Keywords: Binary hypothesis testing, decision, estimation, likelihood theory, multivariate normal distribution, thresholding, signal processing, statistical test. Résumé Décision, distribution multidimensionelle Gaussienne, estimation, rapport de vraisemblance, test d'hypothèses, test statistique, traitement du signal

This research report is an extended version of two previous reports, namely, "A limit theorem for sequences of independent random vectors with unknown distributions and its applications to non-parametric detection", RR-2004001-SC, Collection des Rapports de Recherche de l'ENST Bretagne, November 2004, written by D. Pastor and R. Gay, and "On the detection of signals with unknown distributions and priors in white Gaussian noise", RR-2006001-SC, Collection des Rapports de Recherche de l'ENST Bretagne, January 2006, written by D. Pastor.

In this new version, the proof of the main theoretical result is improved, more application fields and experimental results are given. Moreover, some typing faults have been corrected.

Avant-propos

Ce rapport de recherche est une version étendue de deux rapports antérieurs, à savoir, "A limit theorem for sequences of independent random vectors with unknown distributions and its applications to non-parametric detection", RR-2004001-SC, Collection des Rapports de Recherche de l'ENST Bretagne, novembre 2004, écrit par D. Pastor et R. Gay, et "On the detection of signals with unknown distributions and priors in white Gaussian noise", RR-2006001-SC, Collection des Rapports de Recherche de l'ENST Bretagne, janvier 2006, écrit par D. Pastor.

Dans cette nouvelle version, la démonstration du résultat théorique est améliorée, les applications ne se limitent pas à la détection de signaux et de nouveaux résultats expérimentaux sont présentés, quelques fautes de frappe ont été corrigées.

A limit theorem for processing signals that have unknown probability distributions and unknown probabilities of presence in white Gaussian noise

Un théorème limite pour le traitement de signaux dont les distributions de probabilité et les probabilités de

Introduction

In signal processing applications, we often meet sequences of d-dimensional real random observations that satisfy the following assumptions: each observation is either the sum of some random signal and independent noise or noise alone; noise is assumed to be Gaussian distributed with null mean and covariance matrix equal to σ 2 0 I d where I d is the d × d identity matrix. Summarizing these assumptions, we say that noise is "white and Gaussian" with standard deviation σ 0 and that each observation results from some signal randomly present or absent in independent and Additive White Gaussian Noise (AWGN).

In many applications, very little is known about the signals or most of their describing parameters ([3, Section I]), so that the probability distributions of the signals are partially or definitely unknown. This issue is met in Electronic (Warfare) Support Measure (ESM) systems faced with noncooperant communications. This is also the case with passive sensors such as sonar systems that receive signals resulting from noise generated by motors and hull vibrations transmitted through a fluctuating environment. Another example is that of an echo received by a radar system from a target since this echo results from some kind of convolution between a known transmitted pulse and an unknown environment.

In situations such as those described above, the detection of the signals cannot be achieved via standard likelihood theory because the usual Bayes, minimax and Neyman-Pearson criteria require full knowledge of the probability distributions of these signals. Alternative solutions are then Wald's tests [START_REF] Wald | Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large[END_REF], which have best constant (resp. average) power on a family of surfaces for making a decision between simple (resp. composite) hypotheses, non-parametric and robust detection ([10, section III.E]), as well as Generalized Likelihood Ratio Tests ( [START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF]). Constant False Alarm Rate (CFAR) systems ( [START_REF] Minkler | The Principles of Automatic Radar Detection in Clutter[END_REF]), standardly used in radar processing for detecting targets with a specified false alarm rate, typically derive from such alternative approaches.

In addition to our lack of prior knowledge about the signals, the noise standard deviation is sometimes unknown and must be estimated in order to process the observations. In general, the estimation is achieved by resorting to the physics of the problem. Two examples thereof are the following ones.

A radar target is detected when the observation is above a threshold height. In order to guarantee a specified false alarm probability, the threshold height must be chosen as a function of the noise standard deviation. When the latter can fluctuate, it is estimated so as to adjust the threshold height. By resorting to the physics of the radar, standard CFAR systems select observations that are considered as signal-free and called "reference cells"; the noise standard deviation is then computed on the basis of these reference cells. The smaller the number of reference cells that contain radar echoes and not noise only, the better the estimate. Selecting the reference cells is no easy task ([7]).

A second example is the denoising of speech signals corrupted by independent AWGN since most standard filtering techniques must be adjusted with an estimate of the noise standard deviation when the latter is unknown. A standard solution involves estimating the noise standard deviation on the basis of periods of time considered as signal-free by a Voice Activity Detector, which, of course, can make errors. The noise standard deviation can also be estimated by computing the smallest eigenvalues of the noisy speech autocorrelation matrix but the computation of the eigenvalues may prove unstable depending on the matrix size.

Motivated by practical issues such as those described above, this paper introduces a theoretical result and, on the basis of this result, addresses the estimation of the noise standard deviation and the detection of the signals when the probability distributions of the signals are unknown.

More specifically, the theoretical result, namely theorem 2.1, is established in section 2. It concerns any sequence of mutually independent observations that are d-dimensional real random vectors where signals with unknown probability distributions are randomly present or absent with unknown probabilities of presence in independent AWGN. Theorem 2.1 does not assume that the available observations are identically distributed. It derives from a corollary of Kolmogorov's classical strong limit theorem but not from usual generalizations of the central limit theorem such as the Lindeberg and the Lyapunov theorems.

On the basis of theorem 2.1, an estimate of the noise standard deviation is introduced in section 3. This estimate is called the Essential Supremum Estimate (ESE) for the crucial role played by the essential supremum norm in theorem 2.1. In the same section, and as a continuation of the results previously published in [START_REF] Pastor | A Sharp Upper-Bound for the Probability of Error of the Likelihood Ratio Test for Detecting Signals in White Gaussian Noise[END_REF], we address the particular case where the amplitudes of the signals are above or equal to some known α ∈ [0, ∞) and the signals are less present than absent in the sense that their probabilities of presence are less than or equal to one half. Under these assumptions, we propose a test for the detection of these signals in AWGN with unknown standard deviation. This test, the Essential Supremum Test (EST), derives from the ESE and [START_REF] Pastor | A Sharp Upper-Bound for the Probability of Error of the Likelihood Ratio Test for Detecting Signals in White Gaussian Noise[END_REF]Theorem VII.1].

Section 4 presents the empirical bias and the empirical Mean-Square Error (MSE) of the ESE, as well as the Binary Error Rate (BER) of the EST when the observations, and thus the signals and noise, are two-dimensional random vectors and the signals are uniformly distributed on the circle centred at the origin with radius α. This model is relevant in many signal processing applications because the two components of each signal can be regarded as the in-phase and quadrature components of a modulated sinusoidal carrier whose amplitude is α and whose phase is uniformly distributed in [0, 2π].

Section 5 concludes this paper by providing several suggestions regarding theoretical extensions and various applications, especially in radar processing, electronic warfare and speech processing.

2 A limit theorem for sequences of independent random vectors that have unknown probability distributions

Notations and terminology

Every random vector and every random variable encountered hereafter is assumed to be defined on the same probability space (Ω, B, P) and for every ω ∈ Ω by setting this random vector or variable to 0 on any negligible subset where it could be undefined. As usual, if a property P holds true almost surely, we write P (a-s).

As above, d stands for some fixed natural number and every random vector considered below is d-dimensional. The set of all the sequences of d-dimensional random vectors defined on Ω is denoted by S.

In what follows, • is the standard Euclidean norm in R d . For every given random vector Y : Ω → R d and any τ ∈ R, the notation I( Y ≤ τ ) stands for the indicator function of the event { Y ≤ τ }.

Given some positive real number σ 0 , we say that a sequence X = (X k ) k∈N of d-dimensional real random vectors is a d-dimensional white Gaussian noise (WGN) with standard deviation σ 0 if the random vectors X k , k ∈ N, are mutually independent and identically Gaussian distributed with null mean vector and covariance matrix σ 2 0 I d . For the sake of brevity, we will henceforth write independent instead of mutually independent.

The minimum amplitude of an element Λ = (Λ k ) k∈N of S is defined as the supremum a(Λ) of the set of those α ∈ [0, ∞] such that, for every natural number k, Λ k is larger than or equal to α (a-s):

a(Λ) = sup {α ∈ [0, ∞] : ∀k ∈ N, Λ k ≥ α (a-s)} . (1)
For every given Λ = (Λ k ) k∈N , the reader will easily verify that a(Λ) is finite, that Λ k ≥ a(Λ) for every k ∈ N and that, given α ∈ [0, ∞], a(Λ) ≥ α if and only if, for every k ∈ N, Λ k ≥ α (a-s).

If f is some map of S into R, we will say that the limit of f is ∈ R when a(Λ) tends to ∞ and write that lim a(Λ)→∞ f (Λ) = if, for any positive real value η, there exists some α 0 ∈ (0, ∞) such that, for every α ≥ α 0 and every Λ ∈ S such that a(Λ) ≥ α, we have |f (Λ) -| ≤ η.

Given some non-negative real number a, let L a (Ω, R d ) stand for the set of those d-dimensional real random vectors Y : Ω → R d for which E[ Y a ] < ∞. We hereafter deal with the set of those elements Λ = (Λ k ) k∈N of S such that Λ k ∈ L a (Ω, R d ) for every k ∈ N and sup k∈N E[ Λ k a ] is finite. A suitable and standard notation for this set is

∞ (N, L a (Ω, R d )) because, although ∞ (N, L 0 (Ω, R d )) is trivially S and L a (Ω, R d ) is a Banach space only if 1 ≤ a < ∞, L a (Ω, R d )
remains a complete metric space even when 0 < a < 1 ( [START_REF] Köthe | Topological Vector Spaces-I[END_REF]). Now, let L ∞ (Ω, R d ) be the Banach space whose elements are the real random vectors

Y : Ω → R d whose essential supremum norm Y ∞ is finite. Consider the set ∞ (N, L ∞ (Ω, R d )) of those sequences Λ = (Λ k ) k∈N of S such that Λ k ∈ L ∞ (Ω, R d ) for every k ∈ N and sup k∈N Λ k ∞ is finite. It is a Banach space once it is endowed with the norm • ∞ defined for any element Λ = (Λ k ) k∈N of S by Λ ∞ = sup k∈N Λ k ∞ . For every given k ∈ N, Λ k is then less than or equal to Λ ∞ (a-s). If α ∈ [0, ∞], Λ ∞ ≤ α if and only if, for every k ∈ N, Λ k ≤ α (a-s). For every k ∈ N, a(Λ) ≤ Λ k ≤ Λ ∞ (a-s). If f is a map of ∞ (N, L ∞ (Ω, R d ))
into R, we say that the limit of f is ∈ R when Λ ∞ tends to 0 and write that lim Λ ∞ →0 f (Λ) = if, given any positive real value η, there exists some α 0 ∈ (0, ∞) such that, for every real value α ≤ α 0 and every Λ ∈

∞ (N, L ∞ (Ω, R d )) such that Λ ∞ ≤ α, we have |f (Λ) -| ≤ η.
Given two (possibly extended) real numbers a and b such that 0

≤ a ≤ b ≤ ∞, note that ∞ (N, L b (Ω, R d )) ⊂ ∞ (N, L a (Ω, R d )). (2)
We call a thresholding function any non-decreasing continuous and positive real function θ : [0, ∞) → (0, ∞) such that θ(ρ) = Cρ + γ(ρ) where 0 < C < 1, γ(ρ) is positive for sufficiently large values of ρ and lim ρ→∞ γ(ρ) = 0; given any q ∈ [0, ∞), Υ q will stand for the map of [0, ∞) into [0, ∞) defined for every x ∈ [0, ∞) by Υ q (x) = x 0 t q+d-1 e -t 2 /2 dt.

(3)

The theoretical result

Let ε = (ε k ) k∈N be a sequence of random variables valued in {0, 1}, Λ = (Λ k ) k∈N stand for some sequence of d-dimensional real random vectors and X = (X k ) k∈N be some d-dimensional WGN with standard deviation σ 0 . Suppose that, for every k ∈ N, Λ k , X k and ε k are independent. Theorem 2.1 stated below concerns the sequence U = εΛ + X, that is, the sequence U = (U k ) k∈N such that, for every k ∈ N, U k = ε k Λ k + X k . In the theorem statement, the random vectors U k , k ∈ N, are assumed to be independent. Before stating the result, it is worth mentioning the following.

The sequence U basically models a sequence of independent observations where random signals are either present or absent in independent AWGN: noise is modelled by the sequence X; given k ∈ N, Λ k stands for some possible random signal and ε k models the possible occurrence of Λ k . The two possible values for ε k form a hypothesis pair according to which the observation is distributed: the null hypothesis is that ε k = 0 and the alternative one is ε k = 1. Even though this model introduces a priori probabilities of presence and absence, the importance of these probabilities will be significantly reduced by theorem 2.1 since this result assumes that these probabilities are upperbounded by a positive real number less than one.

We now state the following theorem whose proof is given in appendix .

Theorem 2.1 Let U = (U k ) k∈N be some element of S such that U = εΛ + X where Λ = (Λ k ) k∈N , X = (X k ) k∈N and ε = (ε k ) k∈N are an element of S, some d-dimensional WGN with standard deviation σ 0 and a sequence of random variables valued in {0, 1} respectively.

Assume that (Ass. 1) for every k ∈ N, Λ k , X k and ε k are independent;

(Ass. 2) the random vectors U k , k ∈ N, are independent;

(Ass. 3) the set of priors {P({ε k = 1}) : k ∈ N} has an an upper bound p ∈ [0, 1) and the random variables ε k , k ∈ N, are independent;

(Ass. 4) there exists some

ν ∈ (0, ∞] such that Λ ∈ ∞ (N, L ν (Ω, R d )).
Let r and s be any two non-negative real numbers such that 0 ≤ s < r ≤ ν/2. Given some natural number m and any pair (σ, T ) of positive real numbers, define the random variable ∆ m (σ, T ) by

∆ m (σ, T ) = m k=1 U k r I( U k ≤ σT ) m k=1 U k s I( U k ≤ σT ) -σ r-s Υ r (T ) Υ s (T ) . (4)
Then, given any thresholding function θ, (i) σ 0 is the unique positive real number σ such that, for every

β 0 ∈ (0, 1], lim a(Λ)→∞ lim sup m ∆ m (σ, βθ(a(Λ)/σ)) ∞ = 0 ( 5
)
uniformly in β ∈ [β 0 , 1]; (ii) if, in addition, ν = ∞, σ 0 is such that, for every β 0 ∈ (0, 1], lim Λ ∞→0 lim sup m ∆ m (σ 0 , βθ( Λ ∞ /σ 0 )) ∞ = 0 (6) uniformly in β ∈ [β 0 , 1].
Throughout the rest of the paper, the notations introduced in the foregoing statement are used with the same meaning and always under assumptions (Ass. [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF][START_REF] Amehraye | On the Application of Recent Results in Statistical Decision and Estimation Theory to Perceptual Filtering of Noisy Speech Signals[END_REF][START_REF] Kailath | Detection of Stochastic Processes[END_REF][START_REF] Kay | Fundamentals of Statistical Signal Processing[END_REF].

For τ ∈ [0, ∞), the ratio m k=1 U k r I( U k ≤ τ )/ m k=1 U k s I( U k ≤ τ
) is defined everywhere for the following reason. Let x 1 , x 2 , . . . , x m be m real numbers. If there exists at least one natural number k ∈ {1, . . . , m} such that x k = 0, the finiteness of the ratio m k=1 |x k | r / m k=1 |x k | s is trivial. Since r > s, the definition of this ratio is then extended by continuity by setting

m k=1 |x k | r / m k=1 |x k | s = 0 if (x 1 , . . . , x m ) = (0, . . . , 0
). Theorem 2.1 concerns the positive solutions of Eq. ( 5) only because σ = 0 trivially satisfies Eq. ( 5) regardless of the specific convergence involved. Straightforwardly, Eq. ( 5) is also satisfied for all σ ∈ [0, ∞) when r = s ≥ 0. This explains why it is assumed that r > s ≥ 0. Finally, statement (i) is of more interest than statement (ii). First, the former specifies the uniqueness of the solution in σ to Eq. ( 5) whereas this uniqueness is not stated by the latter and remains an open question; second, the assumption ν = ∞ is strong since, according to Eq. ( 2

), ∞ (N, L ∞ (Ω, R d )) is a subset of every ∞ (N, L ν (Ω, R d )), ν ∈ [0, ∞).
3 Application to the estimation of the noise standard deviation and the detection of the signals

The Essential Supremum Estimate

Basically, Eq. ( 5) means the following. Let ν ∈ (0, ∞), β 0 ∈ (0, 1] and consider some positive real value η. Then, there exists some positive real number α 0 with the following property: for every real number α larger than or equal to α 0 , every element Λ of L ν (Ω, R d ) whose minimum amplitude is larger than or equal to α, and every β ∈ [β 0 , 1], there exists a negligible subset Ω β of Ω such that, for every

ω ∈ Ω \ Ω β , ∆ m (σ 0 , βθ(α/σ 0 )) (ω) ≤ η if the sample size m is large enough. Consider now a finite subset I of [β 0 , 1]
, a real number α ≥ α 0 and an element Λ of L ν (Ω, R d ) whose minimum amplitude is larger than or equal to α. According to the foregoing, every β ∈ I can be assigned to a negligible subset Ω β ⊂ Ω outside which ∆ m (σ 0 , βθ(α/σ 0 )) is less than or equal to η when m is large enough. The finite union of these negligible subsets is also a negligible subset of Ω and, outside this union, max β∈I ∆ m (σ 0 , βθ(α/σ 0 )) is less than or equal to η and, thus, small.

The foregoing suggests proceeding as follows to estimate σ 0 , given m observations

U k = ε k Λ k + X k where Λ ∈ L ν (Ω, R d )
with a(Λ) ≥ α: choose a thresholding function θ, two values r and s such that 0 ≤ s < r ≤ ν/2, a finite subset I of (0, 1] and a search interval included in (0, ∞); estimate σ 0 by a possibly local minimum σ 0 of the discrete cost max β∈I ∆ m (σ, βθ(α/σ)) when σ ranges over the search interval; the value σ 0 is henceforth called the Essential Supremum Estimate (ESE) of the noise standard deviation. In the rest of this paper, the notation σ 0 will always stand for this estimate.

A great number of such estimates exist, depending on our choice for the thresholding function θ, the set I, the values r and s and the search interval. Henceforth, the ESE is computed with the following choices for r, s and I.

In most applications of importance, the signals have "finite energy" in the sense that the second order moments E[ Λ k 2 ], k ∈ N, are finite. With the notations used so far, this corresponds to the assumption that ν = 2. Since the values r and s must verify the inequalities 0 ≤ s < r ≤ 1, a rather natural choice is s = 0 and r = 1.

We set I = {β = /L : = 1, 2, . . . , L} where L is some natural number. On the basis of preliminary experimental results such as those of section 4, we take L = m because it seems a reasonable trade-off between computational load and accuracy.

Under the "finite energy" assumption and with the choice proposed above for I, the ESE considered henceforth is a possibly local minimum σ 0 of sup

∈{1,...,L}            m k=1 U k I( U k ≤ β σθ(α/σ)) m k=1 I( U k ≤ β σθ(α/σ)) -σ β θ(α/σ) 0 t d e -t 2 /2 dt β θ(α/σ) 0 t d-1 e -t 2 /2 dt            (7)
when σ ranges over a search interval to specify. Any minimization routine for scalar bounded non-linear functions is suitable for the computation of σ 0 . For instance, the experimental results presented in the sequel were obtained with the MATLAB routine fminbnd.m based on parabolic interpolation ( [START_REF] Press | Numerical recipes in C, The Art of Scientific Computing[END_REF]). In the next section, we propound a thresholding function and a search interval under the assumption that the signals are less present than absent in the sense that their probabilities of presence are less than or equal to one half.

3.2

The case of signals less present than absent: a thresholding function, a test for the signal detection and a search interval

A thresholding function

For every given ρ ∈ [0, ∞), let ξ(ρ) be the unique positive solution for x in the equation

0 F 1 (d/2; ρ 2 x 2 /4) = e ρ 2 /2 , (8)
where 0 F 1 is the generalized hypergeometric function ([6, p. 275]).

The map ξ is a thresholding function with C = 1/2. This follows from the three following facts: first, the increasingness of the generalized hypergeometric function 0 F 1 (d/2 ; •) and that of the function of ρ in the right-hand side of Eq. ( 8) imply the increasingness of ξ; second, ξ is a positive map because ξ(0

) = √ d; third, ξ(ρ) = (ρ/2) + (log 2/ρ)(1 + ψ 1 (ρ)) if d = 1 and ξ(ρ) = (ρ/2) + (d -1)(log ρ/ρ)(1 + ψ d (ρ)) otherwise, with lim ρ→+∞ ψ d (ρ) = 0 for all d ≥ 1 ([8, Lemma VI.1]).
When the signals are less present than absent, the use of this thresholding function for the computation of the ESE is particularly relevant. We now explain why.

Given a thresholding function θ and under the assumptions of theorem 2.1, it follows from lemma C.1 that the probability P({ Λ k + X k ≤ σ 0 θ(a(Λ)/σ 0 )}) tends to 0 uniformly in k when a(Λ) tends to infinity and that the faster the convergence of P({ Λ k + X k ≤ σ 0 θ(a(Λ)/σ 0 )}) to 0, the better. Now, assume that the signals are less present than absent. We keep this assumption in the rest of this section. Given h ∈ [0, ∞), let the thresholding test T h with threshold height h stand for the map I [h,∞) ( • ), that is, the map such that T h (u) = 1 if u ≥ h and T h (u) = 0 otherwise. Then, according to [8, Theorem VII-1], for every k ∈ N, the probability of error of the test T σ 0 ξ(a(Λ)/σ 0 ) for the decision about the value of ε k is less than or equal to V (a(Λ)/σ 0 ) where, for every ρ ∈ [0, ∞),

V (ρ) = e -ρ 2 /2 2 d/2 Γ(d/2) ξ(ρ) 0 e -t 2 /2 t n-1 0 F 1 (d/2 ; ρ 2 t 2 /4)dt + 1 2 1 - 1 2 d/2-1 Γ(d/2) ξ(ρ) 0 e -t 2 /2 t d-1 dt , (9)
and where Γ is, as usual, the standard (complete) gamma function. The function V (ρ) decreases very rapidly when ρ increases. Therefore, when a(Λ) increases, the probability of error of the test T σ 0 ξ(A/σ 0 ) and thus the probability P({ Λ k + X k ≤ σ 0 ξ(a(Λ)/σ 0 )}) tend to 0 uniformly in k and, above all, very rapidly. This is what we need. Thus, we recommend and use the thresholding function ξ to compute the ESE when signals are less present than absent.

The Essential Supremum Test

Given α ∈ [0, ∞), the test I ( • ≥ σ 0 ξ(α/ σ 0 )) assigns, to a given d-dimensional observation, the value 1 if the norm of the observation is above or equal to σ 0 ξ(α/ σ 0 ) and 0 otherwise. This test is henceforth called the Essential Supremum Test (EST). Strictly speaking, it is not a thresholding test since the threshold height to which the norm of a given observation is compared is the random variable σ 0 ξ(α/ σ 0 ). However, with a slight abuse of language, we will denote it by T σ 0 ξ(α/ σ 0 ) . Assume that the signals are less present than absent and have norms larger than or equal to α. If α and the sample size m are large enough, the ESE σ 0 should be a good estimate of σ 0 ; therefore, we can expect that the performance of the EST approaches that of the test T σ 0 ξ(α/σ 0 ) and, thus, that the probability of error of the EST does not significantly exceed V (α/σ 0 ). If the probabilities of presence of the signals all equal one half and the signals are uniformly distributed on αS d-1 , which is the sphere centred at the origin with radius α in R d , the probability of error of the test T σ 0 ξ(α/σ 0 ) is V (α/σ 0 ) for the detection of any of these signals (see [START_REF] Pastor | A Sharp Upper-Bound for the Probability of Error of the Likelihood Ratio Test for Detecting Signals in White Gaussian Noise[END_REF]Theorem VII.1]). Hence, in this particular case, the probability of error of the EST should become close to V (α/σ 0 ) when m and α are large enough.

The computation of the probability of error of the EST is still an open issue and the foregoing claims remain intuitive only. However, the experimental results presented in section 4 agree with them.

A search interval

We conclude this section by presenting a search interval for the computation of the ESE. This search interval is defined so that the EST satisfies two constraints. In what follows, U [k] , k = 1, 2, . . . , m, stands for the sequence of observations U 1 , U 2 , . . . , U m , sorted by increasing norm.

The left endpoint of the search interval is chosen so as to guarantee that the EST accepts a minimum number of times the null hypothesis, i.e. the hypothesis that no signal is present. More specifically, let Q be some positive real number less than or equal to 1 -m 4(m/2-1) 2 . According to Bienaymé-Chebyshev's inequality, the probability that the number of observations due to noise alone is above

k min = m/2 -hm is larger than or equal to Q if h = 1/ 4m(1 -Q). The lower bound of the search interval is then set to σ min = U [k min ] / √ d.
For every non-negative real value ρ, ξ(ρ) ≥ √ d ( [START_REF] Pastor | A Sharp Upper-Bound for the Probability of Error of the Likelihood Ratio Test for Detecting Signals in White Gaussian Noise[END_REF]). Therefore, an ESE above or equal to σ min guarantees a minimum of k min acceptances of the null hypothesis by the EST, whatever the value of the minimum amplitude of the signals.

The right endpoint is computed so as to guarantee that the EST accepts at least once the alternative hypothesis, i.e. the hypothesis that some signal is present. Hence, the upper bound of the search interval is fixed to

σ max = U [m] / √ d.
An estimate σ 0 larger than σ max would imply that U k ≤ σ 0 ξ(α/ σ 0 ) for every k = 1, 2, . . . , m, since ξ(ρ) ≥ √ d for all ρ ∈ [0, ∞). The outcome of the EST could then be that no signal is present whereas the full absence of signals amongst m observations is hardly probable when m is large, provided that the probabilities of presence are not too small. The experimental results presented in the sequel were obtained by choosing

Q = (1 - m 4(m/2-1)
2 ) × 1000 /1000 where x rounds the real value x to the nearest integer smaller than or equal to x.

4 Application to modulated sinusoidal carriers in AWGN: empirical bias and empirical MSE of the ESE, non-coherent detection of the carriers via the EST

The statistical properties of the ESE and the probabilities of error of the first and second kind of the EST are not known yet. Hence, in this section, we will restrict ourselves to some experimental results concerning the following case study.

The observations considered below, and thus the signals and noise, are two-dimensional random vectors (d = 2), or equivalently, complex random variables. We assume that the norms of the random signals all equal some known value α, and that each random signal is, in fact, uniformly distributed on the circle αS 1 centred at the origin with radius α. Therefore, the two components of each signal can be regarded as the in-phase and quadrature components of some modulated sinusoidal carrier with amplitude α and phase uniformly distributed in [0, 2π]. This model is relevant in many applications.

In what follows, we address the case where the probabilities of presence of the carriers all equal some unknown p ∈ (0, 1/2]. Consequently, the carriers are less present than absent and we make use of the results of section 3.2. In particular, note that the expression of the thresholding function ξ introduced in paragraph 3.2.1 simplifies in the two-dimensional case considered in this section: according to [1, Eq. 9.6.47, p. 377], for every x ∈ [0, ∞),

I 0 (x) = 0 F 1 (1; x 2 /4), (10)
where I 0 is the zeroth-order modified Bessel function of the first kind. Therefore, by setting d = 2 in Eq. ( 8), we obtain, in the two-dimensional case and for every given ρ ∈ [0, ∞)

ξ(ρ) = I -1 0 (e ρ 2 /2 )/ρ. (11)
In the next subsection, we present the absolute value |Bias| of the empirical bias and the empirical MSE of the ESE when this estimate is obtained by minimizing Eq. ( 7) when the search interval is computed according to paragraph 3.2.3 and the thresholding function is given by Eq. [START_REF] Press | Numerical recipes in C, The Art of Scientific Computing[END_REF].

In subsection 4.2, the EST of paragraph 3.2.2 is used to make a decision about the value of every given ε k , k = 1, 2, . . . , m. This amounts to achieving the so-called "non-coherent detection of the modulated sinusoidal carriers" ([10, Example III.B.5, p. 65]), the modulated carriers being represented by the random signals Λ k , k = 1, 2, . . . , m. This detection problem is standard and arises in many applications.

Empirical bias and MSE of the ESE when observations are noisy modulated sinusoidal carriers

Given N independent copies of the sequence U 1 , U

Application of the EST to the non-coherent detection of the carriers

Since we assume that each signal is uniformly distributed on αS 1 and that the probabilities of presence of the signals all equal some p ∈ (0, 1/2], the probability of error of the test T σ 0 ξ(α/σ 0 ) is less than or equal to V (α/σ 0 ) for making a decision about the value of ε k , given k ∈ N (see paragraph 3.2.2).

The following fact, which can easily be derived from [ uniformly distributed on αS 1 , the test T σ 0 ξ(α/σ 0 ) , where ξ is given by Eq. ( 11), is the Bayesian likelihood ratio test guaranteeing the least probability of error among all the possible tests for deciding about the value of ε k . The probability of error of this test is then

V (ρ) = 1 2 e -ρ 2 /2 ξ(ρ) 0 te -t 2 /2 I 0 (ρt)dt + 1 2 e -ξ(ρ) 2 /2 , (12)
which follows from Eqs. ( 9) with d = 2, ( 10) and [START_REF] Press | Numerical recipes in C, The Art of Scientific Computing[END_REF].

Since the empirical bias and the empirical MSE of the ESE presented above suggest that the ESE is a reasonably good estimate of σ 0 when α and m are large enough, we can reasonably expect that, for large values of α and m, the EST behaves as described in section 3.2.2: if P({ε k = 1}) ≤ 1/2, the probability of error of the EST should not exceed V (α/σ 0 ) and become rather close to V (α/σ 0 ) when P({ε k = 1}) = 1/2.

In the absence of theoretical results about the probability of error of the EST, even under the assumptions made in this section, the foregoing claim remains intuitive only. This is why, for different values of α, m and p, we computed the BER of the EST via a Monte-Carlo simulation: for every k ∈ {1, 2, . . . , m}, the EST was used to make a decision about the value of ε k and, since the m observations and the m successive decisions made by the EST are independent, we calculated the BER of the EST by counting the number of errors made by the EST. More specifically, the several Monte-Carlo simulations yielding the results presented below were achieved according to the following protocol, standardly employed to assess telecommunication systems.

Given α, m and p, independent trials of m observations each were carried out until two conditions were fulfilled. First, the total number N e of errors made by the EST had to be above or equal to a specified number N . Second, a minimum number M of trials had to be achieved. The reason for this second condition is the following one. Inasmuch as the decision about the presence or the absence of signals is made on the observations used for estimating σ 0 , the accuracy of the estimate affects m decisions at one go. This effect is then reduced by performing a minimum number of trials. If j is the first trial number larger than or equal to M and for which the total number of errors N e becomes larger than or equal to N , the BER is then defined as the ratio N e /(j × m). The simulations presented below were performed with σ 0 = 1. The pre-specified number of errors was fixed to N = 400 and the minimum number of trials was set to M = 150.

When the signals have the same amplitude α and the same probability of presence p ≤ 1/2, the BER's of the EST for different values of α, p and m are those of figure 1. These results show that, under the experimental conditions considered in this section, the EST behaves as expected: when p = 1/2, the larger α and m, the smaller the difference between the BER of the EST and V (α/σ 0 ); yet for p = 1/2, for every tested sample size, the difference between the BER of the EST and V (α/σ 0 ) becomes rapidly small when the value of α increases; when p is less than one half, the BER of the EST remains less than or equal to V (α/σ 0 ). Now, assume that the amplitudes of the signals all equal κα where κ is some real number larger than 1 and that the probabilities of presence of these signals all equal some p ∈ (0, 1/2]. The expected behaviour of the EST (see section 3.2.2) is that the probability of error of the EST remains less than or equal to V (α/σ 0 ) if α and m are large enough. Monte-Carlo simulations of the same type as those described above were carried out for different values of α and p. When κ = 1.1220, that is, when the actual norms of the signals are one dB larger than the value α with respect to which the EST is adjusted, the results obtained are then those of figure 2: the BER of the EST remains, as expected, less than or equal to V (α/σ 0 ) for every α, every m and every p.

Perspectives and extensions

This paper has presented a limit theorem, namely theorem 2.1, that concerns independent observations deriving from the random presence or absence of signals whose probability distributions and prior probabilities of presence are unknown in independent AWGN. On the basis of this theorem, we have introduced the Esssential Supremum Estimate (ESE), an estimate of the noise standard deviation. The Essential Supremum Test (EST) is a binary hypothesis test also introduced in this paper for the detection of signals that are less present than absent and whose norms are equal to or above some known α ∈ [0, ∞). The ESE and the EST have been tested in the particular case, relevant in many applications, where the signals are two-dimensional random vectors representing modulated sinusoidal carriers whose amplitudes all equal some known α ∈ [0, ∞) and whose phases are uniformly distributed in [0, 2π]. Experimental results obtained when the signals are less present than absent suggest that, under the assumptions made for the signals, the convergence stated by theorem 2.1 is fast. This is the reason why complementary theoretical studies in progress are aimed at analysing to what extent the asymptotic conditions of theorem 2.1 can actually be relaxed, as well as studying the statistical properties of the ESE and the performance of the EST. In particular, the influence of L, r, s and the thresholding function must be addressed. In fact, for signals less present than absent, the author's feeling is that theorem 2.1 and the ESE can be exploited to derive an estimate that requires no lower bound for the norms of the signals. Attempts of this type are presented in [START_REF] Amehraye | On the Application of Recent Results in Statistical Decision and Estimation Theory to Perceptual Filtering of Noisy Speech Signals[END_REF] and [START_REF] Pastor | From non parametric statistics to speech denoising[END_REF]. In these papers, methods for denoising speech signals corrupted by independent AWGN are adjusted by an estimate of the noise standard deviation. This estimate is performed on the basis of the ESE obtained by setting α = 0, a trivial lower bound for the amplitudes of the signals. The performance of the denoising so adjusted is very close to that achieved by using the exact value of the noise standard deviation.

In order to state results that extend theorem 2.1 and give better insight into the behaviour of the ESE and the EST, we are strongly interested in establishing links between theorem 2.1, [8, Theorem VII.1] and Wald's fundamental paper [START_REF] Wald | Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large[END_REF].

Although theorem 2.1 is stated for signals whose probabilities of presence are upper-bounded by p ∈ [0, 1), the EST and the experiments of section 4 concern the particular case p = 1/2 and the thresholding function ξ. Can the thresholding function ξ be used even when the upper bound for the probabilities of presence exceeds one half? From a more general point of view, we are looking for thresholding functions other than those proposed in the paper and, to deal with situations where signals may have probabilities of presence above one half, we plan to analyse the behaviour of the ESE adjusted with such thresholding functions so as to derive extensions of the EST.

As far as practical applications of the contents of this paper are concerned, speech denoising has already be mentioned as a promising field of application of the foregoing results (see [START_REF] Amehraye | On the Application of Recent Results in Statistical Decision and Estimation Theory to Perceptual Filtering of Noisy Speech Signals[END_REF] and [START_REF] Pastor | From non parametric statistics to speech denoising[END_REF]). A rather natural application of the approach presented in this paper is the design of Constant False Alarm Rate (CFAR) systems for the detection of radar targets. Our intention is to study to what extent theorem 2.1, the ESE and the EST are complementary to standard results and algorithms such as those described in [START_REF] Minkler | The Principles of Automatic Radar Detection in Clutter[END_REF]. ESM is also a possibly relevant field of application where the ESE and the EST could contribute to the interception of non-cooperant communications. Proximity sensing aimed, for instance, at informing a robot that it is approaching an object or that something is near it, can also be regarded as a potential application area where our results could apply. Finally, another possibly relevant and exciting field of theoretical and practical application of the material presented in this paper is the design and analysis of distributed detection systems (see [START_REF] Varshney | Distributed Detection and Data Fusion[END_REF]), where the absence of prior knowledge about the statistics of the observations is one of the main issues to overcome.

Acknowledgement

The author is very grateful to the reviewers for their insightful remarks that helped to significantly improve this paper. In particular, their suggestions strongly encourage the author to extend his approach in relation to Wald's theory and to address applications such as distributed detection and proximity sensing.

The author is also very thankful to Janet Ormrod for her help in improving the English and the organization of the paper.

Many thanks to Roger Gay and Albert 'Bert' Groenenboom for their keen suggestions and remarks.

Appendix

The notations introduced in theorem 2.1 are kept with the same meaning throughout this proof.

After stating some preliminary results in the next subsection, our first purpose is to prove that the noise standard deviation satisfies Eqs. ( 5) and ( 6). This will be achieved in three steps, namely subsections B, C and D.

More specifically, in subsection B, the behaviour of m k=1 U k q I( U k ≤ T ) is studied when m is large, T is a positive real value and q is any nonnegative real number. A corollary of Kolmogorov's classical strong limit theorem shows the almost sure convergence of the ratio

m k=1 U k q I( U k ≤ T )/E m k=1 U k q I( U k ≤ T )
to 1. From this almost sure convergence, we derive that of the ratio

m k=1 U k r I( U k ≤ T )/ m k=1 U k s I( U k ≤ T ) m k=1 E [ U k r I( U k ≤ T )] / m k=1 E [ U k s I( U k ≤ T )] when 0 ≤ r < s ≤ ν/2 and ν ∈ (0, ∞].
In subsection C, we then analyse the behaviour of the denominator of the ratio above when

T is βθ(α), Λ ∈ ∞ (N, L ν (Ω, R d )), a(Λ) ≥ α if ν = ∞ and Λ ∞ ≤ α otherwise.
We will then see that this ratio can be approximated by Υ r (βθ(α))/Υ s (βθ(α)).

In subsection D, the two types of convergence studied previously will be combined to conclude that σ 0 actually satisfies Eqs. ( 5) and [START_REF] Lebedev | Special Functions and their Applications[END_REF].

In subsection E, we complete the proof of statement (i) by showing that the noise standard deviation is the sole solution in σ to Eq. [START_REF] Köthe | Topological Vector Spaces-I[END_REF].

From now on, we assume that σ 0 = 1 and carry out the proof in this case. The reader will easily verify that this simplification infers no loss of generality.

A Preliminary results

For every k ∈ N, X k and Λ k are assumed to be independent and the distribution of X k is absolutely continuous with respect to Lebesgue's measure in R d . Therefore, given k ∈ N, the distribution of the random vector Λ k + X k is absolutely continuous with respect to Lebesgue's measure in R n . Its density, denoted hereafter by f Λ k +X k , is given by

f Λ k +X k (u) = 1 (2π) d/2 R d e -u-λ 2 /2 P Λ k (dλ), u ∈ R d (13)
where the probability distribution P Λ k is the positive measure such that P Λ k (E) = P(Λ -1 k (E)) for any Borel set E of R d . For every k ∈ N, the distribution of Λ k + X k is absolutely continuous with respect to Lebesgue's measure. The density f Λ k +X k of this distribution is given, for any non-negative real number x, by

f Λ k +X k (x) = 2π d/2 Γ(d/2) x d-1 S d-1 f Λ k +X k (xu)Σ(du), (14)
where Σ stands for the rotation-invariant positive Borel measure on the sphere S d-1 = {u ∈ R d : u = 1} for which Σ(S d-1 ) = 1 and Γ is the usual complete Gamma function. It then follows from Eqs. ( 13) and ( 14), and Fubini's theorem, that, for any non-negative real number x,

f Λ k +X k (x) = e -x 2 /2 x d-1 2 d/2-1 Γ(d/2) R d e -λ 2 /2 S d-1 e x(λ|u) Σ(du) P Λ k (dλ), (15)
where u|v

= d i=1 u i v i , u, v ∈ R d , is the usual scalar product in R d . It is then known ([8, Lemma A.1]) that S d-1 e x(λ|u) Σ(du) = 0 F 1 (d/2 ; x 2 λ 2 /4),
where 0 F 1 is the generalized hypergeometric function ([6, p. 275]). We thus derive from Eq. (15) that

f Λ k +X k (x) = e -x 2 /2 x d-1 2 d/2-1 Γ(n/2) R d e -λ 2 /2 0 F 1 (d/2; x 2 λ 2 /4) P Λ k (dλ). ( 16
)
For any non-negative real numbers q and T , the following equalities are trivial but will prove very useful in the sequel. To begin with, we have

E [ Λ k + X k q I( Λ k + X k ≤ T )] = T 0 x q f Λ k +X k (x)dx (17)
and

E [ X k q I( X k ≤ T )] = T 0 x q f d (x)dx, (18)
where

f d (x) = 1 2 d/2-1 Γ(d/2) e -x 2 /2 x d-1 , x ≥ 0. ( 19
)
is the probability density function of the square root of any centred chi-2 distribution with d degrees of freedom. With p k and q k henceforth standing for P({ε k = 1}) and 1 -p k respectively, we also can write that

E [ U k q I( U k ≤ T )] = p k E [ Λ k + X k q I( Λ k + X k ≤ T )] + q k T 0 x q f d (x)dx. (20)

B The case of large sample sizes

The behaviour of m k=1 U k r I( U k ≤ T )/ m k=1 U k s I( U k ≤ T ) is given in lemma B.2 when m is large and T is any positive real value. The starting point is the following corollary ([12, chapter 5, p 108, corollary 3]) of Kolmogorov's classical strong limit theorem.

Lemma [Rao] B.1 If Y 1 , Y 2 , . . .
U k q I( U k ≤ T )/E m k=1 U k q I( U k ≤ T ) = 1 (a-s).
Proof: The proof involves establishing that (i') the sequence (V ar ( U k q I( U k ≤ T ))) k∈N is upper-bounded;

(ii') the sequence (E [ U k q I( U k ≤ T )]) k∈N is bounded away from 0.

Actually, the two conditions (i') and (ii') are sufficient to guarantee that the sequence ( U k q I( U k ≤ T )) k∈N of independent variables fulfils conditions (i) and (ii) of lemma B.1 and, thereby, to conclude.

Clearly, X ∈ ∞ (N, L 2q (Ω, R d )); on the other hand, it follows from Eq. (2) that ∞ (N, L ν (Ω, R d )) is a subset of ∞ (N, L 2q (Ω, R d )). Therefore, Λ+X is an element of ∞ (N, L 2q (Ω, R d )). Condition (i') then holds true as a consequence of the inequality V ar( U

k q I( U k ≤ T )) ≤ E [ Λ k + X k 2q ] + E [ X k 2q ]
, which straightforwardly follows from Eq. (20).

Condition (ii') derives from the non-vanishing behaviour of the probabilities of absence q k , k ∈ N, the fact that T is positive and the easy inequality

E[ U k q I( U k ≤ T )] ≥ (1 -p) T 0 x q f d (x)dx,
which also follows from Eq. ( 20).

The subsequent result straightforwardly derives from the foregoing.

Corollary B.3 Under assumptions (Ass. 2), (Ass. 3) and (Ass. 4), for every pair (r, s) of real numbers such that 0 ≤ s < r ≤ ν/2 and every positive real number T ,

lim m→∞       m k=1 U k r I( U k ≤ T ) m k=1 U k s I( U k ≤ T ) × E m k=1 U k s I( U k ≤ T ) E m k=1 U k r I( U k ≤ T )       = 1 (a-s).

C The case of large and small amplitudes

The thresholding function θ is of most importance in the present section where, given β ∈ [0, 1], we derive the behaviour of the ratio

E m k=1 U k r I( U k ≤ βθ(a(Λ))) / E m k=1 U k s I( U k ≤ βθ(a(Λ)))
when a(Λ) tends to ∞ and that of

E m k=1 U k r I( U k ≤ βθ( Λ ∞ )) / E m k=1 U k s I( U k ≤ βθ( Λ ∞ ))
when Λ ∞ tends to 0. The starting point of the analysis is the following result.

Lemma C.1 Under assumption (Ass. 1), for every given q ∈ [0, ∞),

(i) lim a(Λ)→∞ E [ Λ k + X k q I( Λ k + X k ≤ βθ(a(Λ)))] = 0, uniformly in k and β ∈ [0, 1]. (ii) If Λ ∈ ∞ (N, L ∞ (Ω, R d ))
, then, in addition to statement (i),

lim Λ ∞→0 E [ Λ k + X k q I( Λ k + X k ≤ βθ( Λ ∞ ))] -E X k q I( X k ≤ βθ( Λ ∞ )) = 0, uniformly in k and β ∈ [0, 1].
Proof: We start by proving statement (i). For the sake of simplifying notations, we put ρ = a(Λ). Given any β ∈ [0, 1], we easily derive from Eq. ( 17) that E

[ Λ k + X k q I( Λ k + X k ≤ βθ(ρ))
] is less than or equal to θ(ρ) q P ({ Λ k + X k ≤ θ(ρ)}). Our proof involves showing that the latter quantity tends to 0 uniformly with k when ρ tends to ∞.

We derive from the properties of θ that, for ρ large enough, Cρ < θ(ρ) ≤ ρ. According to Eq. ( 13) and since, for any natural number k, Λ k ≥ ρ (a-s), the probability that Λ k + X k lies inside the ball of radius θ(ρ) can be written as follows:

P ({ Λ k + X k ≤ θ(ρ)}) = 1 (2π) d/2 x ≤Cρ λ ≥ρ e -x-λ 2 /2 P Λ k (dλ)dx, + 1 (2π) d/2 Cρ< x ≤θ(ρ) λ ≥ρ e -x-λ 2 /2 P Λ k (dλ)dx. (21)
Because x -λ ≥ ρ(1 -C) for x ≤ Cρ and λ ≥ ρ, the first term in the right-hand side of Eq. ( 21) is bounded by a constant multiplied with

ρ d e -(1-C) 2 2 ρ 2 . For Cρ < x ≤ θ(ρ) and λ ≥ ρ, x -λ is larger than or equal to ρ(1 -C) -γ(ρ).
(ρ) Cρ ) d -1 ρ n e -(1-C)-γ(ρ)/ρ 2 (ρ 2 /2) .
In addition, θ(ρ) ∼ Cρ when ρ tends to +∞. It thus follows from the foregoing that, for every non-negative real number q, θ(ρ) q P ({ Λ k + X k < θ(ρ)}) tends to 0 uniformly with k when ρ = a(Λ) tends to ∞, which completes the proof of statement (i).

We now turn our attention to the proof of statement (ii). Still for notational sake, we set ρ = Λ ∞ . An easy consequence of Eq. ( 17) is that, for any β ∈ [0, 1], the absolute value of the difference

E Λ k + X k q I( Λ k + X k ≤ βθ(ρ)) -E X k q I( X k ≤ βθ(ρ))
is less than or equal to θ (ρ) q θ(ρ)

0 |f Λ k +X k (x) -f d (x)|dx.
Our intention is then to prove that the latter quantity tends to 0 uniformly in k when ρ tends to 0. The hypergeometric function 0 F 1 (d/2 ; •) is continuous, increasing and 0 F 1 (d/2 ; 0) = 1. Hence, for λ ≤ ρ and x ∈ [0, θ(ρ)],

1 ≤ 0 F 1 d/2 ; x 2 λ 2 /4 ≤ 0 F 1 d/2 ; ρ 2 θ(ρ) 2 /4 .
From the inequality above, Eqs. ( 16) and ( 19), and the fact that for every k ∈ N, Λ k is less than or equal to ρ (a-s), it follows that, for all

x ∈ [0, θ(ρ)], e -x 2 /2 x d-1 2 d/2-1 Γ(d/2) e -ρ 2 /2 -1 ≤ f Λ k +X k (x) -f d (x)
and

f Λ k +X k (x) -f d (x) ≤ e -x 2 /2 x d-1 2 d/2-1 Γ(d/2) 0 F 1 d/2 ; ρ 2 θ(ρ) 2 /4 -1 .
The two functions of ρ that bracket f Λ k +X k (x) -f d (x) in the inequalities above both trivially tend to 0 independently of k when ρ tends to 0. Since θ is continuous at the origin, we have lim ρ→0

I [0,θ(ρ)] (x)θ(ρ) q f Λ k +X k (x) -f d (x) = 0, uniformly in k. It follows from Lebesgue's dominated convergence theorem that θ(ρ) q θ(ρ) 0 | f Λ k +X k (x) -f d (x)
| dx tends to 0 uniformly in k when ρ tends to 0, which completes the proof.

Proposition C.2 Under assumptions (Ass. 1), (Ass. 2) and (Ass. 3), for every given β 0 ∈ (0, 1] and every given pair (r, s) of non-negative real numbers such that 0 ≤ s < r,

lim a(Λ)→∞       E m k=1 U k r I ( U k ≤ βθ(a(Λ))) E m k=1 U k s I ( U k ≤ βθ(a(Λ))) - Υ r (βθ(a(Λ))) Υ s (βθ(a(Λ)))       = 0 (22) uniformly in m and β ∈ [β 0 , 1]. Furthermore, if Λ belongs to ∞ (N, L ∞ (Ω, R d )), lim Λ ∞ →0       E m k=1 U k r I( U k ≤ βθ( Λ ∞ )) E m k=1 U k s I( U k ≤ βθ( Λ ∞ )) - Υ r (βθ( Λ ∞ )) Υ s (βθ( Λ ∞ ))       = 0 (23)
uniformly in m and β ∈ [β 0 , 1].

Proof: We prove Eqs. ( 22) and ( 23) together. As above, let ρ stand for either a(Λ) or Λ ∞ . In the latter case, it is implicitly assumed that

Λ ∈ ∞ (N, L ∞ (Ω, R d )).
Let q be any non-negative real value, β 0 any element of (0, 1] and β any element of [β 0 , 1]. It follows from Eq. ( 20) that

E m k=1 U k q I( U k ≤ βθ(ρ)) = m k=1 p k E [ Λ k + X k q I( Λ k + X k ≤ βθ(ρ)) ] -δ ,0 E [ X k q I( X k ≤ βθ(ρ)) ] + m k=1 q k + δ ,0 m k=1 p k βθ(ρ) 0 x q f d (x)dx, (24)
where = ∞ if ρ = a(Λ) and = 0 otherwise and δ a,b , a, b ∈ R, stands for the usual Kronecker symbol: δ a,b = 1 if a = b and δ a,b = 0 otherwise. According to lemma C.1, if ρ = a(Λ) (resp. ρ = Λ ∞ ), there exists some positive real number α 0 such that, for every α ≥ α 0 (resp. α ≤ α 0 ), every Λ such that a(Λ) ≥ α (resp. Λ ∞ ≤ α) and every β ∈ [β 0 , 1], each factor of p k in the right-hand side of Eq. ( 24) is less than or equal to η. By taking into account that each p k is smaller than or equal to p < 1 and that

m k=1 q k + δ ,0 m k=1 p k remains positive, it follows that E m k=1 U k q I( U k ≤ βθ(ρ)) m k=1 q k + δ ,0 m k=1 p k - βθ(ρ) 0 x q f d (x)dx ≤ p 1 -p η
for ρ large enough (resp. small enough). The right-hand side of the latter inequality does not depend on m. Moreover, lim ρ→ βθ(ρ) Proof: By differentiating Υ r (x)/Υ s (x) with respect to x ≥ 0, it follows that Υ r /Υ s is a non-decreasing map of [0, ∞] into [0, Υ r (∞)/Υ s (∞)] with Υ r (∞)/Υ s (∞) < ∞.

0 x q f d (x)dx = βθ( ) 0 x q f d (x)dx, whether = 0 or = ∞ since θ(0) > 0, θ(∞) = ∞ and β ≥ β 0 > 0. Thereby, we obtain lim ρ→ E m k=1 U k q I( U k ≤ βθ(ρ)) m k=1 q k + δ ,0 m k=1 p k = βθ( ) 0 x q f d (x)dx ∈ (0, ∞), uniformly in m and β ∈ [β 0 , 1]. Given two real numbers r > s ≥ 0, it straightforwardly follows that lim ρ→ E m k=1 U k r I( U k ≤ βθ(ρ)) E m k=1 U k s I( U k ≤ βθ(ρ)) = βθ( ) 0 x r f d (x)dx βθ( ) 0 x s f d (x)dx
On the other hand, since θ(x) ≥ θ(0) > 0 for every non-negative real number x, a consequence of Eqs. ( 17) and ( 20) is that, for all β > 0, where I K stands for the indicator function of a given set K. According to corollary B.3, for every ω in some measurable subset with measure 1 of Ω, the second factor in the right-hand side of the inequality above can be made arbitrarily small for m large enough, which completes the proof for a(•). The reader will then easily obtain the same result for • ∞ by mimicking the foregoing.

We now combine the almost everywhere convergence established by the latter lemma with the uniform convergence stated in proposition C.2 so as to complete the proof that σ = 1 satisfies Eqs. ( 5) and [START_REF] Lebedev | Special Functions and their Applications[END_REF]. In fact, we prove Eq. ( 5) only, because the proof of Eq. ( 6) can be achieved in exactly the same way as below by replacing a(•) with • ∞ .

Let β 0 ∈ (0, 1] and η be any positive real number. Let α 1 be a positive real value such that, for all α ∈ [α 1 , ∞), all Λ ∈ ∞ (N, L ν (Ω, R d )) such that a(Λ) ≥ α and every β ∈ [β 0 , 1], Eq. (25) holds true with ρ = a(Λ).

We derive from proposition C.

Summarizing, for any given real number β 0 ∈ (0, 1] and any positive real number η, we have found a positive real number α 0 such that, for any α ∈ [α 0 , ∞), any Λ ∈ ∞ (N, L ν (Ω, R d )) that satisfies a(Λ) ≥ α and any β ∈ [β 0 , 1], Eq. ( 27) holds true. This means that σ = 1 satisfies Eq. ( 5).

E Uniqueness of the solution to equation

Assume the existence of two positive real numbers σ 1 ≥ σ 2 > 0 that both satisfy Eq. ( 5). For notational sake, put ρ i = a(Λ)/σ i , i ∈ {1, 2} and set β i = θ(ρ i )/ρ i , i = 1, 2. For every positive real number x, 0 < Υ r (x)/Υ s (x) < ∞. Therefore, for any given pair (m, ω) ∈ N × Ω, we can write that

Figure 1 :

 1 Figure 1: BER of the EST for different values of m and α in comparison to V (α/σ 0 ) .

Figure 2 :

 2 Figure 2: BER of the EST versus V (α/σ 0 ) for different values for m and α and when the amplitudes of the signals all equal κα with κ = 1.1220, that is when the amplitudes of the signals are one dB larger than expected.

  Hence, by taking into account the volume of the shell {x ∈ R d : Cρ < x ≤ θ(ρ)}, the second term in the right-hand side of Eq.

(

  21) is less than a constant times (1 + γ

,

  which concludes the proof. D Combining large sample sizes and large or small amplitudes Corollary B.3 holds true for any threshold T and real values r and s such that 0 ≤ s < r ≤ ν/2. Proposition C.2 is valid for all non-negative real values r and s such that s < r and thresholds deriving from the thresholding function θ. Hence, the following lemma D.1 particularizes corollary B.3 to such thresholds. The rest of the section then combines proposition C.2 and lemma D.1.Lemma D.1 Under assumptions (Ass. 1), (Ass. 2), (Ass. 3), (Ass. 4), let r and s be two real numbers such that 0 ≤ s < r ≤ ν/2. Then, for any β 0 ∈ (0, 1], there exists some positive real number α 0 (resp. α 1 ) such that, for any α in [α 0 , ∞) (resp. (0,α 1 ]), any Λ ∈ ∞ (N, L ν (Ω, R d )) (resp. Λ ∈ ∞ (N, L ∞ (Ω, R d ))) such that a(Λ) ≥ 1 (resp. Λ ∞ ≤ α) and any β ∈ [β 0 , with ρ = a(Λ) (resp. ρ = Λ ∞ ).

U

  U k ≤ βθ(a(Λ))) /E m k=1 U k s I( U k ≤ βθ(a(Λ))) > 0Let β 0 be an element of (0, 1] and η be some positive real number. It follows from proposition C.2 and the remarks above that there exists some positive real number α 0 such that, for any real number α larger than or equal to α 0 , any Λ ∈ ∞ (N,L ν (Ω, R d )) such that a(Λ) ≥ α, any β ∈ [β 0 , 1] and any m ∈ N, U k ≤ βθ(a(Λ))) E m k=1 U k s I( U k ≤ βθ(a(Λ))) ≤ Υ r (∞) Υ s (∞) + η. If α ∈ [α 0 , ∞) and β ∈ [β 0 , 1], the latter inequality implies that, for every ω ∈ Ω and every m ∈ N,m k=1 U k (ω) r I [0,βθ(a(Λ))] ( U k (ω) ) m k=1 U k (ω) s I [0,βθ(a(Λ))] ( U k (ω) ) k (ω) r I [0,βθ(a(Λ))] ( U k (ω) ) m k=1 U k (ω) s I [0,βθ(a(Λ))] ( U k (ω) ) U k ≤ βθ(a(Λ))) -1 ,

  2 the existence of another positive real number α 2 such that, for any α∈ [α 2 , ∞), any Λ ∈ ∞ (N, L ν (Ω, R d )) with a(Λ) ≥ α, any β ∈ [β 0 , 1] and any natural number m, U k ≤ βθ(a(Λ))) -Υ r (βθ(a(Λ))) Υ s (βθ(a(Λ))) ≤ η.(26)For any α larger than or equal to α 0 = max(α 1 , α 2 ), any Λ in ∞ (N, L ν (Ω, R d )) such that a(Λ) ≥ α and any β ∈ [β 0 , 1], Eqs. (25) and (26) imply thatlim sup U k ≤ βθ(a(Λ))) -Υ r (βθ(a(Λ))) Υ s (βθ(a(Λ))) ≤ η (a-s), U k ≤ βθ(a(Λ))) m k=1 U k s I( U k ≤ βθ(a(Λ))) -Υ r (βθ(a(Λ))) Υ s (βθ(a(Λ))) ∞ ≤ η.

σ r-s 1 Υ 2 Υ 1 Υ 2 Υ

 1212 r (β 2 θ(ρ 1 )) Υ s (β 2 θ(ρ 1 )) -σ r-s r (β 1 θ(ρ 2 )) Υ s (β 1 θ(ρ 2 )) ≤ m k=1 U k r I [0,β 2 σ 1 θ(ρ 1 )] ( U k (ω) ) m k=1 U k s I [0,β 2 σ 1 θ(ρ 1 )] ( U k (ω) ) -σ r-s r (β 2 θ(ρ 1 )) Υ s (β 2 θ(ρ 1 )) + m k=1 U k r I [0,β 2 σ 1 θ(ρ 1 )] ( U k (ω) ) m k=1 U k s I [0,β 2 σ 1 θ(ρ 1 )] ( U k (ω) ) -σ r-s r (β 1 θ(ρ 2 )) Υ s (β 1 θ(ρ 2 )).

  Notations and terminology . . . . . . . . . . . . . . . . . . . . 2.2 The theoretical result . . . . . . . . . . . . . . . . . . . . . . . 3 Application to the estimation of the noise standard deviation and the detection of the signals 3.1 The Essential Supremum Estimate . . . . . . . . . . . . . . . 3.2 The case of signals less present than absent: a thresholding function, a test for the signal detection and a search interval . 3.2.1 A thresholding function . . . . . . . . . . . . . . . . . 3.2.2 The Essential Supremum Test . . . . . . . . . . . . . . 3.2.3 A search interval . . . . . . . . . . . . . . . . . . . . . The case of large and small amplitudes . . . . . . . . . . . . . D Combining large sample sizes and large or small amplitudes . E Uniqueness of the solution to equation . . . . . . . . . . . . .

	Collection Contents
	des Rapports 1 Introduction
	de Recherche A limit theorem for processing signals that 2 A limit theorem for sequences of independent random vec-de l'ENST Bretagne tors that have unknown probability distributions have unknown probability distributions and unknown probabilities of presence 2.1
	in white Gaussian noise
	RR-2006007-SC
	Un théorème limite pour le traitement de
	signaux dont les distributions de probabilité et
	les probabilités de présence sont inconnues
	dans un bruit blanc Gaussien
	Rapport Interne GET / ENST Bretagne
		Dominique Pastor
	Ecole Nationale Supérieure des Télécommunications de Bretagne,
		CNRS TAMCIC (UMR 2872),
		Technopôle de Brest Iroise, CS 83818
	29238 BREST Cedex, FRANCE inconnues dans un bruit blanc présence sont e-mail: dominique.pastor@enst-bretagne.fr
		Gaussien
		Rapport Interne
	2006	GET / ENST Bretagne
		Dominique PASTOR

4 Application to modulated sinusoidal carriers in AWGN: empirical bias and empirical MSE of the ESE, non-coherent detection of the carriers via the EST 4.1 Empirical bias and MSE of the ESE when observations are noisy modulated sinusoidal carriers . . . . . . . . . . . . . . . 4.2 Application of the EST to the non-coherent detection of the carriers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Perspectives and extensions 6 Acknowledgement Appendix A Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . B The case of large sample sizes . . . . . . . . . . . . . . . . . . C

Table 1 :

 1 2 , . . . , U m , where the observations U 1 , U 2 , . . . , U m obey the assumptions introduced above, we can obtain N samples of the ESE and, then, calculate the empirical bias and the empirical MSE of the ESE. In this respect, tables 1, 2 and 3 display the absolute value of the empirical bias and the empirical MSE on the basis of N = 500 samples of the ESE, when the signals have the same amplitude α and the same probability of presence p equal to 0.1, 0.3 and 0.5, respectively. These results comply with theorem 2.1 since the larger the amplitude α, the smaller the bias and the MSE of the estimate. These experimental results also show that the ESE quite rapidly becomes a reasonably good estimate of σ 0 with increasing values for α and m. Absolute value of the empirical bias and empirical MSE of the ESE when signals are uniformly distributed on the circle centred at the origin with radius α and have a probability of presence equal to 0.1.

	α	0.5	1	1.5	2	2.5	3	3.5	4
	m = 100	|Bias| 0.212 0.119 0.095 0.063 0.044 0.039 0.038 0.048 MSE 0.166 0.095 0.068 0.038 0.025 0.020 0.015 0.012
	m = 200	|Bias| 0.136 0.114 0.048 0.012 0.011 0.028 0.019 0.021 MSE 0.088 0.056 0.041 0.037 0.026 0.014 0.011 0.008
	m = 400	|Bias| 0.101 0.056 0.023 0.019 0.016 0.003 0.001 0.009 MSE 0.051 0.034 0.019 0.011 0.007 0.005 0.004 0.003
	m = 800	|Bias| 0.060 0.044 0.022 0.001 0.013 0.001 0.003 0.013 MSE 0.020 0.015 0.009 0.004 0.004 0.003 0.002 0.002

Table 2 :

 2 Absolute value of the empirical bias and empirical MSE of the ESE when signals are uniformly distributed on the circle centred at the origin with radius α and have a probability of presence equal to 0.3.

	10, Example II.E.1,

Table 3 :

 3 Absolute value of the empirical bias and empirical MSE of the ESE when signals are uniformly distributed on the circle centred at the origin with radius α and have a probability of presence equal to 0.5.

  is a sequence of independent random variables with non-negative means α 1 , α 2 , . . . , and variances σ 2 1 , σ 2 2 , . . . , such that

	+∞ m=1 α m = +∞ (i) and +∞ m=1 σ 2 m /a 2 m < +∞ (ii), where a m = m k=1 α k , then lim m (1/a m ) m k=1 Y m = 1 (a-s).
	Proposition B.2 Under assumptions (Ass. 2), (Ass. 3) and (Ass. 4),
	for every non-negative real number q less than or equal to ν/2 and every po-
	sitive real number T ,
		m
	lim m→+∞	k=1

We derive from this inequality that σ r-s


Dominique Pastor
email: dominique.pastor@enst-bretagne.fr

On the coverage probability of the Clopper-Pearson confidence interval Sur la probabilité de couverture de l'intervalle de confiance de Clopper-Pearson Rapport Interne GET / ENST Bretagne

Keywords: Binomial proportion, Clopper-Pearson Confidence Interval, Confidence interval, Coverage probability, Incomplete beta function Intervalle de confiance, Intervalle de confiance de Clopper-Pearson, Fonction beta incomplète, Proportion binomiale, Probabilité de couverture

Given α ∈ (0, 1), the coverage probability of the 100(1-α)% Clopper-Pearson Confidence Interval (CPCI) for estimating a binomial parameter p is proved to be larger than or equal to 1 -α/2 for sample sizes less than a bound that depends on p. This is a mathematical evidence that, as noticed in recent papers on the basis of numerical results, the CPCI coverage probability can be much higher than the desired confidence level and thence, that the Clopper-Pearson approach is mostly inappropriate for forming confidence intervals with coverage probabilities close to the desired confidence level.

Soit α ∈ (0, 1). Considérons l'intervalle de confiance de Clopper-Pearson de niveau de confiance 100(1-α)%. On prouve que pour estimer une proportion binomiale p, la probabilité de couverture de cet intervalle est supérieure ou égale à 1 -α/2 lorsque la taille de l'échantillon est inférieure à une borne qui dépend de p. Ceci montre que, comme l'ont remarqué certains auteurs sur la base de résultats numériques, la probabilité de couverture de l'intervalle de confiance de Clopper-Pearson peut être largement supérieure au niveau de confiance 1 -α. Aussi, l'intervalle de confiance de Clopper-Pearson n'est pas adéquat dès que l'on veut garantir une probabilité de couverture proche du niveau de confiance spécifié.

Introduction

Interval estimation of a binomial proportion p is one of the most basic and methodologically important problems in practical statistics. A considerable literature exists on the topic and manifold methods have been suggested for bracketing a binomial proportion within a confidence interval.

Among those methods, the Clopper-Pearson Confidence Interval (CPCI), originally introduced in 1934 by Clopper and Pearson ([4]), is often referred as the "exact" procedure by some authors for it derives from the binomial distribution without resorting to any approximation to the binomial. However, despite this "exacteness", the CPCI is basically conservative because of the discreteness of the binomial distribution: given α ∈ (0, 1), the coverage probability of the 100(1 -α)% CPCI is above or equal to the nominal confidence level (1 -α).

In this paper, it is proved that this coverage probability is actually larger than or equal to 1 -α/2 if the sample size is less than ln(α/2)/ ln(max(p, 1p)). This bound basically depends on the proportion itself. Thence, as suggested by numerical results presented in [START_REF] Agresti | Approximate is better than "exact" for interval estimation of binomial proportions[END_REF] and [START_REF] Brown | Interval Estimation for a Binomial Distribution[END_REF], the CPCI is not suitable in practical situations when getting a coverage probability close to the specified confidence level is more desirable than guaranteeing a coverage probability above or equal to the said confidence level.

Theoretical results

Throughout the rest of this paper, α stands for some real value in the interval (0, 1) and n for some natural number.

We start by giving a description of the 100(1 -α)% CPCI, that is the CPCI with level of confidence (1 -α) where α ∈ (0, 1). This description is purposely brief for it focuses only on the material needed for stating proposition 1 below. For further details on the construction of the CPCI, the reader can refer to numerous papers and textbooks on statistics ( [START_REF] Agresti | Approximate is better than "exact" for interval estimation of binomial proportions[END_REF], [START_REF] Brown | Interval Estimation for a Binomial Distribution[END_REF], [?], [START_REF] Stuart | Kendall's Advanced Theory of Statistics, Fifth Edition[END_REF] amongst many others).

Let ( k ) k∈{0,...,n} be the sequence defined as follows. We put

0 := 0. ( 1
)
For k ∈ {1, . . . , n}, k is defined as the unique solution in (0, 1) for θ in the equation

n i=k n i θ i (1 -θ) n-i = α/2. (2)
Similarly, the sequence (u k ) k∈{0,...,n} is defined as follows. We set

u n := 1 (3)
and, for k ∈ {0, . . . , n -1}, u k is the unique solution in (0, 1) for θ in the equation

k i=0 n i θ i (1 -θ) n-i = α/2. ( 4
)
Let X henceforth stand for a binomial variate for sample size n and binomial parameter p ∈ [0, 1]. In other words, X stands for the total number of successes in n independent trials with constant probability p of success. Since X is valued in {0, 1, . . . , n}, the random variables X and u X are well-defined. The 100(1 -α)% CPCI for size n is then the random interval ( X , u X ) whose lower and upper endpoints are X and u X respectively. The CPCI coverage probability is defined as the probability P ({ X < p < u X }) that the CPCI actually brackets the true value p of the proportion. The 100(1 -α)% CPCI is known to be conservative in the sense that its coverage probability is always above or equal to the confidence level ([2], [START_REF] Brown | Interval Estimation for a Binomial Distribution[END_REF], [START_REF] Clopper | The use of confidence of fiducial limits illustrated in the case of the binomial[END_REF]).

This standard result is refined by the following proposition whose proof is given in section 3.

Proposition 2.1 With the same notations as above, (i) If n < -ln(α/2)/ ln(2), then

P ({ X < p < u X }) = 0 for p = 0, ≥ 1 -α/2 for 0 < p ≤ (α/2) 1/n , = 1 for (α/2) 1/n < p < 1 -(α/2) 1/n , ≥ 1 -α/2 for 1 -(α/2) 1/n ≤ p < 1, = 0 for p = 1. (ii) If n ≥ -ln(α/2)/ ln(2), then P ({ X < p < u X }) = 0 for p = 0, ≥ 1 -α/2 for 0 < p < 1 -(α/2) 1/n , ≥ 1 -α for 1 -(α/2) 1/n ≤ p ≤ (α/2) 1/n , ≥ 1 -α/2 for (α/2) 1/n < p < 1, = 0 for p = 1.
The following result then easily derives from the proposition above.

Lemma 2.2 With the same notations as above, for every proportion p ∈ (0, 1) and every natural number n less than ln(α/2)/ ln(max(p, 1 -p)), the CPCI coverage probability is larger than or equal to 1 -α/2.

Proof: Since max(p, 1 -p) is larger than or equal to 1/2, -ln(α/2)/ ln(2) is less than ln(α/2)/ ln(max(p, 1 -p)). If n < -ln(α/2)/ ln(2), the result then follows from proposition 1, statement (i). If -ln(α/2)/ ln(2) ≤ n < ln(α/2)/ ln(max(p, 1 -p)), p is either larger than (α/2) 1/n or smaller than 1 -(α/2) 1/n and the result follows from proposition 1, statement (ii).

The theoretical results above are thus mathematical evidences that, as suggested in [START_REF] Agresti | Approximate is better than "exact" for interval estimation of binomial proportions[END_REF] and [START_REF] Brown | Interval Estimation for a Binomial Distribution[END_REF], the CPCI is inaccurate in the sense that "[...]its actual coverage probability can be much larger than 1 -α unless the sample size n is quite large."([3, Sec. 4.2.1]). 4.2.1).

Figures 1, 2 and 3 illustrate the foregoing by displaying the coverage probability of the 95% CPCI for p = 0.1, p = 0.05 and p = 0.01 respectively and sample sizes ranging from 1 to 500. In each figure, the value of ln(α/2)/ ln(max(p, 1 -p)) is represented by a vertical red line. On the left hand side of this line, coverage probabilities are all larger than or equal to 97.5%; on the right hand side of this same line, coverage probabilities can be less than 97.5% and even close to 95%.

Proof

Given θ ∈ [0, 1], let P θ stand for the distribution of a binomial variate for sample size n with binomial parameter θ and let F θ be the distribution function defined for every real value x by

F θ (x) = P θ ((-∞, x]). It is then convenient to set G θ (x) = P θ ([x, ∞))
and

F u k (k) = α/2 for every k ∈ {0, . . . , n -1}. (6)
According to [1, Eq. 6.6.4, p. 263], for every given θ ∈ [0, 1],

G θ (k) = I θ (k, n -k + 1) for k ∈ {1, . . . , n}, (7)
and

F θ (k) = 1 -I θ (k + 1, n -k) for k ∈ {0, . . . , n -1}. (8)
Proof: Set F = {k ∈ {0, . . . , n} : u k ≤ p}. It follows from statement (ii) of lemma 2 that F is empty if and only if 0 ≤ p < 1 -(α/2) 1/n . Therefore, if 0 ≤ p < 1 -(α/2) 1/n , u X is larger than p and statement (i) holds true. Under the condition 1 -(α/2) 1/n ≤ p < 1, F is not empty. According to statement (ii) of lemma 2 and (3), we obtain that F = {0, . . . , L} with L ≤ n -1. Therefore, the event {u X ≤ p} is the event {X ≤ L} so that P ({u X ≤ p}) = F p (L). Since u L ≤ p, it follows from statement (i) of lemma 1 that F p (L) ≤ F u L (L). According to [START_REF] Johnson | Distributions in Statistics, Discrete Distributions[END_REF], the right hand side in this inequality equals α/2 and statement (ii) follows.

Statement (iii) trivially holds true since u X ≤ 1.

We now complete the proof of proposition 2.1. According to lemma 3.2, statement (iii), X < u X . Thereby, {p ≤ X } ⊂ {p < u X } so that P ({ X < p < u X }) = P ({p < u X }) -P ({p ≤ X }) = 1 -P ({p ≥ u X }) -P ({p ≤ X })

Since the condition 1-(α/2) 1/n ≤ (α/2) 1/n is equivalent to n ≥ -ln(α/2)/ ln(2), proposition 1 straightforwardly follows from lemmas 3.3 and 3.4.

  /2)/ln(max(p,1-p)) = 35.0120

Figure 1 :

 1 Figure 1: The 95% CPCI-BS Coverage probabilities for proportion p = 0.1

  for every real value x. According to the definitions /2)/log(max(p,1-p)) = 71.9174

Figure 2 :

 2 Figure 2: The 95% CPCI-BS Coverage probabilities for proportion p = 0.05

Figure 3 :

 3 Figure 3: The 95% CPCI-BS Coverage probabilities for proportion p = 0.01

where I x (a, b) stands for the Incomplete Beta Function ([1, Eq. 6.2.1, p. 258, Eq. 6.6.2, p. 263, Eq. 26.5.1, p. 944]). Trivially, the maps θ ∈ [0, 1] -→ I θ (k + 1, n -k) and θ ∈ [0, 1] -→ I θ (k, n -k + 1) are strictly increasing. Thereby, we can straightforwardly state the following result.

The subsequent lemma states useful properties concerning the real values k and u k .

Lemma 3.2 With the same notations as above, (i) the sequence ( k ) k∈{0,...,n} is strictly increasing and n = (α/2) 1/n ;

(ii) the sequence (u k ) k∈{0,...,n} is strictly increasing and

(iii) for every given k ∈ {0, . . . , n}, k < u k .

Proof:

Proof of statement (i). Given k ∈ {1, . . . , n}, k ∈ (0, 1) and is therefore larger than 0 = 0. It then suffices to prove that k < k+1 for k ∈ {1, . . . , n -1} to establish the strict increasingness of the sequence

For every θ ∈ [0, 1], G θ (n) = θ n . It then follows from (5) that n n = α/2, which completes the proof of statement (i).

Proof of statement (ii). The strict increasingness of the sequence (u k ) k∈{0,...,n} derives from the same type of arguments as those used for proving statement (i). Given k ∈ {0, . . . , n -1}, u k belongs to (0, 1) and is therefore less than u n = 1. It then suffices to show that u k < u k+1 for every k ∈ {0, . . . , n -2} to establish the strict increasingness of the sequence.

Let k ∈ {0, . . . , n -2}. Trivially,

The result then follows from lemma 1, statement (i).

Given θ ∈ (0, 1), F θ (0) = (1 -θ) n . Therefore, by [START_REF] Johnson | Distributions in Statistics, Discrete Distributions[END_REF], we obtain that

Proof of statement (iii). According to (1), ( 3) and the values of n and u 0 , statement (iii) trivially holds true for k = 0 and k = n.

Consider any k ∈ {1, . . . , n -1}. We can write that

This follows from the definition of G θ , ( 5) and the fact that α < 1. Since F k (k) > F k (k -1) and α/2 = F u k (k) according to [START_REF] Johnson | Distributions in Statistics, Discrete Distributions[END_REF], we finally obtain that F k (k) > F u k (k). The fact that k < u k then straightforwardly derives from lemma 1, statement (i).

Lemma 3.3 With the same notations as above,

As far as statements (ii) and (iii) are concerned, it is convenient to introduce the set E = {k ∈ {0, . . . , n} : k ≥ p} . Statement (i) of lemma 2 implies the following facts. First, E is non empty if and only if p ≤ (α/2) 1/n ; second, if 0 < p ≤ (α/2) 1/n , E = {m, . . . , n} where m ≥ 1 according to [START_REF] Abramowitz | Handbook of Mathematical Functions[END_REF].

Thereby, X ≥ p if and only if X ≥ m. We therefore have that P ({ X ≥ p}) = G p (m). Now, since p ≤ m , it follows from lemma 1, statement (ii), that G p (m) ≤ G m (m). According to [START_REF] Johnson | Distributions in Statistics[END_REF], the right hand side in this inequality equals α/2. Therefore, we obtain that P ({ X ≥ p}) ≤ α/2.

If (α/2) 1/n < p ≤ 1, E is empty. Therefore, X < p and statement (iii) follows.

Lemma 3.4 With the same notations as above, (i) if 0 ≤ p < 1 -(α/2) 1/n , P u X ≤ p = 0, (ii) if 1 -(α/2) 1/n ≤ p < 1, P u X ≤ p ≤ α/2, (iii) if p = 1, P u X ≤ p = 1.


Keywords: Automatic speech recognition, spectrally reduced speech, amplitude modulation, frequency modulation

This report investigates the behavior of automatic speech recognition (ASR) system with spectrally reduced speech (SRS) synthesized from subband amplitude modulations (AMs) and frequency modulations (FMs). Acoustic analysis shows that the resynthesis of SRS from only AM components helps alleviate certain non-linguistic variabilities in the original speech signal. When the SRS spectral resolution is sufficiently good, this alleviation not only has no consequence but also yields comparable or even better ASR word accuracy compared to that attained with original clean speech signal. In contrast, FM components support human speech recognition but yield no significant improvement in terms of ASR word accuracy when the SRS spectral resolution is sufficiently good.

Résumé

Ce rapport étudie le comportement de système de reconnaissance automatique de la parole (ASR) avec la parole réduite spectralement (SRS) synthétisée à partir des modulations d'amplitude (AM) et de fréquence (FM). Les analyses d'acoustique montrent que la resynthèse de la SRS à partir des composants AM permet à s'affranchir certaines variabilités non-linguistiques dans le signal de parole original. Lorsque la résolution spectrale de la SRS est suffisamment bonne, cette simplification n'a aucune conséquence tout en gardant le précision de mot (word accuracy) de l'ASR comparable ou meilleure que celui atteinte par le signal de parole propre original. En revanche, les composants FM supportent la reconnaissance de la parole de l'être humain, mais n'apportent pas d'amélioration significative en terme de précision de mot de l'ASR lorsque la résolution spectrale de la SRS est suffisamment bonne.

Mots-clés

Reconnaissance automatique de la parole, parole réduite spectralement, modulation d'amplitude, modulation de fréquence.

Introduction

Speech communication among humans is stable despite the great variability in the speech signal. There are many sources of acoustic variance in the speech signal that are not directly associated with the linguistic message, including (1) acoustic degradations (e.g., constant or slowly varying additive noise, microphone frequency response, talker or microphone movement, etc.) and (2) speech production variations (e.g., accent and dialect, speaking style, acoustic variability due to specific states of health and mental state, etc.) [START_REF] Morgan | Speech processing in the auditory system[END_REF].

The spectrum of sound is considered to be an important correlate of phonetic quality of speech sounds [START_REF] Hermansky | Mel cepstrum, deltas, double-deltas,.. what else is new? Robust Methods for Speech Recognition in Adverse Conditions[END_REF]. However, the prime carrier of the linguistic information are changes of the spectral envelopes of the speech signal [START_REF] Hermansky | Should recognizers have ears?[END_REF]. Indeed, the relatively narrow bandwidth of the spectral envelope modulations is created by the relatively slow motion of the vocal tract during speech production. In this view, the "message" (the signal containing information of vocal tract motions with a narrow bandwidth) modulates the "carrier" signal (high frequency) analogous to the amplitude modulation (AM) used in radio communications. The major linguistically significant information in speech is contained in the details of this low frequency vocal tract motion (i.e., the spectral envelope of the speech signal) [START_REF] Avendano | Speech processing in the auditory system[END_REF].

In an automatic speech recognition (ASR) system, the speech analysis module performs feature extraction using signal processing techniques to compensate for variability in speaker characteristics as well as acoustic environment. It can help alleviate non-linguistic components of speech signal and improve reliability of ASR in realistic environments [START_REF] Hermansky | Mel cepstrum, deltas, double-deltas,.. what else is new? Robust Methods for Speech Recognition in Adverse Conditions[END_REF]. Most state-ofthe-art ASR systems use Mel cepstrum [START_REF] Davis | Comparison of parametric representations for monosyllabic word recognition in continuous spoken sentences[END_REF] or Perceptual Linear Predictive (PLP) analysis [START_REF] Hermansky | Perceptual linear predictive (plp) analysis of speech[END_REF] of speech in the speech analysis module. In the calculation of the feature vectors, the speech spectrum at a given time instant is derived from a segment of speech which is short enough (of the order of 10-20 ms) to be assumed to result from a stationary process. The phase of such a shortterm spectrum is typically discarded. The fine structure of this short-term spectrum (which carries information about the voice source) is most often discarded as well, and it is the spectral envelope of the speech short-term spectrum which provides a starting point for most speech features used in ASR [START_REF] Hermansky | Mel cepstrum, deltas, double-deltas,.. what else is new? Robust Methods for Speech Recognition in Adverse Conditions[END_REF].

So what will happen if certain speech non-linguistic variabilities can be reduced before the resulting signal being fed to the speech analysis module of an ASR system? Will this reduction have any influence on the performance of an ASR system and if so, how can we adjust the degree of this influence? We thus resynthesize a new kind of speech, called spectrally reduced speech (SRS), from original clean speech and then use this SRS in an ASR system (Fig. 1). By resynthesizing SRS from original clean speech signal, certain types of speech non-linguistic variabilities can be alleviated. Shannon et al., Figure 1: Using SRS in ASR. SRS is first synthesized from original speech signal and then is used in ASR.

[7] synthesized SRS by using amplitude modulations (AMs) extracted from 3 or 4 frequency bands of speech to modulate white noise and then summed up these subband signals to obtain SRS. Despite the great reduction of spectral information, perceptive tests have shown that this kind of speech could support human speech recognition in quiet environment. Recently, Nie et al., [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF] have suggested that the slowly varying frequency modulation (FM) should be integrated in the SRS described in [START_REF] Shannon | Speech recognition with primarily temporal cues[END_REF] to support human speech recognition in noisy environment. Indeed, acoustic analysis have shown that slowly varying FM components preserve dynamic information regarding speech formant transitions and fundamental frequency movements; these may help human recognize speech in noise [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]. In this letter, we will use the SRSs proposed in [START_REF] Shannon | Speech recognition with primarily temporal cues[END_REF] and [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF] for testing with ASR because these SRSs make it possible to reduce certain non-linguistic variabilities in the resynthesized speech signal. Above a certain SRS spectral resolution, we will see that the ASR word accuracy [START_REF] Young | The HTK book (for HTK version 3.2.1)[END_REF] attained with SRS synthesized from only AM components, is surprisingly comparable to or even better than that attained with the original clean speech signal. This result suggests many prospects for future applications.

SRS Synthesis Using AM-FM Model of Speech

A speech signal s(t) can be approximated by a sum of N band-limited com- ponents, s k (t), k = 1, • • • , N . s(t) ∼ = N k=1 s k (t) (1)
Each subband component s k (t) can be considered as an AM-FM signal [START_REF] Maragos | Energy separation in signal modulations with application to speech analysis[END_REF] which contains both amplitude and frequency modulations

s k (t) = m k (t) cos 2πf ck t + 2π t 0 g k (τ )dτ + θ k (2)
where the signals m k (t) and g k (t) are the amplitude modulation and the frequency modulation in the k th subband whereas f ck and θ k are the k th subband central frequency and the initial phase, respectively. In this paper, we use the term AM-only SRS to designate SRS synthesized by using only the AM components in the subbands of the decomposed speech signal. To synthesize AM-only SRS, the speech signal is first decomposed into several frequency bands and AMs are extracted from the subband signals by full wave rectification followed by lowpass filtering. The extracted AMs of the subband signals are then used to modulate either white noise or sinusoids whose frequencies equal the central frequencies of the frequency subbands [START_REF] Shannon | Speech recognition with primarily temporal cues[END_REF]. The modulated subband signals from all the frequency bands are then summed up to construct the AM-only SRS. The mathematical expression s AM (t) of AM-only SRS can be written as follows

s AM (t) = N k=1 m k (t) cos(2πf ck t) (3)
where m k (t) is the lowpass filtered AM in the k th subband. Besides, we use the acronym AM+FM SRS to designate SRS synthesized from both AM and FM components. The frequency amplitude modulation encoding (FAME) strategy [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF], which is used to synthesize AM+FM SRS, involves two steps, analysis and synthesis (Fig. 2). In the analysis stage, the speech signal is first decomposed into several frequency bands by using a critical filter bank having logarithmically increasing central frequencies to mimic the human cochlear filters. Then, two independent parallel pathways are employed to extract the slowly varying AM and FM cues from each of the filtered subband signal. In the FM extraction pathway, the extracted FM is band-limited and further lowpass filtered to obtain the slowly varying FM. In the synthesis, the slowly varying FM signal is added to the original central frequency f ck , then integrated to recover the original phase information and, finally, multiplied by the slowly varying AM signal m k (t) extracted from the same subband to recover the original subband signal [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]. Finally, by summing the recovered subband signals from all the subbands, AM+FM SRS is obtained. This signal, s AM+FM (t), can be written as follows

s AM+FM (t) = N k=1 m k (t) cos 2π t 0 ( g k (τ ) + f ck )dτ ( 4
)
where g k (t) stands for the slowly varying FM in the k th subband.

Figure 2: Functional block diagram of speech processing in the FAME strategy for AM+FM SRS synthesis [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF].

Detailing further the FM extraction algorithm of FAME in Fig. 3, we can see that this is a conventional FM extraction diagram followed by two blocks "Band limit" and "LPF 3" to control the FM bandwidth and the FM rate, respectively. The subband signal s k (t) is first multiplied with the outcome of a quadrature oscillator with central frequency f ck . This manipulation is equivalent to shifting the spectrum of s k (t) to 0 and 2f ck in the frequency domain. After the lowpass filterings performed by LPF 2 and LPF 2', the in-phase signal, a, and the out-of-phase signal, b, of the subband signal s k (t), are extracted. In the discrete implementation, the FM signal is calculated by the following formula [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]

FM = b∆a -a∆b 2π(a 2 + b 2 ) × T s ( 5
)
where ∆ is the differentiation and T s is the sampling period. The mathematical development of the FM extraction process can be found in [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]. "Band limit" is used to avoid undesirable cross-talk between adjacent bands whereas LPF 3 is used to set the FM rate lower than 400 Hz so that it may be perceived by human cochlear-implant listeners [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]. Using the slowly varying AM signal m k (t) to modulate the FM full frequency range (FR) signal g k (t) taken at c (Fig. 3), we synthesize the AM+FM SRS with full frequency range FM component, termed FR AM+FM SRS. In the same way, by using the slowly varying FM signal g k (t) taken at d, we can synthesize the AM+FM SRS with FM slowly varying (SV) component, termed SV AM+FM SRS. The analysis filter bank in this work contains bandpass filters whose central frequencies and bandwidths match those of the critical bands [START_REF] Gunawan | Speech enhancement using temporal masking and fractional bark gammatone filters[END_REF]. Each bandpass filter is constructed using a fourth-order elliptic bandpass filter having a stop-band attenuation of 50 dB and a 2 dB ripple in the pass-band to be consistent with [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]. The original speech signal s(t) is thus decomposed into subband signals s k (t), k = 1, • • • , N with N taking values in {4, 6, 8, 10, 16, 24, 32}. Subsequently, fourth-order elliptic lowpass filter with 50 Hz cutoff frequency is used in the AM extraction. In the FM extraction, the two filters used to determine the FM depth or the FM bandwidth, LPF 2 and LPF 2', are also fourth-order lowpass Bessel filters. Their cutoff frequencies are set to either 500 Hz or the analysis subband filter bandwidth whenever this bandwidth is less than 500 Hz [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF]. Finally, the cutoff frequency of the FM-rate filter, LPF 3, is set to 400 Hz. After AM and FM extractions, the corresponding AM-only SRS, FR AM+FM SRS and SV AM+FM SRS are synthesized. Fig. 4 shows the frequency response of an analysis filter bank containing 16 fourth-order elliptic bandpass filters. The blue line in each spectrogram, estimated by using Praat software [START_REF] Boersma | Praat, a system for doing phonetic by computer[END_REF], represents the speech fundamental frequency (f 0 ) as a function of time. The range of the fundamental frequency is figured on the right-hand side vertical axis of each spectrogram. Using the extracted f 0 contour of the original clean speech as the reference, we can remark that the f 0 information is not correctly estimated from the AM-only SRSs, whether we use 4 or 16 subbands. Another remark concerns the f 0 information in the AM+FM SRSs. The f 0 information is not correctly estimated in the 4-subband SV AM+FM SRS but its estimation in the 4-subband FR AM+FM SRS is sufficiently good. This does not happen with the 16-subband SV AM+FM SRS (6(c)) and the 16-subband FR AM+FM SRS (6(d)) where Praat estimates correctly f 0 . We can therefore conclude that FM components, especially the rapidly varying ones, contain information regarding the speech fundamental frequency which is a speaker-dependent information. However, even in case of missing FM rapidly varying components, increasing the SV AM+FM SRS spectral resolution makes it possible to retrieve the f 0 information.

Acoustic Analysis of Synthesized SRS

Testing SRS with an ASR System

We construct a speaker-independent ASR system for testing the behavior of ASR with SRSs. The TIDIGITS, which is a small vocabulary, continuous speech database and the HTK speech recognition software [START_REF] Young | The HTK book (for HTK version 3.2.1)[END_REF] are used for the training of the phoneme acoustic models. Each phoneme is modeled by a context-dependent three-state left-to-right triphone Hidden Markov Model (HMM). The output distributions are modeled by multivariate Gaussian Mixtures Densities. Each mixture consists of 16 multivariate Gaussian components. Feature vectors consist of 13 Mel Frequency Cepstral Coefficients (MFCCs) in which the first coefficient C 0 is used as the energy component [START_REF] Young | The HTK book (for HTK version 3.2.1)[END_REF]. The MFCCs are calculated from every Hamming windowed speech frame of 25 ms length. The overlap between two adjacent frames is 15 ms. The delta and acceleration coefficients [START_REF] Young | The HTK book (for HTK version 3.2.1)[END_REF] are appended to the MFCCs to provide 39-dimensional feature vectors. 250 utterances from the TIDIGITS clean speech test database are selected for the tests. From this set of original clean speech signals, we synthesize the AM-only SRSs, the SV AM+FM SRSs, and the FR AM+FM SRSs. Table 1 shows the ASR word accuracies of synthesized SRSs by using the triphone HMMs trained on the TIDIGITS clean speech training database.

The recognition results show that among the three kinds of SRS studied, the AM-only SRSs give the best ASR word accuracies when the SRSs are synthesized on the basis of AM and FM components of at least 16 subbands. We thus use the model of 16-subband AM-only SRS, the AM-only SRS who gave the best ASR word accuracy in Table 1, to resynthesize the TIDIGITS AM-only SRSs give the best ASR word accuracies when the SRSs spectral resolution is sufficiently good (10 subbands in this case).

Conclusions

We have investigated the behavior of ASR with SRSs synthesized from subband AMs and FMs. For the small vocabulary, continuous speech database (TIDIGITS) studied in this letter, the following conclusions can be formulated:

• Above a certain SRS spectral resolution, AM-only SRS gives the best ASR word accuracy among the three kinds of SRS studied.

• We can use AM-only SRSs synthesized from 10 subbands or more to achieve comparable or even better ASR word accuracies compared to those attained with original clean speech (see Table 1 and Table 2).

• FM components support human speech recognition [START_REF] Nie | Encoding frequency modulation to improve cochlear implant performance in noise[END_REF] but yield no significant improvement in terms of ASR word accuracy when the SRSs spectral resolution is sufficiently good.

The acoustic analysis of synthesized SRSs has shown that even though the formant transitions are not clearly visible and the fundamental frequency information is absent in the AM-only SRS, this kind of SRS still provides the best ASR word accuracy when the SRS spectral resolution is sufficiently good. This suggests that certain non-linguistic variabilities in the speech signal can actually be alleviated via SRS resynthesis. Using appropriate synthesized SRSs in ASR can help achieve comparable or even better ASR word accuracies compared to those attained with original clean speech. Future work will extend the study to large vocabulary and focus on finding new kinds of SRS that make it possible to reduce more speech acoustic variances while yielding good ASR word accuracy.

Figure 3 :

 3 Figure 3: Algorithm for extracting FM components in the k th subband of the FAME strategy [8].

Fig. 5

 5 Fig. 5 shows spectrograms of a continuous speech utterance "one oh four six", pronounced by a female speaker, from the TIDIGITS speech database. The spectrogram of the original clean speech utterance is shown in 5(a) whereas 5(b), 5(c), and 5(d) show the spectrograms of the 4-subband synthesized SRSs; 5(b): 4-subband AM-only SRS, 5(c): 4-subband SV AM+FM SRS, and 5(d): 4-subband FR AM+FM SRS. Similarly, Fig. 6 presents the

Figure 4 :

 4 Figure 4: Frequency response of an analysis filter bank containing 16 fourthorder elliptic bandpass filters used for speech signal decomposition. The speech signal is sampled at 8 kHz.

Figure 5 :

 5 Figure 5: Spectrograms of synthesized SRSs of the speech utterance "one oh four six" from the TIDIGITS database, pronounced by a female speaker. (a) original clean speech, (b) 4-subband AM-only SRS, (c) 4-subband SV AM+FM SRS, (d) 4-subband FR AM+FM SRS. Arrows indicate formant transitions from vowel /o/ in "oh" to consonant /f/ in "four". These formant transitions are absent in (b), the spectrogram of the 4-subband AM-only SRS. The blue lines, estimated by Praat [12], represent the speech fundamental frequency f 0 . The f 0 frequency range is [75 Hz -500 Hz] (see the right-hand side vertical axis).

Figure 6 :

 6 Figure 6: Spectrograms of the synthesized SRSs of the same utterance as in Fig. 5. (a) original clean speech, (b) 16-subband AM-only SRS, (c) 16subband SV AM+FM SRS, (d) 16-subband FR AM+FM SRS.

Table 1 :

 1 ASR word accuracies (in %) of synthesized SRSs using triphone HMMs trained on the TIDIGITS clean speech training database. The ASR word accuracy computed on 250 original clean speech utterances is 99.76%. Using the triphone HMMs trained on this new training database for recognition of the testing SRSs, we obtain the ASR word accuracies of Table2. These results confirm the conclusion mentioned above:

				Number of frequency bands
	SRS	4	6	8	10	16	24	32
	AM-only	35.57	41.05	84.23	98.48	99.82	99.70	99.57
	SV AM+FM	64.43	97.87	98.42	99.09	99.33	98.36	96.41
	FR AM+FM	82.95	92.81	93.61	94.15	94.76	94.88	94.70
	training database.

Table 2 :

 2 ASR word accuracies (in %) of synthesized SRSs using triphone HMMs trained on the TIDIGITS training database resythesized from the 16-subband AM-only SRS model. The ASR word accuracy computed on the same 250 original clean speech utterances mentioned above is 99.45%.

				Number of frequency bands
	SRS	4	6	8	10	16	24	32
	AM-only	34.59	39.77	89.89	99.39	99.76	99.70	99.70
	SV AM+FM	40.13	85.08	95.43	97.99	98.96	94.58	87.76
	FR AM+FM	68.94	88.86	91.60	91.72	94.28	94.40	93.85

Acknowledgement

The authors would like to acknowledge M.Vijay Kumar for his assistance during the summer 2008.


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  On testing the norm of a random vector in Gaussian noise by tests with uniformly best constant conditional power (UBCCP) on spheres Dominique Pastor, Laurent Dejean To cite this version: Dominique Pastor, Laurent Dejean. On testing the norm of a random vector in Gaussian noise by tests with uniformly best constant conditional power (UBCCP) on spheres. [Research Report] Laboratoire en sciences et technologies de l'information, de la communication et de la connaissance (UMR CNRS 6285 -Télécom Bretagne -Université de Bretagne Occidentale -Université de Bretagne Sud -ENSTA Bretagne -Ecole Nationale d'ingénieurs de Brest); Dépt. Signal et Communications (Institut Mines-Télécom-Télécom Bretagne-UEB). 2009, pp.14. hal-02316500