New Insights on Inverse Problems: Multidimensional Strategies for Deconvolution or Regression, and Ruin Probability Estimation

Overview & Synopsis

In this thesis, we are interested in three inverse problems in nonparametric statistics. We study the estimation of functions of several variables on non-compact domains: R^d and R_+^d. We use projection estimators on orthonormal bases constructed by tensorization of the Hermite basis (in the case of R^d) and the Laguerre basis (in the case of R_+^d). These two bases are formed from orthogonal polynomials, and they have the property of being non-compactly supported. This avoids issues regarding the choice of the support, that one encounters with bases supported on an interval [a, b] for example. In order to ensure that our estimators perform well, the dimension of the projection space must be carefully chosen. We use two procedures: the penalized model selection procedure and the Goldenshluger and Lepski's procedure. These procedures allow us to construct adaptive estimators on regularity spaces associated to the used bases: the Sobolev–Laguerre spaces and the Sobolev–Hermite spaces. The first problem we study is the deconvolution of a density on R_+^d, that is, the problem of estimating a density on R_+^d from noisy observations. The second problem is the estimation of the Gerber–Shiu function in the Cramér–Lundberg risk model. The model describes the evolution of the financial reserve of an insurance company, and the Gerber–Shiu function is a quantity associated to the company risk of ruin. The last problem is the nonparametric regression with a random design. In this problem, we observe pairs (X, Y) satisfying Y = b(X) + error, and we want to recover the function b. Our contribution is to consider estimation domains in R^p that are non-compact, and to provide a theoretical study of the risk of the estimator relative to a norm weighted by the distribution of the design. We propose a model selection procedure in which the model collection is random and takes into account the discrepancy between the empirical norm and the norm associated with the distribution of design.