Frits H. Ruymgaart

Convergence Rates of General Regularization Methods for Statistical Inverse Problems and Applications

Previously, the convergence analysis for linear statistical inverse problems has mainly focused on spectral cut-off and Tikhonov-type estimators. Spectral cut-off estimators achieve minimax rates for a broad range of smoothness classes and operators, but their practical usefulness is limited by the fact that they require a complete spectral decomposition of the operator. Tikhonov estimators are simpler to compute but still involve the inversion of an operator and achieve minimax rates only in restricted smoothness classes. In this paper we introduce a unifying technique to study the mean square error of a large class of regularization methods (spectral methods) including the aforementioned estimators as well as many iterative methods, such as

Statistical inverse estimation in Hilbert scales

\nu

Applications of empirical characteristic functions in some multivariate problems

-methods and the Landweber iteration. The latter estimators converge at the same rate as spectral cut-off but require only matrix-vector products. Our results are applied to various problems; in particular we obtain precise convergence rates for satellite gradiometry,

A robust principal component analysis

L^2

Asymptotic minimax rates for abstract linear estimators

-boosting, and errors in variable problems.

Theoretical aspects of ill-posed problems in statistics

The recovery of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable. For greater flexibility the general problem is embedded in an abstract Hilbert scale. In the applications Sobolev scales are used. For the construction of estimators we employ preconditioning along with regularized operator inversion in the appropriate inner product, where the operator is bounded but not necessarily compact. A lower bound to certain minimax rates is included, and it is shown that in generic examples the proposed estimators attain the asymptotic minimax rate. Examples include errors-in-variables (deconvolution) and indirect nonparametric regression. Special instances of the latter are estimation of the source term in a differential equation and the estimation of the initial state in the heat equation.

Nonparametric estimation of ruin probabilities given a random sample of claims

In this note we extend univariate tests for normality and symmetry based on empirical characteristic functions to the multivariate case. RESUME Cette note vise a etendre au cas multidimensionnel certains tests de normalite et de symetrie univaries construits a partir de fonctions caracteristiques experimentales.

The almost sure behavior of maximal and minimal multivariate k n -spacings

A robust principal component analysis for samples from a bivariate distribution function is described. The method is based on robust estimators for dispersion in the univariate case along with a certain linearization of the bivariate structure. Besides the continuity of the functional defining the direction of the suitably modified principal axis, we prove consistency of the corresponding sequence of estimators. Asymptotic normality is established under some additional conditions.

The delta method for analytic functions of random operators with application to functional data

Abstract Abstract linear estimation concerns the estimation of an abstract parameter that depends on the underlying density via a linear transformation. An important subclass is the class of inverse problems where this transformation is naturally described as the inverse of some bounded operator. Suitable preconditioning allows us to restrict ourselves to the inverse of some Hermitian operator, which does not remain restricted to the class of compact operators. A lower bound to the minimax risk is obtained for the class of all estimators satisfying a natural moment condition and certain submodels. To establish the bound we use the Bayesian van Trees inequality and systems of (pseudo) eigenvectors of the operator involved. We also briefly sketch a general construction method for estimators, based on a regularized inverse of the operator involved, and show that these estimators attain the asymptotic minimax rate in interesting examples.

Regularized Deconvolution on the Circle and the Sphere

Ill-posed problems arise in a wide variety of practical statistical situations, ranging from biased sampling and Wicksells problem in stereology to regression, errors-in-variables and empirical Bayes models. The common mathematics behind many of these problems is operator inversion. When this inverse is not continuous a regularization of the inverse is needed to construct approximate solutions. In the statistical literature, however, ill-posed problems are rather often solved in an ad hoc manner which obccures these common features. It is our purpose to place the concept of regularization within a general and unifying framework and to illustrate its power in a number of interesting statistical examples. We will focus on regularization in Hilbert spaces, using spectral theory and reduction to multiplication operators. A partial extension to a Banach function space is briefly considered.