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## Über dieses Buch

With this collections volume, some of the important works of Willem van Zwet are moved to the front layers of modern statistics. The selection was based on discussions with Willem, and aims at a representative sample. The result is a collection of papers that the new generations of statisticians should not be denied. They are here to stay, to enjoy and to form the basis for further research.

The papers are grouped into six themes: fundamental statistics, asymptotic theory, second-order approximations, resampling, applications, and probability. This volume serves as basic reference for fundamental statistical theory, and at the same time reveals some of its history.

The papers are grouped into six themes: fundamental statistics, asymptotic theory, second-order approximations, resampling, applications, and probability. This volume serves as basic reference for fundamental statistical theory, and at the same time reveals some of its history.

## Inhaltsverzeichnis

Open Access

### Convex transformations: A new approach to skewness and kurtosis

Abstract
In dit artikel een tweetal orde-relaties voor waarschijinlijikheidsverdelingen voorgestld, die - beter dan de kalassieke maten gebseerd op derde en vierde momenten - aangeven wanneer een verdeling een geotere scheefhied of kurtosis bezit dan een andere verdeling.
W. R. von Zwet

Open Access

### Some Remarks on the Two-Armed Bandit

Abstract
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.
J. Fabius, W. R. van Zwet

Open Access

### Van De Hulst on Robust Statistics: A Historical Note

Abstract
This paper provides a discussion of an unpuplished set of notes written in 1942 by the Dutch astronomer H.C. Van DE HULST. In these notes VAN DE HULST derives the asymptotic variances of M-estimators as well as trimed means and concludes that the asympotic variances of what is now called (1965). A letter written by D. VAN DANTZIG in 1943 providing a critical evaluation of Van DE HULST’s results, adds interest to this suprisingly early contibution to the theory of robust statiscs.
W. R. van Zwet

Open Access

### Chapter 4 Discussion of three statistics papers by Willem van Zwet

Abstract
I discuss three statistics papers of willem van Zwet: van Zwet(1964a), Fabius and van Zwet (1970), and van Zwet (1985).
Jon A. Wellner

Open Access

### Asymptotic Normality of Nonparametric Tests for Independence

Abstract
Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established under fixed alternatives. A generalization of a result of Bhuchongkul [I) is obtained both with respect to the conditions concerning the orders of magnitude of the score functions and with respect to the smoothness conditions on these functions.
F. H. Ruymgaart, G. R. Shorack, W. R. van Zwet

Open Access

### A Note on Contiguity and Hellinger Distance

Abstract
For n = 1, 2, … let (X n1, A n1), …, (X nn , A nn ) be arbitrary measurable spaces. Let P ni and Q ni be probability measures defined on (X ni , A ni ), i = 1, …, n; n = 1, 2, …, and let $$P_n^{\left( n \right)} = \prod\limits_{i = 1}^n {{P_{ni}}}$$ and $$Q_n^{\left( n \right)} = \prod\limits_{i = 1}^n {{Q_{ni}}}$$ denote the product probability measures. For each i and n let X ni be the identity map from X ni onto X ni . Then P ni and Q ni represent the two possible distributions of the random element X ni as well as the probability measures of the underlying probability space. Obviously X n1, …, X nn are independent under both $$P_n^{\left( n \right)}$$ and $$Q_n^{\left( n \right)}$$ (n = 1, 2, …).
J. Oosterhoff, W. R. van Zwet

Open Access

### On Estimating a Parameter and Its Score Function

Abstract
We consider the problem of estimating a real-valued parameter θ in the presence of an abstract nuisance parameter η, such as an unknown distributional shape. Attention is restricted to the case in which the score functions for θ and η are orthogonal, so that fully asymptotically efficient estimation is not a priori impossible. For fixed sample size, we provide a bound of Cramér-Rao type. The bound differs from the classical one for known η by a term involving the integrated mean square error of an estimator of a multiple of the score function for θ for the case where θ is known. This implies that an estimator of θ can only perform well over a class of shapes η if it is possible to estimate the score function for θ accurately over this class.
C. A. J. Klaassen, W. R. van Zwet

Open Access

### On Estimating a Parameter and Its Score Function, II

Abstract
A bound of Cramer-Rao type is provided for an estimator of a real-valued parameter θ in the presence of an abstract nuisance parameter η, such as an unknown distributional shape, on the basis of N i.i.d. observations. The bound consists of the reciprocal of the effective Fisher information in the sample, plus a term involving the integrated mean squared error of an estimator of a multiple of the so-called conditional score function for θ, for the case where θ is known. This implies that an estimator of θ can only perform well over a class of shapes η if it is possible to estimate the conditional score function for θ accurately over this class. For the special case where fully adaptive estimation may be possible, this result was given in a companion paper (Klaassen and van Zwet (1985)).
C. A. J. Klaassen, A. W. van der Vaart, W. R. van Zwet

Open Access

### A Remark on Consistent Estimation

Abstract
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].
E. W. van Zwet, W. R. van Zwet

Open Access

### Chapter 10 Finite samples and asymptotics

Abstract
Willem van Zwet is a scientist and a scholar with a broad spectrum of research interests. This is reflected by the five papers in this section, which study very different fundamental problems and which have four of his PhD students and his youngest son as coauthor.
Chris A. J. Klaassen

Open Access

### Asymptotic Expansions for the Power of Distributionfree Tests in the Two-Sample Problem

Abstract
Asymptotic expansions are established for the power of distributionfree tests in the two-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann for distributionfree tests with respect to their parametric competitors and for the estimators of shift associated with these tests.
P. J. Bickel, W. R. van Zwet

Open Access

### On Efficiency of First and Second Order

Summary
It has been noted by a number of authors that if two tests are asymptotically efficient for the same testing problem, then typically their powers will not only agree to first but also to second order. A general result of this type was given by Pfanzagl (1979) in a paper entitled ’First order efficiency implies second order efficiency’. Because of their technical nature, however, these contributions give little insight into the nature of this phenomenon. The purpose of the present paper is to provide an intuitive understanding of the phenomenon by proving a simple theorem of this kind under mild assumptions.
P. J. Bickel, D. M. Chibisov, W. R. van Zwet

Open Access

### A Berry-Esseen Bound for Symmetric Statistics

Summary
The 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 ($$O\left( {{N^{ - \frac{1}{2}}}} \right)$$). This theorem generalizes many known results for special cases and two examples are given. Possible further extensions are indicated.
W. R. van Zwet

Open Access

### The Edgeworth Expansion for U-Statistics of Degree Two

Abstract
An Edgeworth expansion with remainder o(N−1) is established for a U-statistic with a kernel h of degree 2. The assumptions involved appear to be very mild; in particular, the common distribution of the summands h(X i , X j ) is not assumed to be smooth.
P. J. Bickel, F. Götze, W. R. van Zwet

Open Access

### Chapter 15 Entropic instability of Cramer’s characterization of the normal law

Abstract
We establish instability of the characterization of the normal law in Cramer’s theorem with respect to the total variation norm and the entropic distance. Two constructions of counter-examples are provided.
S. G. Bobkov, G. P. Chistyakov, F. Götze

Open Access

### Resampling: Consistency of Substitution Estimators

Abstract
On the basis of N i.i.d. random variables with a common unknown distribution P we wish to estimate a functional τ N (P). An obvious and very general approach to this problem is to find an estimator N of P first, and then construct a so-called substitution estimator τ N (P̂ N ) of τ N (P). In this paper we investigate how to choose the estimator N so that the substitution estimator τ N ( N ) will be consistent.
Although our setup covers a broad class of estimation problems, the main substitution estimator we have in mind is a general version of the bootstrap where resampling is done from an estimated distribution N . We do not focus in advance on a particular estimator N , such as, for example, the empirical distribution, but try to indicate which resampling distribution should be used in a particular situation. The conclusion that we draw from the results and the examples in this paper is that the bootstrap is an exceptionally flexible method which comes into its own when full use is made of its flexibility. However, the choice of a good bootstrap method in a particular case requires rather precise information about the structure of the problem at hand. Unfortunately, this may not always be available.
Hein Putter, Willem R. van Zwet

Open Access

### Resampling Fewer Than n Observations: Gains, Losses, and Remedies for Losses

Abstract
We discuss a number of resampling schemes in which m = o(n) observations are resampled. We review nonparametric bootstrap failure and give results old and new on how the m out of n with replacement and without replacement bootstraps work. We extend work of Bickel and Yahav (1988) to show that m out of n bootstraps can be made second order correct, if the usual nonparametric bootstrap is correct and study how these extrapolation techniques work when the nonparametric bootstrap does not.
P. J. Bickel, F. Götze, W. R. van Zwet

Open Access

### 20 On a Set of the First Category

Abstract
In an analysis of the bootstrap Putter & van Zwet (1993) showed that under quite general circumstances, the bootstrap will work for “most” underlying distributions. In fact, the set of exceptional distributions for which the bootstrap does not work was shown to be a set D of the first category in the space P of all possible underlying distributions, equipped with a topology Π. Such a set of the first category is usually "small" in a topological sense. However, it is known that this concept of smallness may sometimes be deceptive and in unpleasant cases such "small" sets may in fact be quite large.
Here we present a striking and hopefully amusing example of this phenomenon, where the “small” subset D equals all of P. We show that as a result, a particular version of the bootstrap for the sample minimum will never work, even though our earlier results tell us that it can only fail for a “small” subset of underlying distributions. We also show that when we change the topology on P—and as a consequence employ a different resampling distribution—this paradox vanishes and a satisfactory version of the bootstrap is obtained. This demonstrates the importance of a proper choice of the resampling distribution when using the bootstrap.
Hein Putter, Willem R. van Zwet

Open Access

### Chapter 19 Discussion of three resampling papers

Abstract
Discussion of: Putter H., and van Zwet W.R. (1996) . Resampling: Consistency of substitution estimators, Annals of Statistics 24 2297-2318. Putter H., and van Zwet W.R. (1997). On a set of the first category. Festschrift for Lucien Le Cam, Springer Verlag , 315-324. Bickel P., Gotze F. and van Zwet W.R. (1997). Resampling fewer than n observations: Gains, Losses and Remedies for Losses, Statistica Sinica 1 1-31.
Peter J. Bickel

Open Access

### A Non-Markovian Model for Cell Population Growth: Tail Behavior and Duration of the Growth Process

Abstract
De Gunst has formulated a stochastic model for the growth of a certain type of plant cell population that initially consists of n cells. The total cell number N n (t) as predicted by the model is a non-Markovian counting process. The relative growth of the population, n −1(N n (t) - n), converges almost surely uniformly to a nonrandom function X. In the present paper we investigate the behavior of the limit process X(t) as t tends to infinity and determine the order of magnitude of the duration of the process N n (t). There are two possible causes for the process N n to stop growing, and correspondingly, the limit process X(t) has a derivative X’(t) that is the product of two factors, one or both of which may tend to zero as t tends to infinity. It turns out that there is a remarkable discontinuity in the tail behavior of the processes. We find that if only one factor of X’(t) tends to zero, then the rate at which the limit process reaches its final limit is much faster and the order of magnitude of the duration of the process N n is much smaller than when both occur approximately at the same time.
Mathisca C. M. de Gunst, Willem R. van Zwet

Open Access

### Parameter estimation for the supercritical contact process

Abstract
Contact processes - and, more generally, interacting particle processes - can serve as models for a large variety of statistical problems, especially if we allow some simple modifications that do not essentially complicate the mathematical treatment of these processes. We begin a statistical study of the supercritical contact process that starts with a single infected site at the origin and is conditioned on survival of the infection. We consider the statistical problem of estimating the parameter λ of the process on the basis of an observation of the process at a single time t. We propose an estimator of λ and show that it is consistent and asymptotically normal as t → ∞.
Marta Fiocco, Willem R. van Zwet

Open Access

### On The Minimal Travel Time Needed to Collect n Items on a Circle

Abstract
Consider n items located randomly on a circle of length 1. The locations of the items are assumed to be independent and uniformly distributed on [0, 1). A picker starts at point 0 and has to collect all n items by moving along the circle at unit speed in either direction. In this paper we study the minimal travel time of the picker. We obtain upper bounds and analyze the exact travel time distribution. Further, we derive closed-form limiting results when n tends to infinity. We determine the behavior of the limiting distribution in a positive neighborhood of zero. The limiting random variable is closely related to exponential functionals associated with a Poisson process. These functionals occur in many areas and have been intensively studied in recent literature.
Nelly Litvak, Willem R. van Zwet

Open Access

### Chapter 23 Applications: simple models and difficult theorems

Abstract
In this short article I will discuss three papers written by Willem van Zwet with three different co-authors: Mathisca de Gunst, Marta Fiocco, and myself. Each of the papers focuses on one particular application: growth of the number of biological cells [3], spreading of an infection [7], and the optimal travel time in warehousing carousel systems [8]. To my opinion, each of these papers displays the attitude that I personally value a lot in mathematics. An application is the strong starting point for each of the papers. Further, the model is simple and transparent. Yet, the analysis involves advanced mathematics and brings to the results that not only give new insights into the applications but also are of a pure mathematical interest.
Nelly Litvak

Open Access

### A Proof of Kakutani’s Conjecture on Random Subdivision of Longest Intervals

Abstract
Choose a point at random, i.e., according to the uniform distribution, in the interval (0, 1). Next, choose a second point at random in the largest of the two subintervals into which (0, 1) is divided by the first point. Continue in this way, at the nth step choosing a point at random in the largest of the n subintervals into which the first (n - 1) points subdivide (0, 1). Let F n be the empirical distribution function of the first n points chosen. Kakutani conjectured that with probability 1, F n converges uniformly to the uniform distribution function on (0, 1) as n tends to infinity. It is shown in this note that this conjecture is correct.
W. R. van Zwet

Open Access

### A Strong Law for Linear Functions of Order Statistics

Abstract
A strong law of large numbers for linear combinations of order statistics is proved under integrability conditions only. Together with some straightforward extensions, the theorem generalizes previous results of Wellner, Helmers and Sen.
W. R. van Zwet

Open Access

### A Refinement of the KMT Inequality for the Uniform Empirical Process

Abstract
A refinement of the Komlós, Major and Tusnády (1975) inequality for the supremum distance between the uniform empirical process and a constructed sequence of Brownian bridges is obtained. This inequality leads to a weighted approximation of the uniform empirical and quantile processes by a sequence of Brownian bridges dual to that recently given by M. Csoörgő, S. Csörgő, Horváth and Mason (1986). The present theory approximates the uniform empirical process more closely than the uniform quantile process, whereas the former theory more closely approximates the uniform quantile process.
David M. Mason, Willem R. van Zwet

Open Access

### The Asymptotic Distribution of Point Charges on a Conducting Sphere

Abstract
Consider n point charges, each with charge $$\frac{1}{n}$$ in electrostatic equilibrium on the surface S of a conducting sphere. It is shown that as n tends to infinity, the distribution of the total charge 1 on S tends to the uniform distribution on S. Though this is an entirely deterministic result, the proof is probabilistic in nature.
Willem R. van Zwet

Open Access

### Weak Convergence Results for the Kakutani Interval Splitting Procedure

Abstract
This paper obtains the weak convergence of the empirical processes of both the division points and the spacings that result from the Kakutani interval splitting model. In both cases, the limit processes are Gaussian. For the division points themselves, the empirical processes converge to a Brownian bridge as they do for the usual uniform splitting model, but with the striking difference that its standard deviations are about one-half as large. This result gives a clear measure of the degree of greater uniformity produced by the Kakutani model. The limit of the empirical process of the normalized spacings is more complex, but its covariance function is explicitly determined. The method of attack for both problems is to obtain first the analogous results for more tractable continuous parameter processes that are related through random time changes. A key tool in their analysis is an approximate Poissonian characterization that obtains for cumulants of a family of random variables that satisfy a specific functional equation central to the K -model.
Ronald Pyke, Willem R. van Zwet

Open Access

### Chapter 29 A discussion of Willem van Zwet’s probability papers

Abstract
I shall begin my discussion of Willem’s probability papers with his 1978 paper on the Kakutani conjecture. Willem tells me that this is his favorite paper. I can see why. It is not only a fine piece of mathematics; it also displays very well a common feature of many of Willem’s best papers, namely, it begins with a key insight, which lights the way to the solution of a knotty problem.
David M. Mason
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