procedure[END_REF]. However, this method seems not helpful in our sparse setting because π 0,m is very close to 1. As a result, we focus in this paper only on the original (nonadaptive) version of FDR thresholding (15).

3.

Results in the general model. This section presents relations of the form (4) and ( 5) for the BFDR and FDR thresholds. Our first main result deals with the BFDR threshold. Theorem 3.1. Assume (A(F m , τ m )) and consider the BFDR threshold t m at a level α m ∈ ((1 + f m (0 + )/τ m ) -1 , π 0,m ) corresponding to a recovery parameter q m = α -1 m -1. Consider q opt m ≥ 1 the optimal recovery parameter given by [START_REF] Efron | Empirical Bayes analysis of a microarray experiment[END_REF]. Then the following holds:

(i) if α m ≤ 1/2, we have for any m ≥ 2, R m (t m ) -R m (t B m ) ≤ π 1,m {(C m /q m -C m /q opt m ) ∨ γ m }, (16)
where we let γ m = (C m -F m (Ψ -1 m (q m τ m ))) + . In particular, under (BP), if α m → 0 and γ m → 0, the BFDR threshold t m is asymptotically optimal at rate ρ m = α m + γ m .

(ii) we have for any

m ≥ 2, R m (t m ) R m (t B m ) ≥ π 1,m R m (t B m ) (1 -(1 -q -1 m ) + F m (q -1 m τ -1 m )). ( 17
)
In particular, under (BP), if R m (t B m ) ∼ π 1,m (1 -C m ) and if q m is chosen such that lim inf m 1 -(1 -q -1 m ) + F m (q -1 m τ -1 m ) 1 -C m > 1, ( 18
)
t m is not asymptotically optimal.

Theorem 3.1 is proved in Section 7. Theorem 3.1(i) presents an upperbound for the excess risk when choosing q m instead of q opt m in BFDR thresholding. First, both sides of ( 16) are equal to zero when q m = q opt m . Hence, this bound is sharp in that case. Second, assumption "α m ≤ 1/2" in The-orem 3.1(i) is only a technical detail that allows us to get C m /q m instead of 1/q m in the right-hand side of ( 16) (moreover, it is quite natural; see the end of Section 2.4). Third, γ m has a simple interpretation as the difference between the power of Bayes' thresholding and BFDR thresholding. Fourth, bound ( 16) induces the following trade-off for choosing α m : on the one hand, α m has to be chosen small enough to make C m /q m small; on the other hand, γ m increases as α m decreases to zero. Finally note that, in Theorem 3.1(i), the second statement is a consequence of the first one because R m (t B m ) ≥ π 1,m (1 -C m ). Theorem 3.1(ii) states lower bounds which are useful to identify regimes of α m that do not lead to an asymptotically optimal BFDR thresholding; see Corollary 4.4(i) further on.

Our second main result deals with FDR thresholding.

Theorem 3.2. Let ε ∈ (0, 1), assume (A(F m , τ m )) and consider the FDR threshold tFDR

m at level α m > (1 -ε) -1 (π 0,m + π 1,m f m (0 + )) -1 . Then the following holds: for any m ≥ 2, R m ( tFDR m ) -R m (t B m ) ≤ π 1,m α m 1 -α m + m -1 α m (1 -α m ) 2 (19) + π 1,m {γ m ∧ (γ ε m + e -mε 2 (τm+1) -1 (Cm-γ ε m )/4 )} for γ ε m = (C m -F m (Ψ -1 m (q ε m τ m ))) + with q ε m = (α m π 0,m (1 -ε)) -1 -1 and γ m = (C m -F m (α m /m)) + .
In particular, under (BP) and assuming α m → 0: (i) if m/τ m → +∞, γ ε m → 0 and additionally ∀κ > 0, e -κm/τm = o(γ ε m ), the FDR threshold tFDR m is asymptotically optimal at rate ρ m = α m + γ ε m ; (ii) if m/τ m → ∈ (0, +∞) with γ m → 0, the FDR threshold tFDR m is asymptotically optimal at rate ρ m = α m + γ m . Theorem 3.2 is proved in Section 7. The proof mainly follows the methodology of [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF], but is more general and concise. It relies on tools developed in [14-16, 18, 22, 23]. The main argument for the proof is that the FDR threshold tFDR m (α m ) is either well concentrated around the BFDR threshold t m (α m π 0,m ) (as illustrated in the right panel of Figure 3) or close to the Bonferroni threshold α m /m. This argument was already used in [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF].

Let us comment briefly on Theorem 3.2: first, as in the BFDR case, choosing α m such that the bound in [START_REF] Genovese | A stochastic process approach to false discovery control[END_REF] is minimal involves a trade-off because γ ε m and γ m are quantities that increase when α m decreases to zero. Second, let us note that cases (i) and (ii) in Theorem 3.2 are intended to cover regimes where the FDR is close to BFDR (moderately sparse) and where the FDR threshold is close to the Bonferroni threshold (extremely sparse), respectively. In particular, these two regimes cover the case where τ m = m β with β ∈ (0, 1]. Finally, the bounds and convergence rates derived in Theorems 3.1 and 3.2 strongly depend on the nature of F m . We derive a more explicit expression of the latter in the next section, in the particular cases of location and scale models coming from a Subbotin density.

m τ m = Ψ m (t m ) ≥ f m (t m ), which yields γ m ≤ C m -F m (f -1 m (q m τ m )) ∈ [0, 1). ( 20
) When f -1
m is easier to use than Ψ -1 m , it is tempting to use [START_REF] Massart | Concentration Inequalities and Model Selection[END_REF] to upper bound the excess risk in Theorems 3.1 and 3.2. However, this can inflate the resulting upper-bound too much. This point is discussed in Section 10.4 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF] for the case of a Gaussian density (for which this results in an additional log log τ m factor in the bound).

Application to location and scale models.

4.1. The Bayes risk and optimal recovery parameter. A preliminary task is to study the behavior of t B m , R m (t B m ) and q opt m = C m /(τ m t B m ) both in location and scale models. While finite sample inequalities are given in Section 9.2 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF], we only report in this subsection some resulting asymptotic relations for short. Let us define the following rates, which will be useful throughout the paper:

r loc m = (ζ log τ m + |D -1 (C m )| ζ ) 1-1/ζ ; (21) r sc m = ζ log τ m + (D -1 (C m /2)) ζ . ( 22
)
Under (Sp), note that the rates r loc m (resp., r sc m ) tend to infinity. Furthermore, by using Section 9.2 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF], we have µ m = (r loc m ) 1/(ζ-1) -D -1 (C m ) in the location model and σ m ≥ (r sc m ) 1/ζ /(D -1 (C m /2)) in the scale model.

m ∼ r loc m ∼ (ζ log τ m ) 1/ζ and σ m ∼ (r sc m ) 1/ζ /(D -1 (C m /2)) ∼ (ζ log τ m ) 1/ζ /(D -1 (C m /2)) and R m (t B m ) ∼ π 1,m (1 -C m ), ( 23
)
t B m = O(R m (t B m )/r m ), ( 24
)
q opt m ∼        C m d(D -1 (C m )) (ζ log τ m ) 1-1/ζ (location), C m /2 D -1 (C m /2)d(D -1 (C m /2)) ζ log τ m (scale). (25)
From ( 23) and ( 24), by assuming (BP) and (Sp), the probability of a type I error (π 0,m t B m ) is always of smaller order than the probability of a type II error (π 1,m (1 -C m )). The latter had already been observed in [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] in the particular case of a Gaussian scale model. Remark 4.2. From ( 23) and since the risk of null thresholding is R m (0) = π 1,m , a substantial improvement over the null threshold can only be expected in the regime where C m ≥ C -, where C -is "far" from 0.

Finite sample oracle inequalities.

The following result can be derived from Theorem 3.1(i) and Theorem 3.2. It is proved in Section 9.3 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. m [defined by [START_REF] Roquain | Type I error rate control for testing many hypotheses: A survey with proofs[END_REF]] and

K m = 2D -1 (C m /2)d(D -1 (C m /2)
) in the scale model. Let α m ∈ (0, 1/2) and denote the corresponding recovery parameter by q m = α -1 m -1. Consider q opt m ≥ 1 the optimal recovery parameter given by [START_REF] Efron | Empirical Bayes analysis of a microarray experiment[END_REF]. Let ν ∈ (0, 1). Then:

(i) The BFDR threshold t m at level α m defined by [START_REF] Efron | Empirical Bayes methods and false discovery rates for microarrays[END_REF] satisfies that for any m ≥ 2 such that r m ≥ Km Cm(1-ν) (log(q m /q opt m ) -log ν),

R m (t m ) -R m (t B m ) (26) ≤ π 1,m C m q m - C m q opt m ∨ K m log(q m /q opt m ) -log ν r m . (ii) Letting ε ∈ (0, 1), D 1,m = -log(νπ 0,m (1 -ε)) and D 2,m = log(ν -1 × C m τ -1
m m), the FDR threshold tFDR m at level α m defined by [START_REF] Finner | On the false discovery rate and an asymptotically optimal rejection curve[END_REF] satisfies that, for any a ∈ {1, 2}, for any m ≥ 2 such that

r m ≥ Km Cm(1-ν) (log(α -1 m /q opt m ) + D a,m ), R m ( tFDR m ) -R m (t B m ) ≤ π 1,m α m 1 -α m + K m (log(α -1 m /q opt m ) + D a,m ) + r m (27) + α m /m (1 -α m ) 2 + π 1,m 1{a = 1}e -m(τm+1) -1 νε 2 Cm/4 .
Corollary 4.3(ii) contains two distinct cases. The case a = 1 should be used when m/τ m is large, because the remainder term containing the exponential becomes small (whereas D 1,m is approximately constant). The case a = 2 is intended to deal with the regime where m/τ m is not large, because D 2,m is of the order of a constant in that case. The finite sample oracle inequalities [START_REF] Sarkar | A general decision theoretic formulation of procedures controlling FDR and FNR from a Bayesian perspective[END_REF] and ( 27) are useful to derive explicit rates of convergence, as we will see in the next section. Let us also mention that an exact computation of the excess risk of BFDR thresholding can be derived in the Laplace case; see Section 10.2 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. 4.3. Asymptotic optimality with rates. In this section, we provide a sufficient condition on α m such that, under (BP) and (Sp), BFDR/FDR thresholding is asymptotically optimal [according to [START_REF] Blanchard | Adaptive false discovery rate control under independence and dependence[END_REF]], and we provide an explicit rate ρ m . Furthermore, we establish that this condition is necessary for the optimality of BFDR thresholding. (i) The BFDR threshold t m is asymptotically optimal if and only if α m → 0 and log α m = o((log m) γ ), [START_REF] Seeger | A note on a method for the analysis of significances en masse[END_REF] in which case it is asymptotically optimal at rate ρ Remark 4.5 (Lower bound for the Laplace scale model). We can legitimately ask whether the rate ρ m = (log m) -γ can be improved. We show that this rate is the smallest that one can obtain over a sparsity class β ∈ [β -, 1] for some β -∈ (0, 1), in the particular case of BFDR thresholding and in the Laplace scale model; see Corollary 10.2 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. While the calculations become significantly more difficult in the other models, we believe that the minimal rate for the relative excess risk of the BFDR is still (log m) -γ in a Subbotin location and scale models. Also, since the FDR can be seen as a stochastic variation around the BFDR, we may conjecture that this rate is also minimal for FDR thresholding.

m = α m + (log(α -1 m /(log m) γ )) + (log m) γ

4.4.

Choosing α m . Let us consider the sparsity regime τ m = m β , β ∈ (0, 1). Corollary 4.4 suggests to choose α m such that α m ∝ (log m) -γ . This is in accordance with the recommendation of [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] in the Gaussian scale model; see Remark 5.3 therein. In this section, we propose an explicit choice of α m from an priori value (β 0 , C 0 ) of the unknown parameter (β, C m ).

P. NEUVIAL AND E. ROQUAIN

Let us choose a value (β 0 , C 0 ) a priori for (β, C m ). A natural choice for α m is the value which would be optimal if the parameters of the model were (β, C m ) = (β 0 , C 0 ). Namely, by using (13) in Section 2.4, we choose

α m = α opt m (β 0 , C 0 ), where α opt m (β 0 , C 0 ) = (1 + q opt m (β 0 , C 0 )) -1 (29) with q opt m (β 0 , C 0 ) = m -β 0 C 0 /F -1
m,0 (C 0 ) by denoting F m,0 the c.d.f. of the p-values following the alternative for the model parameters (β 0 , C 0 ). For instance:

-Gaussian location:

F -1 m,0 (C 0 ) = Φ({Φ -1 (C 0 ) 2 + 2β 0 log m} 1/2 ); -Gaussian scale: F -1 m,0 (C 0 ) = 2Φ(Φ -1 (C 0 /2)x), where x > 1 is the solution of 2β 0 log m + 2 log x = (Φ -1 (C 0 /2)) 2 (x 2 -1); -Laplace scale: q opt m (β 0 , C 0 ) = y, where y > 1 is the solution of β 0 log m + log y = (y -1) log(1/C 0 ),
where Φ(z) denotes P(Z ≥ z) for Z ∼ N (0, 1).

The above choice of α m does depend on (β 0 , C 0 ), which can be interpreted as a "guess" on the value of the unknown parameter (β, C m ). Hence, when no prior information on (β, C m ) is available from the data, the above choice of α m can appear of limited interest in practice. However, we would like to make the following two points:

• asymptotically, choosing α m = α opt m (β 0 , C 0 ) always yields an optimal (B)FDR thresholding [under (BP)], even if (β 0 , C 0 ) = (β, C m ): by Proposition 4.1, we get α opt m (β 0 , C 0 ) ∝ (log m) -γ and thus the asymptotic optimality is a direct consequence of Corollary 4.4(iii); • nonasymptotically, our numerical experiments suggest that α m = α opt m (β 0 , C 0 ) performs fairly well when we have at hand an a priori on the location of the model parameters: if (β, C m ) is supposed to be in some specific (but possibly large) region of the "sparsity × power" square, choosing any (β 0 , C 0 ) in that region yields a thresholding procedure with a reasonably small risk; see Sections 5 and 14 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF].

Finally, let us note that the choice α m = α opt m (β 0 , C 0 ) is motivated by the analysis of the BFDR risk, not that of the FDR risk. Hence, it might be possible to choose a better α m for FDR thresholding, especially for small values of m for which BFDR and FDR are different. Because obtaining such a refinement appeared quite challenging, and as our proposed choice already performed well, we decided not to investigate this question further. Taking τ m = m β , the parameters of the model are (β, C m ) ∈ (0, 1] × (0, 1). Assume that the parameter sequence (C m ) m satisfies (BP) for some 0 < C -≤ C + < 1. Then Corollary 9.4 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF] provides explicit constants D = D(β, C -, C + , β 0 , C 0 , ν) and M = M (β, C -, C + , β 0 , C 0 , ν) such that the following inequality holds:

(R m (t m ) -R m (t B m ))/R m (t B m ) ≤ D/(log m) 1-1/ζ for any m ≥ M . ( 30
)
As an illustration, in the Gaussian case (ζ = 2), for β = 0.7, C -= 0.5, C + = 0.7, β 0 = C 0 = 0.5 and ν = 0.25, we have M 61.6 and D 2.66. As expected, these constants are over-estimated: for instance, by taking m = 1000, the left-hand side of ( 30) is smaller than 0.1 (see Figure 4 in the next section) while the right-hand side of ( 30) is D/ log(1000) 1.01. Finally, we can check that D becomes large when β is close to 0 or C + is close to 1. These configurations correspond to the cases where the data are almost nonsparse and where the Bayes rule can have almost full power, respectively. They can be seen as limit cases for our methodology.

→ +∞, α opt m (β 0 , C 0 ) ∼ α ∞ m (β 0 , C 0 ), for an equivalent α ∞ m (β 0 , C 0 )
having a very simple form; see Section 10.1 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. Therefore, we could use α ∞ m (β 0 , C 0 ) instead of α opt m (β 0 , C 0 ). Numerical comparisons between the (B)FDR risk obtained according to α opt m (β 0 , C 0 ) and α ∞ m (β 0 , C 0 ) are provided in Section 14 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]. While α ∞ m (β 0 , C 0 ) qualitatively leads to the same results when m is large (say, m ≥ 1000), the use of α opt m (β 0 , C 0 ) is more accurate for a small m.

Numerical experiments.

In order to complement the convergence results stated above, it is of interest to study the behavior of FDR and BFDR thresholding for a small or moderate m in numerical experiments. These experiments have been performed for the inductive risk R m (•) = R I m (•) defined by (8). 5.1. Exact formula for the FDR risk. The BFDR threshold t m can be approximated numerically, which allows us to compute R m (t m ). Computing R m ( tFDR m ) is more complicated because the FDR threshold tFDR m is not deterministic. However, we can avoid performing cumbersome and somewhat imprecise simulations to compute R m ( tFDR m ) by using the approach proposed in [START_REF] Finner | Multiple hypotheses testing and expected number of type I errors[END_REF] and [START_REF] Roquain | Exact calculations for false discovery proportion with application to least favorable configurations[END_REF]. Using this methodology, the full distribution of tFDR m may be written as a function of the joint c.d.f. of the order statistics of i.i.d. uniform variables. Let for any k ≥ 0 and for any (t 1 , . . . , t k ) ∈ [0, 1] k , Ψ k (t 1 , . . . , t k ) = P(U (1) ≤ t 1 , . . . , U (k) ≤ t k ), where (U i ) 1≤i≤k is a sequence of i.i.d. uniform variables on (0, 1) and with the convention Ψ 0 (•) = 1. The Ψ k 's can be evaluated, for example, by using Steck's recursion; see [START_REF] Shorack | Empirical Processes with Applications to Statistics[END_REF], pages 366-369. Then, relation [START_REF] Donoho | Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data[END_REF] in [START_REF] Roquain | Exact calculations for false discovery proportion with application to least favorable configurations[END_REF] entails

R m ( tFDR m ) = m k=0 m k R m α(k ∨ 1) m G m (αk/m) k (31) × Ψ m-k (1 -G m (αm/m), . . . , 1 -G m (α(k + 1)/m)),
where G m (t) = π 0,m t + π 1,m F m (t). For reasonably large m (m ≤ 10,000 in what follows), expression [START_REF] Storey | A direct approach to false discovery rates[END_REF] can be used for computing the exact risk of FDR thresholding tFDR m in our experiment.

5.2. Adaptation to unknown sparsity. We quantify the quality of a thresholding procedure using the relative excess risk

E m ( tm ) = (R m ( tm ) -R m (t B m ))/R m (t B m
). The closer the relative excess risk E m ( tm ) is to 0, the better the corresponding classification procedure is.

Figure 4 compares relative excess risks of different procedures in the Gaussian location model (results for the Gaussian scale and the Laplace scale models are qualitatively similar, see Figures 5 and6 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]). Each row of plots corresponds to a particular procedure, and each column to a particular value of m ∈ {25, 10 2 , 10 3 , 10 4 , 10 5 , 10 6 }. The first row corresponds to the Bayes procedure defined by [START_REF] Donoho | Higher criticism for detecting sparse heterogeneous mixtures[END_REF], where the model parameters are taken as (β, C m ) = (β 0 , C 0 ). It is denoted by Bayes0. Next, we consider BFDR (rows 2 to 5) and FDR (rows 6 to 9) thresholding at level α m , for α m ∈ {0.1, 0.2, 0.25} (independent of m) and for the choice α m = α opt m (β 0 , C 0 ) defined in Section 4.4. For each procedure and each value of m, the behavior of the relative excess risk is studied as the (unknown) true model parameters (β, C m ) vary in [0, 1] × [0, 1], and we arbitrarily choose β 0 and C 0 as the midpoints of the corresponding intervals, that is, (β 0 , C 0 ) = (1/2, 1/2) (similar results are obtained for other values of (β 0 , C 0 ); see Figures 8,9 and 10 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]). Colors reflect the value of the relative excess risk. They range from white

[R m = R m (t B m )] to dark red [R m ≥ 2R m (t B m )].
Black lines represent the level set E m = 0.1, that is, they delineate a region of the (β, C m ) plane in which the excess risk of the procedure under study is ten times less than the Bayes risk. The number at the bottom left of each plot gives the fraction of configurations (β, C m ) for which E m ≤ 0.1. This evaluates the quality of a procedure uniformly across all the (β, C m ) values.

For m = 10 6 , we did not undertake exact FDR risk calculations: they were too computationally intensive, as the complexity of the calculation of function Ψ k used in [START_REF] Storey | A direct approach to false discovery rates[END_REF] is quadratic in m. However, FDR risk is expected to be well approximated by BFDR risk for such a large value of m, as confirmed by the fact that FDR and BFDR plots at a given level α are increasingly similar as m increases.

Bayes0 performs well when the sparsity parameter β is correctly specified, and its performance is fairly robust to C m . However, it performs poorly when β is misspecified, and increasingly so as m increases. The results are markedly different for the other thresholding methods. BFDR thresholding and FDR thresholding are less adaptive to C m than Bayes0, but much more adaptive to the sparsity parameter β, as illustrated by the fact that the configurations with low relative excess risk span the whole range of β.

For α m = α opt m (β 0 , C 0 ), the fraction of configurations (β, C m ) for which E m ≤ 0.1 increases as m increases. This illustrates the asymptotic optimality of (B)FDR thresholding, as stated in Corollary 4.4(iii), because α opt m (β 0 , C 0 ) ∝ (log m) -1/2 . Additionally, observe that the (β, C m )-region around (β 0 , C 0 ) contains only very small values of E m , even for moderate m. This suggests that, nonasymptotically, α opt m (β 0 , C 0 ) is a reasonable choice for α m , when we know a priori that the parameters lie in some specific region of the (β, C m )square.

Next, let us consider the case of (B)FDR thresholding using a fixed value of α m = α. While our theoretical results show that choosing α m fixed with m (and in particular not tending to zero) is always asymptotically suboptimal, the results shown by Figure 4 are less clear-cut. An explanation is that (log m) -1/2 decreases only slowly to zero [e.g., α opt m (β 0 , C 0 ) 0.17 for m = 10 6 ], hence the asymptotic is quite "far" and not fully attained by our experiments.

Hence, from a more practical point of view, in a classical situation where m does not exceed, say, 10 6 , a practitioner willing to use the (B)FDR can consider two different approaches to calibrate α m : the first one is to take some arbitrary value, for example, 0.05, 0.1 or 0.2. The overall excess risk might be small, but the location of the region of smallest excess risk (pictured in white in our figures) is unknown, and depends strongly on α and m (and even ζ). In contrast, the second method α opt m (β 0 , C 0 ) "stabilizes" the region of the (β, C m )-square where the (B)FDR has good performance across all the values of m (and ζ). Thus, while the first method has a clear interpretation in terms of FDR, the second approach is more interpretable w.r.t. the sparsity and power parameters and is recommended when these parameters are felt to correctly parametrize the model.

Finally note that, when considering the weighted mis-classification risk (as formally defined in [START_REF] Tamhane | A generalized step-up-down multiple test procedure[END_REF] and studied in Section 11 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF]), there exists a particular choice of the weight (as a function of m) such that the optimal (B)FDR level α opt m (β 0 , C 0 ) does not depend on m, making (B)FDR thresholding with fixed values of α m asymptotically optimal, as noted by [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]. This point is discussed in Section 6.2.

R m = inf tm sup (β,Cm)∈Θ {R m ( tm )} ,
where the infimum is taken over the set of thresholds that can be written as measurable functions of the p-values. Obviously, R m ≥ sup (β,Cm)∈Θ {R m (t B m )}, where t B m is the Bayes threshold. Hence, by taking the supremum w.r.t. (β, C m ) in our excess risk inequalities, we are able to derive minimax results. However, this requires a precise formulation of ( 5) where the dependence in β of the constant D is explicit. For simplicity, let us consider the Laplace scale model. By using (69) and (74) in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF], and by taking α m ∝ (log m) -1 , we can derive that there exists a constant D > 0 (independent of β -, C -and C + ) such that for a large m,

sup (β,Cm)∈Θ {R m ( tFDR m )} ≤ sup (β,Cm)∈Θ {R m (t B m )} 1 + -log(β -/2) β -(1 -C + ) D log m (32) ≤ R m 1 + -log(β -/2) β -(1 -C + ) D log m .
This entails that tFDR m is asymptotically minimax, that is, sup

(β,Cm)∈Θ {R m ( tFDR m )} ∼ R m .
This property can be seen as an analogue to the asymptotically minimaxity stated in Theorem 1.1 in [START_REF] Abramovich | Adapting to unknown sparsity by controlling the false discovery rate[END_REF] and Theorem 1.3 in [START_REF] Donoho | Asymptotic minimaxity of false discovery rate thresholding for sparse exponential data[END_REF], in an estimation context. Finally, regarding [START_REF] Storey | The positive false discovery rate: A Bayesian interpretation and the q-value[END_REF], an interesting avenue for future research would be to establish whether there are asymptotically minimax rules tm such that sup (β,Cm)∈Θ {R m ( tm )} = R m (1+o(ρ m )) for a rate ρ m smaller than (log m) -1 . 6.2. Extension to weighted mis-classification risk. In our sparse setting, where we assume that there are many more labels "0" than labels "1," one could consider that mis-classifying a "0" is less important than misclassifying a "1." This suggests to consider the following weighted risk:

R m,λm ( tm ) = E(π 0,m tm + λ m π 1,m (1 -F m ( tm ))) (33)
for a known factor λ m ∈ (1, τ m ). This weighted risk was extensively used in [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]. In Section 11 in the supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF], we show that all our results can be adapted to this risk. Essentially, when considering R m,λm instead of R m , our results hold after replacing τ m by τ m /λ m and q m by q m λ m .

As an illustration, let us consider here the case of a ζ-Subbotin density, τ m = m β , β ∈ (0, 1], log λ m = o((log m) γ ), where γ = 1 -ζ -1 and γ = 1 for the location and scale cases, respectively. As displayed in Table 4 in the P. NEUVIAL AND E. ROQUAIN supplemental article [START_REF] Neuvial | Supplement to "On false discovery rate thresholding for classification under sparsity[END_REF], under the (corresponding) assumptions (BP) and (Sp), we show that a sufficient condition for FDR thresholding to be asymptotically optimal for the risk R m,λm is to take q -1 m = O(1), q m λ m → ∞ and log q m = o((log m) γ ). This recovers Theorem 5.3 of [START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] when applied to the particular case of a Gaussian scale model (for which γ = 1). Furthermore, we show that taking q m ∝ q opt m , that is, q m ∝ λ -1 m (log m) γ , leads to the optimality rate ρ m = (log m) -γ for the relative excess risk based on R m,λm . While the order of q opt m is not modified when λ m ∝ 1, it may be substantially different when λ m → ∞. Typically, λ m ∝ (log m) γ leads to q opt m ∝ 1. Hence, when considering R m,λm instead of R m , the value of λ m should be carefully taken into account when choosing α m to obtain a small excess risk.

Conversely, our result states that FDR thresholding with a pre-specified value of α m = α (say, α = 0.05), is optimal over the range of weighted misclassification risks using a λ m satisfying λ m → ∞ and log λ m = o((log m) γ ), and that choosing λ m ∝ (log m) γ leads to the optimality rate ρ m = (log m) -γ .

7. Proofs of Theorems 3.1 and 3.2. The proofs are first established for the misclassification risk R m = R I m defined by [START_REF] Chi | On the performance of FDR control: Constraints and a partial solution[END_REF]. The case of the misclassification risk R T m , defined by ( 7) is examined in Section 7.4.

R m (t m ) -R m (t B m ) ≤ π 1,m C m /q m -π 0,m t B m + π 1,m (1 -q -1 m )γ m ; (35) R m (t m ) -R m (t B m ) ≤ π 1,m (C m /q m -τ m t B m ) ∨ γ m . ( 36
)
Proof. To prove (34), we use F m (t m ) = t m q m τ m and τ m = π 0,m /π 1,m , to write

R m (t m ) -R m (t B m ) = π 0,m t m -π 0,m t B m + π 1,m (C m -F m (t m )) (37)
= π 1,m F m (t m )/q m -π 0,m t B m + π 1,m (C m -F m (t m )). Expression (35) is an easy consequence of (34). Finally, (37) and (34) entail

R m (t m ) -R m (t B m ) ≤ π 1,m C m /q m -π 0,m t B m , if t B m ≤ t m , π 1,m (C m -F m (Ψ -1
m (q m τ m ))), if t B m ≥ t m , which yields (36).

t m = F m (t m )(q m τ m ) -1 ≤ (q m τ m ) -1 , we ob- tain R m (t m ) ≥ π 1,m (1-(1-q -1 m ) + F m (t m )) ≥ π 1,m (1-(1-q -1 m ) + F m (q -1 m τ -1 m )
). This entails [START_REF] Gavrilov | An adaptive step-down procedure with proven FDR control under independence[END_REF] and [START_REF] Genovese | Operating characteristics and extensions of the false discovery rate procedure[END_REF]. . By exchangeability of (p i , H i ) i , we can assume without loss of generality that the p-values corresponding to a label H i = 0 are p 1 , . . . , p m 0 (H) for simplicity. Let us denote tm,0 the thresholding tBH m defined by ( 14), applied to the p-value family p i , 1 ≤ i ≤ m, in which each of the p-value p m 0 (H)+1 , . . . , p m has been replaced by 0. Classically, we have tm,0 = α m (m 1 (H) + km,0 )/m, where km,0 = max{k ∈ {0, 1, . . . , m 0 (H)} : q (k) ≤ α m (m 1 (H) + k)/m}, where (q 1 , . . . , q m 0 (H) ) = (p 1 , . . . , p m 0 (H) ) is the set of p-values corresponding to zero labels; see, for example, Lemma 7.1 in [START_REF] Roquain | Exact calculations for false discovery proportion with application to least favorable configurations[END_REF].

i=1 1{p i ≤ tBH m } H ≤ E m -1 m 0 (H) i=1 1{p i ≤ tm,0 } H = m -1 E( km,0 |H) ≤ π 1,m α m 1 -α m + m -1 α m (1 -α m ) 2
by using (42). Hence, (39) is proved for the risk R T m . Next, the proof for bounding the type II error derives essentially from the following argument, which is quite standard in the multiple testing methodology; see, for example, [START_REF] Ferreira | On the Benjamini-Hochberg method[END_REF][START_REF] Finner | On the false discovery rate and an asymptotically optimal rejection curve[END_REF][START_REF] Roquain | Type I error rate control for testing many hypotheses: A survey with proofs[END_REF][START_REF] Roquain | Exact calculations for false discovery proportion with application to least favorable configurations[END_REF]. Let us denote tm = max{t ∈ [0, 1] : α m G m (t) ≥ t}, where G m (t) = m -1 (1 + m i=2 1{p i ≤ t}) denotes the empirical c.d.f. of the p-values where p 1 has been replaced by 0. Then, for any realization of the p-value family, p 1 ≤ tBH m is equivalent to p 1 ≤ tm ; see, for example, proof of Theorem 2.1 in [START_REF] Ferreira | On the Benjamini-Hochberg method[END_REF] and Section 3.2 of [START_REF] Roquain | Type I error rate control for testing many hypotheses: A survey with proofs[END_REF]. This entails that the type II error is equal to π 1,m (1 -E(F m ( tm ))) [by using the exchangeability of (H i , p i ) 1≤i≤m ]. Finally, since tm ≥ tBH m and tm ≥ α m /m, we have tm ≥ tFDR m . Hence π 1,m (1 -E(F m ( tm ))) ≤ π 1,m (1 -E(F m ( tFDR m ))) and bounds (43) and (44) also hold for the risk R T m .

Fig. 1 .

 1 Fig. 1. Left: illustration of the considered classification problem for the Gaussian location model for the inductive risk (2); density of N (0, 1) (solid line); X k , k = 1, . . . , m (crosses); a new data point Xm+1 to be classified (open circle); Bayes' rule (dotted line); FDR rule ŝFDR m

2. 4 .

 4 BFDR thresholding. Let us consider the following Bayesian quantity:

Remark 3 . 3 (

 33 Conservative upper-bound for γ m ). By the concavity of F m , we have q

Proposition 4 . 1 .

 41 Consider a ζ-Subbotin density (3) with ζ ≥ 1 for a scale model and ζ > 1 for a location model. Let (τ m , C m ) ∈ (1, ∞) × (0, 1) be the parameters of the model. Let r m be equal to r loc m defined by (21) in the location model or to r sc m defined by (22) in the scale model. Then, under (BP) and (Sp), we have µ

Corollary 4 . 3 .

 43 Consider a ζ-Subbotin density (3) with ζ > 1 for a location model and ζ ≥ 1 for a scale model, and let (τ m , C m ) ∈ (1, ∞) × (0, 1) be the parameters of the model. Let r m = r loc m [defined by (21)] and K m = d(0) in the location model or r m = r sc

Corollary 4 . 4 .

 44 Take ζ > 1, γ = 1-ζ -1 for the location case and ζ ≥ 1, γ = 1 for the scale case. Consider a ζ-Subbotin density (3) in the sparsity regime τ m = m β , 0 < β ≤ 1 and under (BP). Then the following holds:

.

  (ii) The FDR threshold tFDR m at a level α m satisfying (28) is asymptotically optimal at rate ρ m = α m + (log(α -1 m /(log m) γ )) + (log m) γ . (iii) Choosing α m ∝ 1/(log m) γ , BFDR and FDR thresholding are both asymptotically optimal at rate ρ m = 1/(log m) γ . In the particular case of a Gaussian scale model (ζ = 2), Corollary 4.4 recovers Corollaries 4.2 and 5.1 of [6]. Corollary 4.4 additionally provides a rate, and encompasses the location case and other values of ζ.

Remark 4 . 6 .

 46 By choosing α m = α opt m (β 0 , C 0 ) as in (29), we can legitimately ask how large the constants are in the finite sample inequalities coming from Corollary 4.3 in standard cases. To simplify the problem, let us focus on the BFDR threshold and consider a ζ-Subbotin loca-tion model with ζ > 1.

Remark 4 . 7 .

 47 By using Proposition 4.1, as m

6. Discussion. 6 . 1 .

 61 Asymptotic minimaxity over a sparsity class. Let us consider the sparsity range τ m = m β , with β -≤ β ≤ 1, for some given β -∈ (0, 1). Assume (BP) with C -and C + defined therein. Denote the set [β -, 1] × [C -, C + ] by Θ for short. The minimax risk is defined by

7. 1 .Proposition 7 . 1 .

 171 Relations for BFDR. Let us first state the following result. Consider the setting and the notation of Theorem 3.1. Then we have for any m ≥ 2,R m (t m ) -R m (t B m ) = π 1,m C m /q m -π 0,m t B m (34) + π 1,m (1 -q -1 m )(C m -F m (t m )). Furthermore, if α m ≤ 1/2,we have for any m ≥ 2,

7. 2 .

 2 Proof of Theorem 3.1. Theorem 3.1(i) follows from (36) because π 0,m t B m = π 1,m C m /q opt m by definition. Let us now prove (ii). First note thatR m (t m ) = π 1,m -π 1,m F m (t m )(1 -q -1 m ). (38)Using (38) and the upper bound

7. 3 .Proposition 7 . 2 . 2 m( 1 -

 37221 Proof of Theorem 3.2. Write tm instead of tFDR m for short. To establish[START_REF] Genovese | A stochastic process approach to false discovery control[END_REF], let us first write the risk of FDR thresholding as R m ( tm ) = T 1,m + T 2,m , with T 1,m = π 0,m E( tm ) and T 2,m = π 1,m (1 -E(F m ( tm ))). In the sequel, T 1,m and T 2,m are examined separately.7.3.1.Bounding T 1,m . The next result is a variation of Lemmas 7.1 and 7.2 in[START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF]. The following bound holds:T 1,m ≤ π 1,m α m 1 -α m + m -1 α m (1 -α m ) 2 . (39)Proof. To prove Proposition 7.2, we follow the proof of Lemma 7.1 in[START_REF] Bogdan | Asymptotic Bayes-optimality under sparsity of some multiple testing procedures[END_REF] with slight simplifications. Recall that we have by definition tm = tBH m ∨ (α m /m). Hence, we have E( tm |H) ≤ α m /m + E( tBH m |H). By integrating w.r.t. the label vector H, it is thus sufficient to proveE( tBH m |H) ≤ π 1,m α m 1 -α m + m -1 α α m ) 2 . (40)Let m 1 (H) = m i=1 H i and m 0 (H) = m -m 1 (H)

7 . 4 .

 74 Since tBH m is nonincreasing in each p-value, setting some p-values equal to 0 can only increase tBH m . This entails E( tBH m |H) ≤ E( tm,0 |H) = α m (m 1 (H) + E( km,0 |H))/m. (41) Proofs for the risk R T m . Let us recall that R T m and R I m are equal for a deterministic threshold and thus also for the BFDR threshold. Hence, Theorem 3.1 also holds for the risk R T m , and we only have to prove Theorem 3.2. First note that since R T m ( tFDR m ) = R T m ( tBH m ), we can work directly with tBH m . Proving the type I error bound (39) can be done similarly: with the same notation, the type I error can be written conditionally on H as E m -1 m 0 (H)

P. NEUVIAL AND E. ROQUAIN

This condition implies that the setting is "noncritical," as defined in[START_REF] Chi | On the performance of FDR control: Constraints and a partial solution[END_REF].

Acknowledgments. We would like to thank Guillaume Lecué and Nicolas Verzelen for interesting discussions. We are also grateful to anonymous referees, an Associated Editor, and an Editor for their very helpful comments and suggestions.

Supported by the French Agence Nationale de la Recherche (ANR Grants ANR-09-JCJC-0027-01, ANR-PARCIMONIE, ANR-09-JCJC-0101-01) and by the French ministry of foreign and European affairs (EGIDE-PROCOPE Project 21887 NJ).

AMS 2000

SUPPLEMENTARY MATERIAL

Supplement to: On false discovery rate thresholding for classification under sparsity (DOI: 10.1214/12-AOS1042SUPP; .pdf). Proofs, additional experiments and supplementary notes for the present paper.