B. Levit

Applications of the van Trees inequality: a Bayesian Cramér-Rao bound

We use a Bayesian version of the Cramer-Rao lower bound due to van Trees to give an elementary proof that the limiting distibution of any regular estimator cannot have a variance less than the classical information bound, under minimal regularity conditions. We also show how minimax convergence rates can be derived in various non- and semi-parametric problems from the van Trees inequality. Finally we develop multivariate versions of the inequality and give applications.

Some new perspectives in best approximation and interpolation of random data

A new notion of universally optimal experimental design is introduced, relevant from the perspective of adaptive nonparametric estimation. It is demonstrated that both discrete and continuous Chebyshev designs are universally optimal in the problem of fitting properly weighted algebraic polynomials to random data. The result is a direct consequence of the well-known relation between Chebyshev’s polynomials and the trigonometric functions.Optimal interpolating designs in rational regression proved particularly elusive in the past. The question can be effectively handled using its connection to elliptic interpolation, in which the ordinary circular sinus, appearing in the classical trigonometric interpolation, is replaced by the Abel-Jacobi elliptic sinus sn(x, k) of a modulus k. First, it is demonstrated that — in a natural setting of equidistant design — the elliptic interpolant is never optimal in the so-called normal case k ∈ (−1, 1), except for the trigonometric case k = 0.However, the equidistant elliptic interpolation is always optimal in the imaginary case k ∈ iℝ. Through a relation between elliptic and rational functions, the result leads to a long sought optimal design, for properly weighted rational interpolants. Both the poles and nodes of the interpolants can be conveniently expressed in terms of classical Jacobi’s theta functions.

Cardinal splines in nonparametric regression

We discuss a nonparametric regression model on an equidistant grid of the real line. A class of kernel type estimates based on the so-called fundamental cardinal splines will be introduced. Asymptotic optimality of these estimates will be established for certain functional classes. This model explains the often mentioned heuristic fact that cubic splines are adequate for most practical applications.

Variance-optimal data approximation using the Abel-Jacobi functions

Conditions are found under which d-dimensional linear interpolating spaces

Minimax revisited. II

\mathcal{L}_d

Asymptotic Optimality of Periodic Spline Interpolation in Non-Parametric Regression

generated by the classical Abel-Jacobi elliptic functions contain constants and are variance optimal with respect to the equidistant d-design χd. The so-called modulus parameter k = k(d) of the Abel-Jacobi functions is assumed to belong to (−1, 1) ∪ iR. The spaces

A Bayesian approach to the estimation of maps between Riemannian manifolds

\mathcal{L}_d

Optimal methods of interpolation in Nonparametric Regression

are optimal if k(d) is restricted to iR when d is odd, with no such restriction needed when d is even.Anotion of universal optimality has been recently introduced into the theory of Optimal Design [6]. Although quite attractive, such optimality is not easy to achieve. As a compromise, a somewhat weaker — but more accessible — variance optimality property is proposed.One of the main results establishes weak universal variance optimality of the sequence of interpolating spaces

A Bayesian Approach to the Estimation of Maps between Riemannian Manifolds. II: Examples

\mathcal{L}_{d_l }

Asymptotically efficient estimation of analytic functions in Gaussian noise

, with respect to the equidistant design χn, provided all dl are divisors of n and the corresponding moduli k(dl) satisfy the above restriction.