Adaptive procedures in convolution models with known or partially known noise distribution

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known or partially known noise density g. In this paper, we focus on statistical procedures, which are adaptive with respect to the smoothness parameter tau of unknown density f, and also (in some cases) to some unknown parameter of the noise density g. In a first part, we assume that g is known and polynomially smooth. We provide goodness-of-fit procedures for the test H_0:f=f_0, where the alternative H_1 is expressed with respect to L_2-norm. Our adaptive (w.r.t tau) procedure behaves differently according to whether f_0 is polynomially or exponentially smooth. A payment for adaptation is noted in both cases and for computing this, we provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In the first case we prove that the payment for adaptation is optimal (thus unavoidable). In a second part, we study a wider framework: a semiparametric model, where g is exponentially smooth and stable, and its self-similarity index s is unknown. In order to ensure identifiability, we restrict our attention to polynomially smooth, Sobolev-type densities f. In this context, we provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f, of the functional \int f^2 and test of the hypothesis H_0. These procedures are adaptive with respect to both s and tau and attain the rates which are known optimal for known values of s and tau. As a by-product, when the noise is known and exponentially smooth our testing procedure is adaptive for testing Sobolev-type densities.Comment: 35 pages + annexe de 8 page

Sparse classification boundaries

Given a training sample of size $m$ from a $d$-dimensional population, we wish to allocate a new observation $Z\in \R^d$ to this population or to the noise. We suppose that the difference between the distribution of the population and that of the noise is only in a shift, which is a sparse vector. For the Gaussian noise, fixed sample size $m$, and the dimension $d$ that tends to infinity, we obtain the sharp classification boundary and we propose classifiers attaining this boundary. We also give extensions of this result to the case where the sample size $m$ depends on $d$ and satisfies the condition $(\log m)/\log d \to \gamma$, $0\le \gamma<1$, and to the case of non-Gaussian noise satisfying the Cram\'er condition

Adaptivity in convolution models with partially known noise distribution

International audienceWe consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density $f$ and some partially known noise density $g$. In this work, $g$ is assumed exponentially smooth with stable law having unknown self-similarity index $s$. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities $f$, with smoothness parameter $\beta$. In this context, we first provide a consistent estimation procedure for $s$. This estimator is then plugged-into three different procedures: estimation of the unknown density $f$, of the functional $\int f^2$ and goodness-of-fit test of the hypothesis $H_0 : f = f_0$, where the alternative $H_1$ is expressed with respect to $\mathbb{L}_{2}$-norm (i.e. has the form $\psi_{n}^{-2}\|f-f_{0}\|_{2}^{2}\ge \mathcal{C}$). These procedures are adaptive with respect to both $s$ and $\beta$ and attain the rates which are known optimal for known values of $s$ and $\beta$. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of s is illustrated on synthetic data

Adaptive goodness-of-fit testing from indirect observations

International audienceIn a convolution model, we observe random variables whose distribution is the convolution of some unknown density $f$ and some known noise density $g$. We assume that $g$ is polynomially smooth. We provide goodness-of-fit testing procedures for the test $H_0:f=f_0$, where the alternative $H_1$ is expressed with respect to $\mathbb{L}_2$-norm (\emph{i.e.} has the form $\psi_{n}^{-2}\|f-f_0\|_2^2 \ge \mathcal{C}$). Our procedure is adaptive with respect to the unknown smoothness parameter $\tau$ of $f$. Different testing rates ($\psi_n$) are obtained according to whether $f_0$ is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a non-uniform Berry-Esseen type theorem for degenerate $U$-statistics. In the case of polynomially smooth $f_0$, we prove that the price for adaptation is optimal. We emphasise the fact that the alternative may contain functions smoother than the null density to be tested, which is new in the context of goodness-of-fit tests

Test on components of mixture densities

International audienceThis paper deals with statistical tests on the components of mixture densities. We propose to test whether the densities of two independent samples of independent random variables $Y_1, \dots, Y_n$ and $Z_1, \dots, Z_n$ result from the same mixture of $M$ components or not. We provide a test procedure which is proved to be asymptotically optimal according to the minimax setting. We extensively discuss the connection between the mixing weights and the performance of the testing procedure and illustrate it with numerical examples

Minimax rates over Besov spaces in ill-conditioned mixture-models with varying mixing-weights

International audienceWe consider ill-conditioned mixture-models with varying mixing-weights. We study the classical homogeneity testing problem in the minimax setup and try to push the model to its limits, that is to say to let the mixture model to be really ill-conditioned. We highlight the strong connection between the mixing-weights and the expected rate of testing. This link is characterized by the behavior of the smallest eigenvalue of a particular matrix computed from the varying mixing-weights. We provide optimal testing procedures and we exhibit a wide range of rates that are the minimax and minimax adaptive rates for Besov balls