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It has been a rare privilege to assemble this volume of Wassily Hoeffding's Collected Works. Wassily was, variously, a teacher, supervisor and colleague to us, and his work has had a profound influence on our own. Yet this would not be sufficient reason to publish his collected works. The additional and overwhelmingly compelling justification comes from the fun­ damental nature of his contributions to Statistics and Probability. Not only were his ideas original, and far-reaching in their implications; Wassily de­ veloped them so completely and elegantly in his papers that they are still cited as prime references up to half a century later. However, three of his earliest papers are cited rarely, if ever. These include material from his doctoral dissertation. They were written in German, and two of them were published in relatively obscure series. Rather than reprint the original articles, we have chosen to have them translated into English. These trans­ lations appear in this book, making Wassily's earliest research available to a wide audience for the first time. All other articles (including those of his contributions to Mathematical Reviews which go beyond a simple reporting of contents of articles) have been reproduced as they appeared, together with annotations and corrections made by Wassily on some private copies of his papers. Preceding these articles are three review papers which dis­ cuss the . impact of his work in some of the areas where he made major contributions.

Inhaltsverzeichnis

Wassily Hoeffding’s Work in the Sixties

The nineteen sixties were a very significant period for Wassily Hoeffding’s research. Never a prolific writer but rather a careful polisher, he published eight papers during this decade. More importantly, he developed a number of significant new ideas. Here we shall discuss four papers, three of which were published in 1963, 1964 and 1967. These three are landmark papers dealing with probability inequalities, optimal tests for the multinomial distribution and large deviations. The fourth paper was never published and only appeared as a technical report in 1961. It introduced what is now called Hoeffding’s decomposition, which may well make this unpublished paper one of his major contributions.

Kobus Oosterhoff, Willem R. van Zwet

The Impact of Wassily Hoeffding’s Work on Sequential Analysis

I was tremendously privileged to have been one of Wassily Hoeffding’s colleagues. While his demeanor and utter clarity of thought could be intimidating at times, it truthfully can be stated that he was always gracious and generous in his assessments of his colleagues, reserving for himself standards that only someone of his exceptional intellectual stature could hope to achieve.

Gordon Simons

The Impact of Wassily Hoeffding’s Research on Nonparametrics

Wassily Hoeffding earned his Ph.D. degree in mathematics from the Berlin University in 1940 for a dissertation in correlation theory which dealt with some aspects of bivariate probability distributions that are invariant under monotone transformations of the marginals. This dissertation was primarily devoted to some (descriptive) studies of certain measures of rank correlations. With the impending Second World War, for Wassily, living in Berlin in the early forties was not that comfortable. Nevertheless, he managed to advance his basic research on nonparametric correlation theory. It was only after his eventual migration to the United States (in the Fall of 1946) that he started to appreciate the full depth of probability theory and statistics (during his sojourn at the Columbia University, New York), and most of his pioneering work emerged during his longtime residence at Chapel Hill (1947–1991). He felt that “…probability and statistics were very poorly represented in Berlin at that time (1936–1945) …”. Notwithstanding this, his early work on correlation theory was not just a landmark in nonparametrics; it also endowed him with a career-long zeal and affection for the pursuit of the most fundamental research in mathematical statistics, probability theory, numerical analysis and a variety of other related areas. In this respect, nonparametrics was indisputedly the “jewel in the crown” of Wassily’s creativity and ingenuity in research. Wassily Hoeffding indeed played a seminal role in stimulating basic research in a broad domain of mathematical statistics and probability theory, and his “collected work” in this volume reflects the genuine depth and immense breadth of his research contributions. In this article, I shall mainly confine myself to describing the profound impact of his work in the general area of nonparametrics, with occasional detours to some other related areas.

Pranab K. Sen

Scale—Invariant Correlation Theory

The problem of correlation may be described as the investigation of those properties of multivariate distributions which characterize these distributions, i.e., do not occur for univariate distributions. These properties depend above all on the relationships of the variables to each other. From the totality of those properties which belong to the topic of correlation one particular class of properties will be considered more closely.

Wassily Hoeffding

Scale—Invariant Correlation Measures for Discontinuous Distributions

In a recently published paper 1 we were concerned with investigating those properties of two-dimensional distributions that do not depend on the scale by which the random variables are measured. In particular, we pointed out that quantities for measuring the degree of correlation or stochastic dependence of two (random) variables should be so constructed as to be insensitive to changes of the scale of these variables, i.e. scale-invariant. In that paper, a change of scale of the original variables ξ, η meant that the transformations L, g were reversibly unique and left the ordering of the transformed values unchanged.

Wassily Hoeffding

Stochastic Dependence and Functional Relationships

§1. If a quantity α is to serve as a measure of the degree of relationship between two random variables X and Y, we will want to place three fundamental conditions on it, among others: I.α should lie between two fixed finite bounds (say 0 and 1).II.α should equal the lower bound if and only if X and Y are stochastically independent.III.α should equal the upper bound if and only if X and Y are functionally dependent.

Wassily Hoeffding

On the Distribution of the Rank Correlation Coefficient τ When the Variates are not Independent

1. Consider a population distributed according to two variates x, y. Two members (x1, y1,) and (x2, y2) of the population will be called concordant if both values of one member are greater than the corresponding values of the other one, that is if

Wassily Höffding

A Class of Statistics with Asymptotically Normal Distribution

Let X1,…Xnbe n independent random vectors, and Φ(x1, …, xm) a function of m(≤n) vectors. A statistic of the form, where the sum ∑″ is extended over all permutations (α1,…, αm) of m different integers, 1 ≤ αi ≥ n, is called a U-statistic. If X1,…Xn have the same (cumulative) distribution function (d.f.) F(x), U is an unbiased estimate of the population characteristic is called a regular functional of the d.f. F(x). Certain optimal properties of U-statistics as unbiased estimates of regular functionals have been established by Halmos [9] (cf. Section 4).

Wassily Hoeffding

The Central Limit Theorem for Dependent Random Variables

The central limit theorem has been extended to the case of dependent random variables by several authors (Bruns, Markoff, S. Bernstein, P. Lévy, Loève). The conditions under which these theorems are stated either are very restrictive or involve conditional distributions, which makes them difficult to apply. In the present paper we prove central limit theorems for sequences of dependent random variables of a certain special type which occurs frequently in mathematical statistics. The hypotheses do not involve conditional distributions.

Wassily Hoeffding, Herbert Robbins

A Non-Parametric Test of Independence

A test is proposed for the independence of two random variables with continuous distribution function (d.f.). The test is consistent with respect to the class Ω′of d.f.’s with continuous joint and marginal probability densities (p.d.). The test statistic D depends only on the rank order of the observations. The mean and variance of D are given and is shown to have a normal limiting distribution for any parent distribution. In the case of independence this limiting distribution is degenerate, and nD has a non-normal limiting distribution whose characteristic function and cumulants are given. The exact distribution of D in the case of independence for samples of size n = 5, 6, 7 is tabulated. In the Appendix it is shown that there do not exist tests of independence based on ranks which are unbiased on any significance level with respect to the class Ω′. It is also shown that if the parent distribution belongs to Ω′ and for some n ≥ 5 the probabilities of the n; rank permutations are equal, the random variables are independent.

Wassily Hoeffding

“Optimum” Nonparametric Tests

The problem of “optimum” tests has two aspects: (1) the choice of a definition of “optimum,” and (2) the mathematical problem of constructing the test. The second problem may be difficult, but at least it is definite once an “optimum” test has been defined. But the definition itself involves a considerable amount of arbitrariness. Clearly, the definition should be “reasonable” from the point of view of the statistician (which is a very vague requirement) and it should be realizable, that is, an “optimum” test must exist, at least under certain conditions (which is trivial). Furthermore, even a theoretically “best” test is of no use if it cannot be brought into a form suitable for applications. When deciding which of two tests is “better” one ought to take into account not only their power functions but also the labor required for carrying out the tests.

Wassily Hoeffding

A Combinatorial Central Limit Theorem

Let (Yn1,…,Ynn be a random vector which takes on the n! permutations of (1,…, n) with equal probabilities. Let cn(i,j), i,j = 1, …, n, be n real numbers. Sufficient conditions for the asymptotic normality of $$S_n = \sum\limits_{i - 1}^n {c_n \left( {i,Y_{ni} } \right)}$$ are given (Theorem 3). For the special case cn(i,j) = an(i)bn(j) a stronger version of a theorem of Wald, Wolfowitz and Noether is obtained (Theorem 4). A condition of Noether is simplified (Theorem 1).

Wassily Hoeffding

The Large-Sample Power of Tests Based on Permutations of Observations

The paper investigates the power of a family of nonparametric tests which includes those known as tests based on permutations of observations. Under general conditions the tests are found to be asymptotically (as the sample size tends to ∞) as powerful as certain related standard parametric tests. The results are based on a study of the convergence in probability of certain random distribution functions. A more detailed summary will be found at the end of the Introduction.

Wassily Hoeffding

On the Distribution of the Expected Values of the Order Statistics

Let X1,X2,… be independent with a common distribution function F(x) which has a finite mean, and let be the ordered values X1, …,Xn. The distribution of the n values EZn1, …, EZnn on the real line is studied for large n. In particular, it is shown that as n→∞, the corresponding distribution function converges to F(x) and any moment of that distribution converges to the corresponding moment of F(x) if the latter exists. The distribution of the values Ef(Znm) for certain functions f(x) is also considered.

Wassily Hoeffding

A Lower Bound for the Average Sample Number of a Sequential Test

A lower bound is derived for the expected number of observations required by an arbitrary sequential test which satisfies conventional conditions regarding the probabilities of erroneous decisions.

Wassily Hoeffding

Bounds for the Distribution Function of a Sum of Independent, Identically Distributed Random Variables

The problem is considered of obtaining bounds for the (cumulative) distribution function of the sum of n independent, identically distributed random variables with k prescribed moments and given range. For n = 2 it is shown that the best bounds are attained or arbitrarily closely approached with discrete random variables which take on at most 2k + 2 values. For nonnegative random variables with given mean, explicit bounds are obtained when n = 2; for arbitrary values of n, bounds are given which are asymptotically best in the “tail” of the distribution. Some of the results contribute to the more general problem of obtaining bounds for the expected value of a given function of independent, identically distributed random variables when the expected values of certain functions of the individual variables are given. Although the results are modest in Scope, the authors hope that this paper will draw attention to a problem of both mathematical and statistical interest.

Wassily Hoeffding, S. S. Shrikhande

The Efficiency of Tests

The efficiency of a family of tests is defined. Methods for evaluating the efficiency are discussed. The asymptotic efficiency is obtained for certain families of tests under assumptions which imply that the sample size is large.

Wassily Hoeffding, Joan Raup Rosenblatt

The Extrema of the Expected Value of a Function of Independent Random Variables

The problem is considered of determining the least upper (or greatest lower) bound for the expected value EK(X1,…, Xn) of a given function K of n random variables X1, …, Xn under the assumption that X1, …, Xn are independent and each Xi has given range and satisfies k conditions of the form for i = 1, …, k. It is shown that under general conditions we need consider only discrete random variables Xi which take on at most k + 1 values.

Wassily Hoeffding

On the Distribution of the Number of Successes in Independent Trials

Let S be the number of successes in n independent trials, and let pi denote the probability of success in the jth trial, j = 1, 2, …, n (Poisson trials). We consider the problem of finding the maximum and the minimum of Eg(S), the expected value of a given real-valued function of S, when ES = np is fixed. It is well known that the maximum of the variance of S is attained when p1 = p2 = … = pn = p This can be interpreted as showing that the variability in the number of successes is highest when the successes are equally probable (Bernoulli trials). This interpretation is further supported by the following two theorems, proved in this paper. If b and c are two integers, 0 ≦,b≦np≦c≦n, the probability P(b ≦S ≦ c) attains its minimum if and only if p1 = p2 = … = pn = p, unless b = 0 and c = n (Theorem 5, a corollary of Theorem 4, which gives the maximum and the minimum of P(S ≦ cc)). If g is a strictly convex function, Eg(S) attains its maximum if and only if p1 = p2 = … = pn = p (Theorem 3). These results are obtained with the help of two theorems concerning the extrema of the expected value of an arbitrary function g(S) under the condition ES = np. Theorem 1 gives necessary conditions for the maximum and the minimum of Eg(S). Theorem 2 gives a partial characterization of the set of points at which an extremum is attained. Corollary 2.1 states that the maximum and the minimum are attained when p1, p2, …, pn take on, at most, three different values, only one of which is distinct from 0 and 1. Applications of Theorems 3 and 5 to problems of estimation and testing are pointed out in Section 5.

Wassily Hoeffding

The Role of Assumptions in Statistical Decisions

In order to obtain a good decision rule for some statistical problem we start by making assumptions concerning the class of distributions, the loss function, and other data of the problem. Usually these assumptions only approximate the actual conditions, either because the latter are unknown, or in order to simplify the mathematical treatment of the problem. Hence the assumptions under which a decision rule is derived are ordinarily not satisfied in a practical situation to which the rule is applied. It is therefore of interest to investigate how the performance of a decision rule is affected when the assumptions under which it was derived are replaced by another set of assumptions.

Wassily Hoeffding

Distinguishability of Sets of Distributions

The case of independent and identically distributed chance variables

Suppose it is desired to make one of two decisions, d1 and d2, on the basis of independent observations on a chance variable whose distribution F is known to belong to a set F. There are given two subsets G and H of F such that decision d1(d2) is strongly preferred if F is in G (H). Then it is reasonable to look for a test (decision rule) which makes the probability of an erroneous decision small when F belongs to G or H, and at the same time exercises some control over the number of observations required to reach a decision when F is in F (not only in G or H).

Wassily Hoeffding, J. Wolfowitz

Lower Bounds for the Expected Sample Size and the Average Risk of a Sequential Procedure

Sections 1–6 are concerned with lower bounds for the expected sample size, E0(N), of an arbitrary sequential test whose error probabilities at two parameter points, θ1 and θ2, do not exceed given numbers, α1 and α2, where E0(N) is evaluated at a third parameter point, θ0. The bounds in (1.3) and (1.4) are shown to be attainable or nearly attainable in certain cases where θ0 lies between θ1 and θ2. In Section 7 lower bounds for the average risk of a general sequential procedure are obtained. In Section 8 these bounds are used to derive further lower bounds for E0(N) which in general are better than (1.3).

Wassily Hoeffding

22. 22 An Upper Bound for the Variance of Kendall’s “Tau” and of Related Statistics

Let X1, X2, …, Xn be independent and identically distributed random variables (real- or vector-valued). Let f(X1, X2) denote a bounded function such that f(X1, X2) =f(X2, X1). With no loss of generality we shall assume that the bounds are Let

Wassily Hoeffding

Lower Bounds for the Expected Sample Size of a Sequential Test

This expository paper is concerned with lower bounds for the expected sample size EO(N) of an arbitrary sequential test whose error probabilities at two parameter points θ1. and θ2, do not exceed given numbers α1. and α2 where EO(N) is evaluated at a third parameter point θ0. The bounds in (1. 3) and (1.4) are shown to be attainable or nearly attainable in certain cases where θ0 lies between θ1. and θ2.

Wassily Hoeffding

On Sequences of Sums of Independent Random Vectors

This paper is concerned with certain properties of the sequence S1, S2,…of the sums Sn = X1 + … + Xn of independent, identically distributed, k-dimensional random vectors X1X1, …, where k ≧ 1. Attention is restricted to vectors Xn with integer-valued components. Let A1, A2, … be a sequence of k-dimensional measurable sets and let N denote the least n for which S1 ∈ A1. The values S0 = 0, S1, S2, … may be thought of as the successive positions of a moving particle which starts at the origin. The particle is absorbed when it enters set A1 at time n, and N is the time at which absorption occurs. Let M denote the number of times the particle is at the origin prior to absorption (the number of integers n, where 0 ≦n < N, for which S1 = 0). For the special case PXn = -1 = PXn = 1} = 1/2 it is found that (1.1)$$E(M) = E\left( {\left| {S_N } \right|} \right)$$ whenenr E(N) < ∞. Thus the expectcd number of times the particle is at the origin prior to absorption equals its cxpected distance from the origin at the moment of absorption, for any time-dependeut absorption boundary such that the expected time of absorption is finite. Some restriction like E(N) < ∞ is essential. Indeed, if N is the least n ≧ 1 such that S n = 0, equation (1.1) would imply 1 = 0. In this case E(N) = ∞.

Wassily Hoeffding

Probability Inequalities for sums of Bounded Random Variables

Upper bounds are derived for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt. It is assumed that the range of each summand of S is bounded or bounded above. The bounds for PrS — ES≥nt depend only on the endpoints of the ranges of the summands and the mean, or the mean and the variance of S. These results are then used to obtain analogous inequalities for certain sums of dependent random variables such as U statistics and the sum of a random sample without replacement from a finite population.

Wassily Hoeffding

On a Theorem of V. M. Zolotarev

Let be a sequence of independent random variables, where has x2 distribution with nr degrees of freedom and let be a strongly decreasing sequence of positive numbers such that. Then the random variable (1)$$\xi = \sum\limits_{r = 1}^\infty {\sigma _r^2 \chi _r^2 }$$ exists with probability 1. V. M. Zolotarev [1] has shown that (2)$$\begin{array}{*{20}{c}} {\lim } \\ {x \to \infty } \\ \end{array} \frac{{{{\mathcal{P}}_{\xi }}(x)}}{{\mathcal{P}\sigma _{1}^{2}x_{1}^{2}\left( x \right)}} = {{\prod\limits_{{r = 2}}^{\infty } {\left( {1 - \frac{{\sigma _{r}^{2}}}{{\sigma _{r}^{2}}}} \right)} }^{{ - {{n}_{r}}/2}}}$$ where Pξ′(x) is the probability density of the random variable ξ′. From (2) one can easily obtain an asymptotic expression for P{ξ > x} for x → ∞.

W. Hoeffding

Asymptotically Optimal Tests for Multinomial Distributions

Tests of simple and composite hypotheses for multinomial distributions are considered. It is assumed that the size αN of a test tends to 0 as the sample size N increases. The main concern of this paper is to substantiate the following proposition: If a given test of size αN is “sufficiently different” from a likelihood ratio test then there is a likelihood ratio test of size ≦αN which is considerably more powerful than the given test at “most” points in the set of alternatives when N is large enough, provided that αN → 0 at a suitable rate. In particular, it is shown that chi-square tests of simple and of some composite hypotheses are inferior, in the sense described, to the corresponding likelihood ratio tests. Certain Bayes tests are shown to share the above-mentioned property of a likelihood ratio test.

Wassily Hoeffding

On Probabilities of Large Deviations

The paper is concerned with the estimation of the probability that the empirical distribution of n independent, identically distributed random vectors is contained in a given set of distributions. Sections 1–3 are a survey of some of the literature on the subject. In section 4 the special case of multinomial distributions is considered and certain results on the precise order of magnitude of the probabilities in question are obtained.

Wassily Hoeffding

Some Recent Developments In Nonparametric Statistics

The main motivation for the development of nonparametric statistics was the need for statistical methods that have desirable properties when little is assumed about the population or populations being sampled. For a number of problems tests were designed whose probability of falsely rejecting the hypothesis was equal or at most equal to a specified constant under little or no assumptions beyond that of random sampling and which were consistent (that is, had error probabilities approaching zero with increasing sample size) in a wide class of alternatives. Classical examples are Smirnov’s two-sample test, which is consistent against all alternatives of the two-sample problem, and Wilcoxon’s two-sample test, whose domain of consistency is more restricted. These two tests depend only on the rank order of the observations and therefore seem to discard much information contained in the sample. It seemed reasonable to expect that a test which is valid under few assumptions, and especially a rank test, could not be nearly as powerful in a parametric class of distributions as an optimal parametric test for that class. It came therefore as a surprise when it was found that often there are nonparametric tests, including rank tests, which compare favorably with corresponding classical parametric tests.

W. Hoeffding

Unbiased Coin Tossing With a Biased Coin

Procedures are exhibited and analyzed for converting a sequence of i.i.d. Bernoulli variables with unknown mean p into a Bernoulli variable with mean 1/2. The efficiency of several procedures is studied.

Wassily Hoeffding, Gordon Simons

Discussion on Hájek’s Paper

Instead of elaborating on Professor Hájek’s interesting and valuable paper, I will, in the spirit of the title of his paper, make a few methodological remarks on the problem of the asymptotic distribution of a linear rank statistic.

W. Hoeffding

Discussion on Witting’s Paper

Professor Witting’s paper brings out the structure of a generalized rank test and the analogous features of a generalized permutation test (test of structure S), in particular of tests of these types that are optimal against specified alternatives. I want to make a few remarks on the relative merits of these two kinds of tests, a topic which Professor Witting deliberately did not elaborate on.

W. Hoeffding

The L 1 Norm of the Approximation Error for Bernstein-Type Polynomials

This paper is concerned with the estimation of the L1 norm of the difference between a function of bounded variation and an associated Bernstein polynomial, and with the analogous problem for a Lebesgue integrable function of bounded variation inside (0, 1). A real-valued function defined in the open interval (0, 1) is said to be of bounded variation inside (0, 1) if it is of bounded variation in every closed subinterval of (0, 1). The class of these functions will be denoted by BV*. To formulate some of the results, we state the following lemma, which is a simple consequence of the well-known canonical representation of a function of bounded variation.

Wassily Hoeffding

On the Centering of a Simple Linear Rank Statistic

Hájek (1968) proved that under weak conditions the distribution of a simple linear rank statistic S is asymptotically normal, centered at the mean ES. He left open the question whether under the same conditions the centering constant ES may be replaced by a simpler constant μ, as was found to be true in the two-sample case and under different conditions by Chernoff and Savage (1958) and Govindarajulu, LeCam and Rhagavachari (1966). In this paper it is shown that the replacement ofES by μis permissible if one of Hájek’s conditions is slightly strengthened.

Wassily Hoeffding

The L 1 Norm of the Approximation Error for Splines with Equidistant Knots

This paper is concerned with the estimation of the L1 norm of the difference between a function f of bounded variation in [0, 1] and the associated variation-diminishing spline function with equidistant knots, Sm,nf, with see Schoenberg [4]. For f bounded on [0, 1] and for integers m, n such that (1.1)$$n \geqslant m \geqslant 2,$$ the function S m,n f is defined by (1.2)$$S_{m.n} f\left( x \right) = \sum\limits_{j = 0}^\iota {f\left( {n^{ - 1} \xi \left( {m,j} \right)} \right)\bar N_{m.j} \left( {nx} \right),}$$ where (1.3)$$l = m + n - 2,$$(1.4)$$\xi \left( {m,j} \right) = \left\{ {\begin{array}{*{20}{c}} {\frac{{\left( {j + 1} \right)j}}{{2\left( {m - 1} \right)}},} \hfill & {j = 0, \ldots ,m - 2} \hfill \\ {j + 1 - \frac{m}{2},} \hfill & {j = m - 1, \ldots ,n - 1,} \hfill \\ {n - \xi \left( {m,l - j} \right),} \hfill & {j = n, \ldots ,l,} \hfill \\ \end{array} } \right.$$(1.5)$${{\tilde{N}}_{{m,j\left( x \right)}}} = \left\{ {\begin{array}{*{20}{c}} {\frac{{j + 1}}{m}{{h}_{{m.j + 1}}}\left( x \right),} \hfill & {j = 0, \ldots ,m - 2,} \hfill \\ {{{h}_{m}}\left( {x + m - j - 1} \right),} \hfill & {j = m - 1, \ldots ,n - 1,} \hfill \\ {{{{\bar{N}}}_{{m.l - j}}}\left( {n - x} \right),} \hfill & {j = n, \ldots ,l,} \hfill \\ \end{array} } \right.$$(1.6)$${{h}_{{m,k}}}\left( x \right) = \frac{m}{{k!}}\sum\limits_{{i = 1}}^{k} {{{{( - 1)}}^{{k - i}}}{{i}^{{k - m}}}} \left( {_{i}^{k}} \right)\left( {i - x} \right)_{ + }^{{m - 1}}{\mkern 1mu} {\text{for }}x0,$$(1.7)$$h_m \left( x \right) = h_{m.m} \left( x \right).$$.

Wassily Hoeffding

Harold Hotelling, 1895–1973

Harold Hotelling, mathematical statistician and mathematical economist, died on 26 December 1973. He was a leader in the field of multivariate statistical analysis and a prominent figire in the development of economic theory.

N. I. Fisher, P. K. Sen

H

Harold Hotelling, a leader In the field of multivariate statistical analysis, played a prominent part in the spectacular growth of mathematical statistics in the United States that began in the 1930s and helped bring about the revival of mathematical economics in the late 1920s.

Wassily Hoeffding

Some Incomplete and Boundedly Complete Families of Distributions

Let Pbe a family of distributions on a measurable space such that, for all P∈P, and which is sufficiently rich; for example, p consists of all distributions dominated by a σ-finite measure and satisfying (†). It is known that when conditions (†) are not present, no nontrivial symmetric unbiased estimator of zero (s.u.e.z.) based on a random sample of any size n exists. Here it is shown that (I) if g(x1,…, xn) is a s.u.e.z. then there exist symmetric functions h1,(x1,…, xn-1), i = 1, …,k, such that $$g\left( {x_1 , \cdots ,x_n } \right) = \sum\nolimits_{i = 1}^k {\sum\nolimits_{i = 1}^n {\left\{ {u_i \left( {x_i } \right) - c_i } \right\}h_i \left( {x_1 , \cdots ,x_{j - 1} ,x_{j + 1} , \cdots ,x_n } \right)} }$$ and (II) if every non trivial linear combination of u1, …, u k is unbounded then no bounded non trivial s.u.e.z. exists. Applieations to unbiased estimation and similar tests are diseussed.

Wassily Hoeffding

More on Incomplete and Boundedly Complete Families of Distributions

Let 2. be a family of distributions (probability measures) on a measurable space [Y,B) and let r be a group of B-measurable transformations of Y. The family 2. is said to be complete relative to r if no nontrivial r-invariant unbiased estimator of zero for 2. exists. (A function is called Γ-invariant if it is invariant under all transformations in Γ.) The family 2. is said to be boundedly complete relative to Γ if no bounded nontrivial Γ-invariant unbiased estimator of zero for 2. exists.

Wassily Hoeffding

A Statistician’s Progress from Berlin to Chapel Hill

I was born in 1914 in Mustamaki, Finland, near St. Petersburg (now Leningrad). Finland was at that time part of the Russian Empire. My father, whose parents were Danish, was an economist and a disciple of Peter Struve, the Russian social scientist and public figure. An uncle of my father’s was Harald Hoeffding, the philosopher. My mother, née Wedensky, had studied medicine. Both grandfathers had been engineers.

Wassily Hoeffding

Unbiased Range-Preserving Estimators

An estimator is said to be range-preserving if its values are confined to the range of what it is to estimate. The property of being range-preserving Is an essential property of an estimator, a sine qua non. Other properties, such as unbiasedness, may be desirable in some situations, but an unbiased estimator that Is not range-preserving should be ruled out as an estimator. (We are not speaking of uses of estimators for purposes other than estimation, for example, as test statistics.)

Wassily Hoeffding

Range Preserving Unbiased Estimators in the Multinomial Case

Consider estimating the value of a real-valued function f(p),p = (p0,P1. …,Pr), on the basis of an observation of the random vector X = (X0, X1, …,Xr) whose distribution is multinomial (n, p). It is known that an unbiased estimator exists if and only if f is a polynomial of degree at most n, in which case the unbiased estimator off(p) is unique. In general, however, this estimator has the serious fault of not being range preserving; that is, its value may fall outside the range of f(p). In this article, a condition on f is derived that is necessary for the unbiased estimator to be range preserving and that is sufficient when n is large enough.

Wassily Hoeffding

Asymptotic Normality

The exact distribution of a statistic is usually highly complicated and difficult to work with. Hence the need to approximate the exact distribution by a distribution of a simpler form whose properties are more transparent. The limit theorems* of probability theory provide an important tool for such approximations. In particular, the classical central limit theorems* state that the sum of a large number of independent random variables is approximately normally distributed under general conditions (see the section “Central Limit Theorems for Sums of Independent Random Variables”). In fact, the normal distribution* plays a dominating role among the possible limit distributions. To quote from Gnedenko and Kolmogorov [18, Chap. 5]: “Whereas for the convergence of distribution functions of sums of independent variables to the normal law only restrictions of a very general kind, apart from that of being infinitesimal (or asymptotically constant), have to be imposed on the sum mands, for the convergence to another limit law some very special properties are required of the summands.” Moreover, many statistics behave asymptotically like sums of independent random variables (see the fifth, sixth, and seventh sections). All of this helps to explain the importance of the normal distribution* as an asvmototic distribution.

W. Hoeffding

Hájek’s Projection Lemma

imation is best in an extensive class of such sums, in a sense explained in the statement of the lemma. Since much is known about asymptotic distributions of sums of independent random variables, the lemma may enable us to find the asymptotic distribution of a more general function of independent random variables (see the Corollary below).

N. I. Fisher, P. K. Sen

Hoeffding’s Independence Test

Let the random vector (X, Y) have the cumulative distribution function * (CDF) F(x, y). Let ℱ be the class of all continuous bivariate CDFs, and ℱ0 be the class of all F ∈ ℱ such that F(x, y) = F(x, ∞) F(∞, y), Assume that F ∈ ℱ. The hypothesis H0 that X and Y are independent is equivalent to the hypothesis that F ∈ ℱ0.

N. I. Fisher, P. K. Sen

Probability Inequalities for Sums of Bounded Random Variables

If S is a random variable with finite rnean and variance, the Bienaymé-Chebyshev inequality states that for x > 0, (1)$$\Pr \left[ {\left| {S - ES} \right| \geqslant x{{{(\operatorname{var} S)}}^{{1/2}}}} \right] \leqslant {{x}^{{ - 2}}}$$ If S is the surn of n independent, identically distributed random variables, then, by the central limit theorem*, as n → ∞, the probability on the left approaehes 2Ф( - x), where Ф(x) is the standard normal distribution function. For x large, Ф( - x) behaves as const. x-1 exp( - x2/2).

N. I. Fisher, P. K. Sen

Range-Preserving Estimators

An estimator is said to be range preserving if its values are confined to the range of what it is to estimate. The property of being range preserving is an essential property of an estimator, a sine qua non. Other properties, such as unbiasedness, may be desirable in some situations, but an unbiased estimator that is not range preserving should be ruled out as an estimator. [We are not speaking of uses of estimators for purposes other than estimation (e.g., as test statistics ).]

N. I. Fisher, P. K. Sen

Book Reviews

The general nature of this book is well described in the following quotation from the author’s preface: “… I have made a selection of basic material in mathematical statistics in accordance with my own preferences and prejudices, with inclinations toward trying to make a unified and systematic presentation of classical results of mathematical statistics, together with some of the more important contemporary results in a framework of modern probability theory, without going into too many ramifications.” An early version of some of the material was issued under the same title in 1943 in lithoprinted form by the Princeton University Press. The book is intended for readers with good undergraduate backgrounds in mathematics. It starts out with a brief account of the foundations of modern probability theory, followed by chapters on distribution functions, mean values and moments, sequences of random variables, characteristic and generating functions, and special distributions. The statistical part begins with sampling theory and asymptotic sampling theory, followed by three chapters on statistical estimation (linear, nonparametric, and parametric) and two on hypothesis testing (parametric and nonparametric). The final chapters deal with sequential analysis, statistical decision functions, time series, and multivariate statistical theory. There are over 400 problems most of which are very helpful to the student and a good bibliography of 19 pages (which serves also as an author index).

William Kruskal

Backmatter

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