N. G. Ushakov

A consistent modification of a test for independence based on the empirical characteristic function

A modification of a test for independence based on the empirical characteristic function is investigated. The initial test is not consistent in the general case. The modification makes the test always consistent and asymptotically distribution free. It is based on a special transformation of the data.

Estimations of the decomposition stability into identical components

The problem of decomposition stability of probability distributions into identical components is investigated. New bounds are obtained for the stability of decomposition of normal and other stable laws.

Inequalities for sums of random variables and the characterization of homogeneity

A number of new inequalities for sums of independent random variables are obtained. Inequalities transform into equalities only for identically distributed variables. Hence, the existence of equalities is a characterization of homogeneity. This can be used in particular to test the hypothesis of homogeneity of several samples.

A note on superkernel density estimators

It is well known that so-called superkernel density estimators have better asymptotic properties than conventional kernel estimators (and generally finite-order estimators) in the case when the density to be estimated is very smooth. In this note, we study asymptotic behavior of the mean integrated square error of superkernel density estimators in the case when the density to be estimated is not very smooth. It turns out that in this case, superkernel estimators still have better asymptotics than finite-order estimators.

Inequalities for mean squared error of multidimensional kernel density estimations

The upper limits for the integral mean squared error of multidimensional kernel density estimations are obtained. In particular, it is showed that under certain conditions of regularity, the real errors are always smaller than the asymptotic.

On bandwidth selection in kernel density estimation

In this paper, we suggest a new method of bandwidth selection in kernel density estimation. The new selector is less subject to the undersmoothing effect than the AMISE (asymptotic mean integrated square error) optimal bandwidth.

On density estimation with superkernels

In this article, we consider the problem of nonparametric density estimation in the case, when the original sample has a large size, but the data are given in a binned form, i.e. in the form of a histogram. Such situations are typical for many physical problems, in particular, in scanning electron microscopy and electron beam lithography. We study how superkernels can be used in such situations. It is shown that superkernels can be essentially superior over conventional kernels not only for very smooth densities. The problem of bandwidth and bin width selection is also considered.

Some inequalities for multivariate characteristic functions

We obtain some new lower and upper bounds for characteristic functions of multivariate distributions that can be useful in various applications.

An estimate of the decomposition stability of the Poisson distribution into identical components

An estimate of the decomposition stability of the Poisson distribution is obtained under the condition that the components are identical. The estimate is much sharper than that in the general case.

Boundaries of the precision of restoring information lost after rounding the results from observations

Lower and upper estimates are obtained for deviations of the limit of a selectedmean from estimated mathematical expectations when rounded data are processed. Different cases of error distribution are considered: normal, Simpson (triangle), and Laplace (double exponential)distributions.