An improvement of the Berry--Esseen inequality with applications to Poisson and mixed Poisson random sums

By a modification of the method that was applied in (Korolev and Shevtsova, 2009), here the inequalities $$\rho(F_n,\Phi)\le\frac{0.335789(\beta^3+0.425)}{\sqrt{n}}$$ and $$\rho(F_n,\Phi)\le \frac{0.3051(\beta^3+1)}{\sqrt{n}}$$ are proved for the uniform distance $\rho(F_n,\Phi)$ between the standard normal distribution function $\Phi$ and the distribution function $F_n$ of the normalized sum of an arbitrary number $n\ge1$ of independent identically distributed random variables with zero mean, unit variance and finite third absolute moment $\beta^3$. The first of these inequalities sharpens the best known version of the classical Berry--Esseen inequality since $0.335789(\beta^3+0.425)\le0.335789(1+0.425)\beta^3<0.4785\beta^3$ by virtue of the condition $\beta^3\ge1$, and 0.4785 is the best known upper estimate of the absolute constant in the classical Berry--Esseen inequality. The second inequality is applied to lowering the upper estimate of the absolute constant in the analog of the Berry--Esseen inequality for Poisson random sums to 0.3051 which is strictly less than the least possible value of the absolute constant in the classical Berry--Esseen inequality. As a corollary, the estimates of the rate of convergence in limit theorems for compound mixed Poisson distributions are refined.Comment: 33 page

On rate of convergence in distribution of asymptotically normal statistics based on samples of random size

In the present paper we prove a general theorem which gives the rates of convergence in distribution of asymptotically normal statistics based on samples of random size. The proof of the theorem uses the rates of convergences in distribution for the random size and for the statistics based on samples of nonrandom size

On asymptotic approximations to the distributions of statistics constructed from samples with random sizes

Due to the stochastic character of the intensities of information flows in high performance information systems, the size of data available for the statistical analysis can be often regarded as random. In the paper general theorem concerning the asymptotic expansions of the distribution function of the statistics based on the sample of random size was proved. Some examples are presented for the cases where the sample size has the negative binomial or discrete Pareto distributions. © ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi,Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors)

On asymptotic approximations to the distributions of statistics constructed from samples with random sizes

Due to the stochastic character of the intensities of information flows in high performance information systems, the size of data available for the statistical analysis can be often regarded as random. In the paper general theorem concerning the asymptotic expansions of the distribution function of the statistics based on the sample of random size was proved. Some examples are presented for the cases where the sample size has the negative binomial or discrete Pareto distributions. © ECMS Zita Zoltay Paprika, Péter Horák, Kata Váradi,Péter Tamás Zwierczyk, Ágnes Vidovics-Dancs, János Péter Rádics (Editors)