W. R. van Zwet

A Berry-Esseen bound for symmetric statistics

SummaryThe rate of convergence of the distribution function of a symmetric function of N independent and identically distributed random variables to its normal limit is investigated. Under appropriate moment conditions the rate is shown to be % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0Jd9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFoe-taaa!413D!

Modelling the growth of a batch culture of plant cells: A corpuscular approach

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A Remark on Consistent Estimation

(N−1/2). This theorem generalizes many known results for special cases and two examples are given. Possible further extensions are indicated.

Miscellanea - The Times of Yuri Vasilyevich Prokhorov

In this paper we consider the following situation: An experimenter has to perform a total of N trial on two Bernoulli-type experiments E1 and E2 with success probabilites α and β respectively, where both α and β are unknown to him.

An Edgeworth expansion for symmetric statistics

Abstract A stochastic model is formulated for the cell cycle of plant cells during batch culture. In the model two types of cells are distinguished: dividing cells and nondividing, differentiating cells. The cell cycle duration is assumed to depend on the substrate concentration in the medium, whereas the number of newly born cells, which will again divide, is supposed to depend on the hormone concentration in the medium. The consequences of the model for the growth of the total plant cell culture in terms of cell numbers are considered. Experiments performed in order to test the model are described and the results are discussed. This approach of describing plant cell population growth by means of a corpuscular mathematical model is a first attempt to obtain a better understanding of the regulation of division, differentiation, growth, and product formation of plant cells grown under different conditions.

Asymptotic Normality of Nonparametric Tests for Independence

Publisher Summary This chapter discusses asymptotic expansions and explains their need. It reviews the classical theory of Edgeworth expansions for sums of independent and identically distributed random variables, and indicates the two main techniques for extending this theory to more general statistics. The chapter presents an account of as yet unpublished results of Bjerve and Helmers who establish Berry–Esseen type bounds for linear combinations of order statistics. For many years, mathematical statisticians have spent a great deal of effort and ingenuity toward applying the central limit theorem in statistics. The estimators and test statistics that interest statisticians are as a rule not sums of independent random variables, and much work went into showing that they can often be approximated sufficiently well by such sums to ensure asymptotic normality. This work can be traced throughout the development of mathematical statistics from the proof of the asymptotic normality of the maximum likelihood estimator to much of the recent work in nonparametric statistics.

A Strong Law for Linear Functions of Order Statistics

In this paper we re-examine two auxiliary results in Putter and va.n Zwet [7]. Viewed in a new light these results provide some insight in two related phenomena, to wit consistency of estimators and local asymptotic equivariance. Though technically quite different, our conclusions will be similar to those in Beran [I] and LeCam a.nd Yang [5].

On the Combination of Independent Test Statistics

Yuri Vasilyevich Prokhorov is the eminence grise of Russian probability theory. Every one of us interested in probability or asymptotic statistics has come across his celebrated weak compactness theorem at one time or another. He was interviewed earlier by Larry Shepp (Statistical Science, 7 (1992), pp. 123--130). That interview dealt largely with his impressive career and scientific work, his international contacts, and the issue of discrimination in the Soviet Union. The world has changed considerably in the intervening years, and our knowledge and perspective of the past has developed accordingly. It seemed natural to us to talk once more with the man who lived through these turbulent times as the intellectual heir of Kolmogorov, and as one who was in a position to observe the inner workings of the powerful Soviet (later Russian) Academy of Sciences, the Steklov Mathematical Institute in Moscow, and the activities of his many colleagues throughout the country and elsewhere. This interview took place be...