Related papers: Asymptotic analysis in multivariate worst case app…

We consider multivariate $\mathbb{L}_2$-approximation in reproducing kernel Hilbert spaces which are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study the minimal worst-case error…

We consider $L^2$-approximation on weighted reproducing kernel Hilbert spaces of functions depending on infinitely many variables. We focus on unrestricted linear information, admitting evaluations of arbitrary continuous linear…

The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of…

We consider a multivariate functional measurement error model $AX\approx B$. The errors in $[A,B]$ are uncorrelated, row-wise independent, and have equal (unknown) variances. We study the total least squares estimator of $X$, which, in the…

This article derives an accurate, explicit, and numerically stable approximation to the kernel quadrature weights in one dimension and on tensor product grids when the kernel and integration measure are Gaussian. The approximation is based…

For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback--Leibler divergence, the Hellinger distance and $\chi^2$…

We are interested in approximation of a multivariate function $f(x_1,\dots,x_d)$ by linear combinations of products $u^1(x_1)\cdots u^d(x_d)$ of univariate functions $u^i(x_i)$, $i=1,\dots,d$. In the case $d=2$ it is a classical problem of…

In the present paper a behavior of the "average case" approximation complexity for d-parametric random fields of tensor-type is studied. It was shown in [Lifshits and Tulyakova, 2006] that for a given approximation accuracy level the…

Gaussian processes (GPs) are ubiquitous tools for modeling and predicting continuous processes in physical and engineering sciences. This is partly due to the fact that one may employ a Gaussian process as an interpolator while facilitating…

We study estimation of a multivariate function $f:{\bf R}^d \to {\bf R}$ when the observations are available from function $Af$, where $A$ is a known linear operator. Both the Gaussian white noise model and density estimation are studied.…

We consider a class of fourth order uniformly elliptic operators in planar Euclidean domains and study the associated heat kernel. For operators with $L^{\infty}$ coefficients we obtain Gaussian estimates with best constants, while for…

We carry out the asymptotic analysis as $n \to \infty$ of a class of orthogonal polynomials $p_{n}(z)$ of degree $n$, defined with respect to the planar measure \begin{equation*} d\mu(z) = (1-|z|^{2})^{\alpha-1}|z-x|^{\gamma}\mathbf{1}_{|z|…

In this paper we investigate the problem of estimating the regression function in models with correlated observations. The data is obtained from several experimental units each of them forms a time series. We propose a new estimator based…

The large-sample behavior of non-degenerate multivariate $U$-statistics of arbitrary degree is investigated under the assumption that their kernel depends on parameters that can be estimated consistently. Mild regularity conditions are…

Kernel expansions are a topic of considerable interest in machine learning, also because of their relation to the so-called feature maps introduced in machine learning. Properties of the associated basis functions and weights (corresponding…

We consider maximum likelihood estimation with data from a bivariate Gaussian process with a separable exponential covariance model under fixed domain asymptotic. We first characterize the equivalence of Gaussian measures under this model.…

We study the worst case error of kernel density estimates via subset approximation. A kernel density estimate of a distribution is the convolution of that distribution with a fixed kernel (e.g. Gaussian kernel). Given a subset (i.e. a point…

We present improved methods for calculating confidence intervals and $p$-values in situations where standard asymptotic approaches fail due to small sample sizes. We apply these techniques to a specific class of statistical model that can…

We consider analytic functions from a reproducing kernel Hilbert space. Given that such a function is of order $\epsilon$ on a set of discrete data points, relative to its global size, we ask how large can it be at a fixed point outside of…

We address the estimation of "extreme" conditional quantiles i.e. when their order converges to one as the sample size increases. Conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian…