The functional linear model with scalar response is a regression model where the predictor is a random function defined on some compact set of R R and the response is scalar. The response is modelled as Y=WðXÞ+e, where W is some linear continuous operator defined on the space of square integrable functions and valued in R R. The random input X is independent from the noise e. In this paper, we are interested in testing the null hypothesis of no effect, that is, the nullity of W restricted to the Hilbert space generated by the random variable X. We introduce two test statistics based on the norm of the empirical cross-covariance operator of (X; Y). The first test statistic relies on a v 2 approximation and we show the asymptotic normality of the second one under appropriate conditions on the covariance operator of X. The test procedures can be applied to check a given relationship between X and Y. The method is illustrated through a simulation study.

In Drosophila melanogaster, Hox genes are organized in an anterior and a posterior cluster, called Antennapedia complex and bithorax complex, located on the same chromosome arm and separated by 10 Mb of DNA. Both clusters are repressed by Polycomb group (PcG) proteins. Here, we show that genes of the two Hox complexes can interact within nuclear PcG bodies in tissues where they are corepressed. This colocalization increases during development and depends on PcG proteins. Hox gene contacts are conserved in the distantly related Drosophila virilis species and they are part of a large gene interaction network that includes other PcG target genes. Importantly, mutations on one of the loci weaken silencing of genes in the other locus, resulting in the exacerbation of homeotic phenotypes in sensitized genetic backgrounds. Thus, the three-dimensional organization of Polycomb target genes in the cell nucleus stabilizes the maintenance of epigenetic gene silencing.

We consider the problem of predicting a real random variable from a functional explanatory variable. The problem is attacked by mean of nonparametric kernel approach which has been recently adapted to this functional context. We derive theoretical results by giving a deep asymptotic study of the behaviour of the estimate, including mean squared convergence (with rates and precise evaluation of the constant terms) as well as asymptotic distribution. Practical use of these results are relying on the ability to estimate these constants. Some perspectives in this direction are discussed. In particular a functional version of wild bootstrapping ideas is proposed and used both on simulated and real functional datasets.

We propose in this work to derive a CLT in the functional linear regression model. The main difficulty is due to the fact that estimation of the functional parameter leads to a kind of ill-posed inverse problem. We consider estimators that belong to a large class of regularizing methods and we first show that, contrary to the multivariate case, it is not possible to state a CLT in the topology of the considered functional space. However, we show that we can get a CLT for the weak topology under mild hypotheses and in particular without assuming any strong assumptions on the decay of the eigenvalues of the covariance operator. Rates of convergence depend on the smoothness of the functional coefficient and on the point in which the prediction is made.

We study prediction in the functional linear model with functional outputs : Y = SX + ε where the covariates X and Y belong to some functional space and S is a linear operator. We provide the asymptotic mean square prediction error with exact constants for our estimator which is based on functional PCA of the input and has a classical form. As a consequence we derive the optimal choice of the dimension k n of the projection space. The rates we obtain are optimal in minimax sense and generalize those found when the output is real. Our main results hold with no prior assumptions on the rate of decay of the eigenvalues of the input. This allows to consider a wide class of parameters and inputs X (·) that may be either very irregular or very smooth. We also prove a central limit theorem for the predictor which improves results by Cardot, Mas and Sarda (2007) in the simpler model with scalar outputs. We show that, due to the underlying inverse problem, the bare estimate cannot converge in distribution for the norm of the function space.

The functional autoregressive model is a Markov model taylored for data of functional nature. It revealed fruitful when attempting to model samples of dependent random curves and has been widely studied along the past few years. This article aims at completing the theoretical study of the model by addressing the issue of weak convergence for estimates from the model. The main difficulties stem from an underlying inverse problem as well as from dependence between the data. Traditional facts about weak convergence in non-parametric models appear: the normalizing sequence is not an O √ n , a bias term appears. Several original features of the functional framework are pointed out.

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. We introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output. The test statistics combine multiple testing ideas and random projections of the input data through functional Principal Component Analysis. Interestingly, the procedures are completely datadriven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions. The levels and powers against local alternatives are assessed in a nonasymptotic setting. This allows us to prove that these procedures are minimax adaptive (up to an unavoidable log log n multiplicative term) to the unknown regularity of the slope. As a side result, the minimax separation distances of the slope are derived for a large range of regularity classes. A numerical study illustrates these theoretical results.