Theory Of U-statistics Research Articles

U-Statistics: Theory and Practice.

In 1946 Paul Halmos studied unbiased estimators of minimum variance, and planted the seed from which the subject matter of the present monograph sprang. The author has undertaken to provide experts and advanced students with a review of the present status of the evolved theory of U-statistics, including applications to indicate the range and scope of U-statistic methods. Complete with over 200 end-of-chapter references, this is an invaluable addition to the libraries of applied and theoretical statisticians and mathematicians.

Statistics of two-dimensional point vortices and high-energy vortex states

Results from the theory ofU-statistics are used to characterize the microcanonical partition function of theN-vortex system in a rectangular region for largeN, under various boundary conditions, and for neutral, asymptotically neutral, and nonneutral systems. Numerical estimates show that the limiting distribution is well matched in the region of major probability forN larger than 20. Implications for the thermodynamic limit are discussed. Vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices. Vortex states for which clustering is generic (in a statistical sense) are shown to result from two modeling processes: the approximation of a continuous inviscid fluid by point vortex configurations; and the modeling of the evolution of a continuous fluid at high Reynolds number by point vortex configurations, with the viscosity represented by the annihilation of close positive-negative vortex pairs. This last process, with the vortex dynamics replaced by a random walk, reproduces quite well the late-time features seen in spectral integration of the 2d Navier-Stokes equation.

Some Theory of Statistical Inference for Nonlinear Science

This article shows how standard errors can be estimated for a measure of the number of excited degrees of freedom (the correlation dimension), and a measure of the rate of information creation (a proxy for the Kolmogorov entropy), and a measure of instability. These measures are motivated by nonlinear science and chaos theory. The main analytical method is central limit theory of U-statistics for mixing processes. The paper takes a step toward formal hypothesis testing in nonlinear science and chaos theory.

A simple asymptotic analysis of fisher's z-statistic

Recently, Hawkins [1989] presented a nice deriviation of the asymptotic normality of Fisher's Z-statistic for a general bivairate sample based on the asymptotic theory of U-statistics. For a bivariate normal parent distribution, Fisher's Z-statistic has asymptotic variance 1/n, independent of the correlation coefficient and all other paremeters defining the parent distribution. Hawkins' result is important, since it shows the dependence, in general, of the asymptotic variance of Z on the correlation coeffient as well as other moments of the underlying bivariate parent distribution. The purpose of this note is to make Hawkins' result more accessible to a wider audience, presenting the derivation without recourse to the U-static theory.

UsingUStatistics to Derive the Asymptotic Distribution of Fisher's Z Statistic

Abstract A simple derivation of the asymptotic distribution of Fisher's Z statistic for general bivariate parent distributions F is obtained using U-statistic theory. This method easily reveals that the asymptotic variance of Z generally depends on the correlation ρ and on certain moments of F. It also reveals the particular structure of F that makes the asymptotic variance of Z independent of ρ, and shows that there are many distributions F with this property. The bivariate normal is only one such F.

Studentized estimation of a certain functional of a probability density function

Abstract This paper is concerned with the estimation of the functional Δ = ∫f 2(x) dx by Δ n = ∫f^ n (x) dFn (x), where f^ n is a kernel-type density estimator of f and Fn is the sample distribution of a sample of size n drawn from f. Relying on the asymptotic theory of U-statistics we derive the asymptotic normality and the Berry-Esseen bound for n 1/2(Δ n -Δ)/2σ where σ 2 = ∫f(x)(f(x)-Δ)2 dx. The main result establishes the rate of approach to normality for n 1/2(Δ n -Δ)/Zn where Zn is a natural estimator of 2σ based on the sample. This result provides the possibility to construct nonparametric approximate confidence intervals for Δ.

Invariance principles for stochastic area and related stochastic integrals

Given an antisymmetric kernel K ( K( z, z′) = − K( z′, z)) and i.i.d. random variates Z n , n⩾1, such that EK 2( Z 1, Z 2)<∞, set A n = ∑ 1⩽ i⩽ j⩽ n K( Z i , Z j ), n⩾1. If the Z n 's are two-dimensional and K is the determinant function, A n is a discrete analogue of Paul Lévy's so-called stochastic area. Using a general functional central limit theorem for stochastic integrals, we obtain limit theorems for the A n 's which mirror the corresponding results for the symmetric kernels that figure in theory of U-statistics.

On Unbiased Estimation for Randomized Response Models

Abstract For quantitative randomized response models, optimal unbiased estimation of regular functionals of distribution functions is considered. In this context, distribution theory of Hoeffding's U-statistics and von Mises' differentiable statistical functions is extended to randomized response models. Estimation of the basic distributions is also considered.

On Unbiased Estimation for Randomized Response Models

Abstract For quantitative randomized response models, optimal unbiased estimation of regular functionals of distribution functions is considered. In this context, distribution theory of Hoeffding's U-statistics and von Mises' differentiable statistical functions is extended to randomized response models. Estimation of the basic distributions is also considered.