Bero Roos

Binomial Approximation to the Poisson Binomial Distribution: The Krawtchouk Expansion

The Poisson binomial distribution is approximated by a binomial distribution and also by finite signed measures resulting from the corresponding Krawtchouk expansion. Bounds and asymptotic relations for the total variation distance and the point metric are given.

Asymptotics and sharp bounds in the Poisson approximation to the Poisson-binomial distribution

The Poisson-binomial distribution is approximated by a Poisson law with respect to a new multimetric (difference metric) unifying a broad class of probability metrics between discrete distributions. The accompanying non-metric situation is also considered leading to moderateand large-deviation results. Using the Charlier B expansion and Fourier arguments, sharp bounds and asymptotic relations are given.

Sharp constants in the Poisson approximation

We present some new sharp bounds for several distances between the Poisson binomial distribution and the Poisson law with the same mean. It is shown that the constants involved cannot be reduced.

Improvements in the Poisson approximation of mixed Poisson distributions

We consider the approximation of mixed Poisson distributions by Poisson laws and also by related finite signed measures of higher order. Upper bounds and asymptotic relations are given for several distances. Even in the case of the Poisson approximation with respect to the total variation distance, our bounds have better order than those given in the literature. In particular, our results hold under weaker moment conditions for the mixing random variable. As an example, we consider the approximation of the negative binomial distribution, which enables us to prove the sharpness of a constant in the upper bound of the total variation distance. The main tool is an integral formula for the difference of the counting densities of a Poisson distribution and a related finite signed measure.

Kerstan's Method in the Multivariate Poisson Approximation: An Expansion in the Exponent

The generalized multinomial distribution is approximated by finite signed measures, resulting from a Poisson-type expansion in the exponent. In the univariate case, this expansion was first used by Kornya and Presman. We apply Kerstans method and present a bound for the total variation distance with explicit constants.

Metric Multivariate Poisson Approximation of the Generalized Multinomial Distribution

The aim of this paper is to introduce the multivariate Charlier B expansion to the metric multivariate Poisson approximation of a generalized multinomial distribution considered by Barbour [J. Appl. Probab., 25 (1988), pp. 175--184] and Deheuvels and Pfeifer [J. Multivariate Anal., 25 (1988), pp. 65--89]. Bounds for the total variation and the point metric are given.

Multinomial and Krawtchouk approximations to the generalized multinomial distribution

The generalized multinomial distribution is approximated by multinomial distributions and also by finite signed measures resulting from the corresponding multivariate Krawtchouk expansion. Bounds for the total variation norm and the

POISSON APPROXIMATION VIA THE CONVOLUTION WITH KORNYA-PRESMAN SIGNED MEASURES ∗

\ell_\infty

A semigroup approach to poisson approximation with respect to the point metric

-norm are presented. The method used is a multivariate extension of that in~[B.~Roos, Theory Probab. Appl., 45 (2000), pp. 258--272], although additional complications occur.

Closeness of convolutions of probability measures

We present an upper bound for the total variation distance between the generalized polynomial distribution and a finite signed measure, which is the convolution of two finite signed measures, one of which is of Kornya--Presman type. In the one-dimensional Poisson case, such a finite signed measure was first considered by K. Borovkov and D. Pfeifer [{\em J. Appl.\ Probab.}, 33 (1996), pp. 146--155].We give asymptotic relations in the one-dimensional case, and, as an example, the independent identically distributed record model is investigated.It turns out that here the approximation is of order