1,720,989 research outputs found
On the lattice point problem for ellipsoids
No full textBentkus V, Götze F. On the lattice point problem for ellipsoids. ACTA ARITHMETICA. 1997;80(2):101-125
ON THE LATTICE POINT PROBLEM FOR ELLIPSOIDS
No full textBentkus V, Götze F. ON THE LATTICE POINT PROBLEM FOR ELLIPSOIDS. DOKLADY AKADEMII NAUK. 1995;343(4):439-440
On minimal moment assumptions in Berry-Esseen theorems for U-statistics
No full textBentkus V, Götze F. On minimal moment assumptions in Berry-Esseen theorems for U-statistics. THEORY OF PROBABILITY AND ITS APPLICATIONS. 1996;40(3):430-445.The rate of convergence for asymptotically normal U-statistics is of order O(n(-1/2)) provided that E\E{h(X(1), X(2))/X(1)}\(3) 0. In this paper we extend the result for statistics of higher orders and with possible nonnormal limit distributions
Optimal rates of convergence in the CLT for quadratic forms
No full textBentkus V, Götze F. Optimal rates of convergence in the CLT for quadratic forms. ANNALS OF PROBABILITY. 1996;24(1):466-490.We prove optimal convergence rates in the central limit theorem for sums in R(k). Assuming a fourth moment, we obtain a Berry-Esseen type bound of O(N-1) for the probability of hitting a ball provided that k greater than or equal to 5. The proof still requires a technical assumption related to the independence of coordinates of sums
ON SMOOTHNESS CONDITIONS AND CONVERGENCE-RATES IN THE CLT IN BANACH-SPACES
No full textBentkus V, Götze F. ON SMOOTHNESS CONDITIONS AND CONVERGENCE-RATES IN THE CLT IN BANACH-SPACES. PROBABILITY THEORY AND RELATED FIELDS. 1993;96(2):137-151.In Banach spaces the rate of convergence in the Central Limit Theorem is of order O(n-1/2) for sets which have 'regular' boundaries with respect to the given covariance structure and which are three times differentiable. We show that in infinite dimensional spaces it is impossible to weaken this differentiability condition in general, whereas in finite dimensional spaces the assumption of convexity suffices. Similar results hold for the expectation of smooth functionals
The Berry-Esseen bound for student's statistic
No full textBentkus V, Götze F. The Berry-Esseen bound for student's statistic. ANNALS OF PROBABILITY. 1996;24(1):491-503.We prove the Berry-Esseen bound for the Student t-statistic. Under the assumption of a third moment this bound coincides (up to an absolute constant) with the classical Berry-Esseen bound for the mean. In general the distribution of the Student statistic converges to the standard normal distribution function at least as fast as the distribution of the mean, and sometimes faster. For example, rates of convergence can be proved if the underlying distribution is in the domain of attraction of the normal law
A Berry-Esseen bound for student's statistic in the non-iid case
No full textBentkus V, Bloznelis M, Götze F. A Berry-Esseen bound for student's statistic in the non-iid case. JOURNAL OF THEORETICAL PROBABILITY. 1996;9(3):765-796.We establish a Berry-Esseen bound for Student's statistic for independent (nonidentically) distributed random variables. In particular, the bound implies a sharp estimate similar to the classical Berry-Esseen bound. In the i.i.d. case it yields sufficient conditions for the Central Limit Theorem for studentized sums. For non-i.i.d. random variables the bound shows that the Lindeberg condition is sufficient for the Central Limit Theorem for studentized sums
A Berry-Esseen bound for M-estimators
No full textBentkus V, Bloznelis M, Götze F. A Berry-Esseen bound for M-estimators. SCANDINAVIAN JOURNAL OF STATISTICS. 1997;24(4):485-502.We prove a Berry-Esseen bound for general M-estimators under optimal regularity conditions on the scare function and the underlying distribution. As an application we obtain Berry-Esseen bounds for the sample median, the L-p-median, p > 1 and Huber's estimator of location
An Edgeworth expansion for symmetric statistics
No full textBentkus V, Götze F, van Zwet WR. An Edgeworth expansion for symmetric statistics. ANNALS OF STATISTICS. 1997;25(2):851-896.We consider asymptotically normal statistics which are symmetric functions of N i.i.d. random variables. For these statistics we prove the validity of an Edgeworth expansion with remainder O(N-1) under Cramer's condition on the linear part of the statistic and moment assumptions for all parts of the statistic. By means of a counterexample we show that it is generally not possible to obtain an Edgeworth expansion with remainder o(N-1) without imposing additional assumptions on the structure of the nonlinear part of the statistic
Berry-Esseen bounds for statistics of weakly dependent samples
No full textBentkus V, Götze F, Tikhomirov A. Berry-Esseen bounds for statistics of weakly dependent samples. BERNOULLI. 1997;3(3):329-349.We prove Berry-Esseen bounds for a general class of asymptotically normal statistics which are functions of N weakly dependent random variables under easily verifiable conditions. In particular, we show, for some delta > 0, the validity of the bound O(N-1/2 log(delta) N) for U-statistics, studentized means, functions of sample means, functionals of empirical distribution functions and linear combinations of order statistics
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