We focus on the estimation of the intensity of a Poisson process in the presence of a uniform noise. We propose a kernel-based procedure fully calibrated in theory and practice. We show that our adaptive estimator is optimal from the oracle and minimax points of view, and provide new lower bounds when the intensity belongs to a Sobolev ball. By developing the Goldenshluger-Lepski methodology in the case of deconvolution for Poisson processes, we propose an optimal data-driven selection of the kernel bandwidth. Our method is illustrated on the spatial distribution of replication origins and sequence motifs along the human genome.

Despite the ubiquity of U-statistics in modern Probability and Statistics, their non-asymptotic analysis in a dependent framework may have been overlooked. In a recent work, a new concentration inequality for U-statistics of order two for uniformly ergodic discrete time Markov chains has been proved. In this paper, we put this theoretical breakthrough into action by pushing further the current state of knowledge in three different active fields of research. First, we establish a new exponential inequality for the estimation of spectra of integral operators with MCMC methods. The novelty is that this result holds for kernels with positive and negative eigenvalues, which is new as far as we know. In addition, we investigate generalization performance of online algorithms working with pairwise loss functions and Markov chain samples. We provide an online-to-batch conversion result by showing how we can extract a low risk hypothesis from the sequence of hypotheses generated by any online learner. We finally give a non-asymptotic analysis of a goodness-of-fit test on the density of the stationary measure of a Markov chain. We identify some classes of alternatives over which our test based on the $L^2$ distance has a prescribed power.

This paper studies the estimation of the conditional density $f(x,\cdot)$ of $Y_i$ given $X_i=x$, from the observation of an i.i.d. sample $(X_i,Y_i)\in \mathbb R^d$, $i\in \{1,\dots,n\}.$ We assume that $f$ depends only on $r$ unknown components with typically $r\ll d$.We provide an adaptive fully-nonparametric strategy based on kernel rules to estimate $f$. To select the bandwidth of our kernel rule, we propose a new fast iterative algorithm inspired by the Rodeo algorithm (Wasserman and Lafferty, 2006) to detect the sparsity structure of $f$. More precisely, in the minimax setting, our pointwise estimator, which is adaptive to both the regularity and the sparsity, achieves the quasi-optimal rate of convergence. Our results also hold for (unconditional) density estimation. The computational complexity of our method is only $O(dn \log n)$. A deep numerical study shows nice performances of our approach.

We consider stationary hidden Markov models with finite state space and nonparametric modeling of the emission distributions. It has remained unknown until very recently that such models are identifiable. In this paper, we propose a new penalized least- squares estimator for the emission distributions which is statistically optimal and practically tractable. We prove a non asymptotic oracle inequality for our nonparametric estimator of the emission distributions. A consequence is that this new estimator is rate minimax adaptive up to a logarithmic term. Our methodology is based on projections of the emission distributions onto nested subspaces of increasing complexity. The popular spectral estimators are unable to achieve the optimal rate but may be used as initial points in our procedure. Simulations are given that show the improvement obtained when applying the least-squares minimization consecutively to the spectral estimation.