We consider the problem of estimating s 2 when s belongs to some separable Hilbert space and one observes the Gaussian process Y t = s t + σL t , for all t ∈ , where L is some Gaussian isonormal process. This framework allows us in particular to consider the classical "Gaussian sequence model" for which = l 2 * and L t = λ≥1 t λ ε λ , where ε λ λ≥1 is a sequence of i.i.d. standard normal variables. Our approach consists in considering some at most countable families of finite-dimensional linear subspaces of (the models) and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 . We prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s, depending on the family of models from which they are built. In particular, in the context of the Gaussian sequence model, a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles, ellipsoids, l p -bodies or Besov bodies. We take special care to describe the conditions under which the penalized estimator is efficient when the level of noise σ tends to zero. Our construction is an alternative to the one by Efroïmovich and Low for hyperrectangles and provides new results otherwise.

Abstract.We consider the problem of estimating the integral of the square of a density f from the observation of a n sample. Our method to estimate R f 2 (x)dx is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for U -statistics of order 2 due to Houdré and Reynaud.Mathematics Subject Classification. 62G05, 62G20, 62J02.

Global sensitivity analysis of complex numerical models can be performed by calculat-

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In this paper we propose a general methodology, based on multiple testing, for testing that the mean of a Gaussian vector in R n belongs to a convex set. We show that the test achieves its nominal level, and characterize a class of vectors over which the tests achieve a prescribed power. In the functional regression model this general methodology is applied to test some qualitative hypotheses on the regression function. For example, we test that the regression function is positive, increasing, convex, or more generally, satisfies a differential inequality. Uniform separation rates over classes of smooth functions are established and a comparison with other results in the literature is provided. A simulation study evaluates some of the procedures for testing monotonicity.

BackgroundIn bacteria, the weak correlations at the genome scale between mRNA and protein levels suggest that not all mRNAs are translated with the same efficiency. To experimentally explore mRNA translational level regulation at the systemic level, the detailed translational status (translatome) of all mRNAs was measured in the model bacterium Lactococcus lactis in exponential phase growth.ResultsResults demonstrated that only part of the entire population of each mRNA species was engaged in translation. For transcripts involved in translation, the polysome size reached a maximum of 18 ribosomes. The fraction of mRNA engaged in translation (ribosome occupancy) and ribosome density were not constant for all genes. This high degree of variability was analyzed by bioinformatics and statistical modeling in order to identify general rules of translational regulation. For most of the genes, the ribosome density was lower than the maximum value revealing major control of translation by initiation. Gene function was a major translational regulatory determinant. Both ribosome occupancy and ribosome density were particularly high for transcriptional regulators, demonstrating the positive role of translational regulation in the coordination of transcriptional networks. mRNA stability was a negative regulatory factor of ribosome occupancy and ribosome density, suggesting antagonistic regulation of translation and mRNA stability. Furthermore, ribosome occupancy was identified as a key component of intracellular protein levels underlining the importance of translational regulation.ConclusionsWe have determined, for the first time in a bacterium, the detailed translational status for all mRNAs present in the cell. We have demonstrated experimentally the high diversity of translational states allowing individual gene differentiation and the importance of translation-level regulation in the complex process linking gene expression to protein synthesis.

The aim of this paper is to establish non-asymptotic minimax rates for goodness-of-fit hypotheses testing in an heteroscedastic setting. More precisely, we deal with sequences (Y j ) j∈J of independent Gaussian random variables, having mean (θ j ) j∈J and variance (σ j ) j∈J . The set J will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to l 2 and l ∞ norms, without assumption on (σ j ) j∈J and on several functions spaces. Our point of view is entirely non-asymptotic.

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