Counselling of parents following the diagnosis of a congenital heart disease should take into account that, in addition of the severity of the congenital heart disease (CHD), ethnicity, gestational age at diagnosis and chromosomal abnormalities influence parental decision regarding pregnancy continuation or interruption.

We study the problem of classification of d-dimensional vectors into two classes (one of which is 'pure noise') based on a training sample of size m. The main specific feature is that the dimension d can be very large. 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 Gaussian noise, fixed sample size m, and dimension d that tends to infinity, we obtain the sharp classification boundary, i.e. the necessary and sufficient conditions for the possibility of successful classification. We propose classifiers attaining this boundary. We also give extensions of the result to the case where the sample size m depends on d and satisfies the condition (log m)/ log d → γ , 0 ≤ γ < 1, and to the case of non-Gaussian noise satisfying the Cramér condition.

We consider the problem of testing hypotheses on the regression function from n observations on the regular grid on [0, 1]. We wish to test the null hypothesis that the regression function belongs to a given functional class (parametric or even nonparametric) against a composite nonparametric alternative. The functions under the alternative are separated in the L 2 -norm from any function in the null hypothesis. We assume that the regression function belongs to a wide range of Hölder classes but as the smoothness parameter of the regression function is unknown, an adaptive approach is considered. It leads to an optimal and unavoidable loss of order log(log n) in the minimax rate of testing compared with the non-adaptive setting. We propose a smoothness-free test that achieves the optimal rate, and finally we prove the lower bound showing that no test can be consistent if in the distance between the functions under the null hypothesis and those in the alternative, the loss is of order smaller than the optimal loss.

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

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