Alexander Korostelev

The Asymptotic Minimax Constant for Sup-Norm Loss in Nonparametric Density Estimation

We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has already been found for nonparametric regression and for signal estimation in Gaussian white noise. H61der classes for arbitrary smoothness index > 0 on the unit interval are considered. The constant involves the value of an optimal recovery problem as in the white noise case, but in addition it depends on the maximum of densities in the function class.

On minimax rates of convergence in image models under sequential design

A binary image model is studied with a Lipschitz edge function. The indicator function of the image is observed in random noise at n design points that can be chosen sequentially. The asymptotically minimax rate as n-->[infinity] is found in estimating the edge function, and an asymptotically optimal algorithm is described.

Rates of convergence for the sup-norm risk in image models under sequential designs

Let G be that portion of the unit square which lies below the graph of a smooth function. Assume that observations of the indicator function of G are available at any points X1,...,Xn in the plane. If each consecutive point Xi can be chosen sequentially, on the basis of all the preceding data, then how accurately can the smooth function be estimated in sup-norm? Using the boundary fragment model, this question translates into a question about the large sample performance of the minimax risks under sup-norm loss. The asymptotic rates of these risks are found. The results are extended to additive noise models and multidimensional images.

Semiparametric diffusion estimation and application to a stock market index

The analysis of diffusion processes in financial models is crucially dependent on the form of the drift and diffusion coefficient functions. A new model for a stock market index process is proposed in which the index is decomposed into an average growth process and an ergodic diffusion. The ergodic diffusion part of the model is not directly observable. A methodology is developed for estimating and testing the coefficient functions of this unobserved diffusion process. The estimation is based on the observations of the index process and uses semiparametric and non-parametric techniques. The testing is performed via the wild bootstrap resampling technique. The method is illustrated on S&P 500 index data.

On a multi-channel change-point problem

A continuous change-point problem is studied in which N independent diffusion processes Xj are observed. Each process Xj is associated with a “channel”, each has an unknown piecewise constant drift and the unit diffusion coefficient. All the channels are connected only by a common change-point of drift. As the result, a change-point problem is defined in which the unknown and unidentifiable drift forms a 2N-dimensional nuisance parameter. The asymptotics of the minimax rate in estimating the change-point is studied as N → ∞. This rate is compared with the case of the known drift. This problem is a special case of an open change-point detection problem in the high-dimensional diffusion with nonparametric drift.

Limit Theorem for the Spread of Branching Diffusion with Stabilizing Drift

In the present paper we derive a formula describing the limiting behavior of R t , the position of the rightmost particle over a time interval [0, t] in the one-dimensional branching diffusion with a stabilizing drift, and generalize the result to a multidimensional case.