A.C.M. van Rooij

Asymptotic minimax rates for abstract linear estimators

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.

Theoretical aspects of ill-posed problems in statistics

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.

On the space of all regular operators between two Riesz spaces

Abstract We prove that for an Archimedean Riesz space E the following two conditions are equivalent. (A) For every Archimedean Riesz space F the regular linear operators of E into F form a Riesz space. (B) E is isomorphic to a direct sum of copies of ℝ.

The Archimedean ℓ-group tensor product

We introduce a construction (inZ F-set theory) for the Archimedean ℓ-group tensor product. We relate this tensor product to the existing ones in the theory of Archimedean vector lattices and ℓ-groups.

The Bornological Tensor Product of two Riesz Spaces

We construct the bornological Riesz space tensor product of two bornological Riesz spaces. This unifies the Archimedean Riesz space tensor product and the projective tensor product, both introduced by Fremlin. We extend the results, even in these special cases, by considering maps of bounded variation rather than positive maps. This note is without proofs, but the proof and complete bornology background of a similar result are discussed elsewhere in this volume.

Non-archimedean harmonic analysis on topological semigroups

Abstract Let K be a complete non-Archimedean valued field, S a commutative topological semigroup (not necessarily locally compact). We study the convolution Banach algebra M(S) of all tight K-valued measures on S and its ideal Ma(S) consisting of all elements μ of M(S) for which the shift map x ↦ x ∗μ is continuous (the analogue of the group algebra). In particular, we show that under reasonable conditions the Fourier-Stieltjes transform is isometric. Furthermore, we describe the images of M(S) and Ma(S) under the Fourier-Stieltjes transform. The results obtained are known for groups but new for topological (semi) lattices.