Adaptive estimation for inverse problems with applications to cell divisions

Abstract : This thesis is divided into two independent parts. In the first one, we consider a stochastic individual-based model in continuous time to describe a size-structured population for cell divisions. This model is motivated by the detection of cellular aging in biology. The random point measure describing the cell population evolves as a piecewise deterministic Markov process. We address here the problem of nonparametric estimation of the kernel ruling the divisions, under two observation schemes. First, we observe the evolution of cells up to a fixed time T and we obtain the whole division tree. We construct an adaptive kernel estimator of the division kernel with a fully data-driven bandwidth selection. We obtain an oracle inequality and optimal exponential rates of convergence. Second, when the whole division tree is not completely observed, we show that, in a large population limit, the renormalized microscopic process describing the evolution of cells converges to the weak solution of a partial differential equation (PDE). Considering an eigenvalue problem related to the asymptotic behavior of the PDE's solutions, we propose an estimator of the division kernel by using Fourier techniques. We prove the consistency of the estimator. The study of rates of convergence is a work in progress. In the second part of this thesis, we consider the nonparametric regression with errors-in-variables model in the multidimensional setting. We estimate the multivariate regression function by an adaptive estimator based on projection kernels defined with multi-indexed wavelets and a deconvolution operator. The wavelet level resolution is selected by the method of Goldenshluger-Lepski. We obtain an oracle inequality and optimal rates of convergence over anisotropic Hölder classes.
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van Ha Hoang. Adaptive estimation for inverse problems with applications to cell divisions. Statistics [math.ST]. Université de Lille 1 – Sciences et Technologies, 2016. English. ⟨tel-01417780⟩

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