Weighted bootstrap for U-statistics

In this paper we investigate the weighted bootstrap for U-statistics and its properties. Under very general choices of random weights and certain regularity conditions, we show that the weighted bootstrap method with U-statistics provides second-order accurate approximations to the distribution of U-statistics. We shall prove this via one-term Edgeworth expansions of weighted U-statistics.     

Asymptotic distributions of non-central studentized statistics

Let X1,...,Xn be independent and identically distributed random variables and Wn = Wn(X1,...,Xn) be an estimator of parameter θ.Denote Tn =(Wn - θ0)/sn,where sn2 is a variance estimator of Wn.In this paper a general result on the limiting distributions of the non-central studen-tized statistic Tn is given.Especially,when s2n is the jacknife estimate of variance,it is shown that the limit could be normal,a weighted χ2 distribution,a stable distribution,or a mixture of normal and stable distribution.Applicati...     

Limit theorems for the canonical von Mises statistics with dependent data

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.     

Generalized Hodges-Lehmann estimators for the analysis of variance

For X_i, …, X_n a random sample and K(·, ·) a symmetric kernel this paper considers large sample properties of location estimator satisfying , . Asymptotic normality of is obtained and two forms of interval estimators for parameter θ satisfying EK(X₁ − θ, X₂ − θ) = 0, are discussed. Consistent estimation of the variance parameters is obtained which permits the construction of asymptotically distribution free procedures. The p-variate and multigroup extension is accomplished to provide generalized one-way MANOVA. Monte Carlo results are included.     

Weak invariance principles for weightedU-statistics

We prove that the distribution of a properly normalized weightedU-statisticU _n in i.i.d. random variables is close to the distribution of a certain functionV _n in i.i.d. standardized Gaussian random variables in the sense that their Lévy-Prokhorov distance tends to zero asn. This property is then used to determine the limit laws ofU _n under special assumptions on the kernel function. This generalizes a method due to Rotar' who proved similar results for random multilinear forms.     

Sampling Properties of U-statistics for a Class of Stationary Nonlinear Processes

We consider the sampling properties of U-statistics based on a sample of realization from a class of stationary nonlinear processes which include, in particular, linear, bilinear and finite order volterra processes. It is shown that if the size n of the realization tends to infinity then certain normalized versions of the U-statistics tend to be distributed normally with zero means and finite variances.     

Bounds for Non-Gaussian Approximations of U-Statistics

Let X, ,X ₁,...,X _n be i.i.d. random variables taking values in a measurable space ( ). Consider U-statistics of degree two

Onm-dependence and Edgeworth expansions

This paper contains two results. The first establishes, under mild assumptions, the validity of an Edgeworth expansion with remaindero(N ^–1/2) for aU-statistic with a kernel of degree two using observations from anm-dependent shift. The second result gives a necessary and sufficient condition for the distribution of a sum ofm-dependent random variables to possess an Edgeworth expansion. This generalizes a result of Bickel and Robinson from the i.i.d. case to them-dependent case.This research was supported in part by National Science Foundation, Grant DMS 89-23071.     

Limits of canonicalU-processes andB-valuedU-statistics

It is shown that the limite law of canonicalU-process is the law of a chaos process which has a versio with bounded and ·₂ continuous paths. This is also true forB-valued canonicalU-statistics with values in a separable Banach space. Some properties of Banach spaces of type 2 related withU-statistics are presented.Research partially supported by NSF Grants No. DMS-9000132 and No. DMS-8505550 and carried out in the University of Connecticut and the MSRI.     

A Note on Edgeworth Expansions for <Emphasis Type="Italic">U</Emphasis>-Statistics under Minimal Conditions

A one-term Edgeworth expansion for U-statistics with kernel h(x, y) was derived by Jing and Wang [3] under optimal moment conditions. In this note, we show that one of the optimal moment conditions E| h(X ₁, X ₂|^5/3 < ∞ can be weakened to lim _t→∞ t ^5/3 P(|h(X ₁, X ₂)| > t) → 0. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 453–440, July–September, 2005.     

On the Asymptotic Behavior of Weighted U-Statistics

A sufficient condition for the convergence of degenerated weighted U-statistics of order k is proved. From this, the previous results by O'Neil and Redner⁽¹⁰⁾ for k = 2 and Major,⁽⁷⁾ which appeared to be very different, are related.     

On the Marcinkiewicz–Zygmund Strong Law for U-Statistics

Sen (when 0<p<1) and Giné and Zin (1<p<2) have given sufficient conditions for a Marcinkiewicz–Zygmund type SLLN to hold for U-statistics. As demonstrated by the latter authors, their requirement is not necessary when U is completely degenerate or 0<p<1. Here, for p in (1, 2), a simplified proof of sufficiency is obtained which weakens their integrability condition and in the nondegenerate case also extends the permissible values of p. This, in turn, leads to improved conditions for Hoeffding's CLT (likewise for the LIL) in the non-degenerate case. Moreover, nonnecessity is shown to hold in the SLLN when U is degenerate but not completely so.     

On integrability in the LIL for degenerateU-statistics

It is shown by example that square integrability of the kennelh of a completely degenerateU-statistic is not a necessary condition for the law of the iterated logarithm to hold. Necessary integrability conditions are given onh which are in a sense the strongest possible. Sufficient conditions for a degenerateU-statistic of order 2 to satisfy the LIL, weaker than square integrability of the kernel, are also given.Research partially supported by NSF Grant No. DMS-93-00725. Part of this author's work was done at the University of Bielefeld, supported by the Deutsche Forschungsgemeinshaft, Sonderforschungbereich 343, Diskrete Structuren in der Mathematik.Research partially supported by the Army Research Office and the National Security Agency.     

Convergence of self-normalized generalizedU-statistics

Conditions are given for almost certain and distribution convergence of self-normalized generalizedU-statistics composed of random variables without particular probabilistic structure. The set of almost certain limit points of some classicalU-statistics is obtained. A variant of theU-statistic involving squares of some of the random variables is also treated. Applications include Martingale differences, stationary sequences, and the classical i.i.d. case where a Marcinkiewicz-Zygmund-type strong law is obtained.     

The LIL for <Emphasis Type="Italic">U</Emphasis>-Statistics in Hilbert Spaces

We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest. R. Adamczak’s research partially supported by MEiN Grant 2 PO3A 019 30. R. Latała’s research partially supported by MEiN Grant 1 PO3A 012 29.     

A Note on the Use of V and U Statistics in Nonparametric Models of Regression

We establish the asymptotic equivalence of V and U statistics when the statistic’s kernel depends on n. Combined with a lemma of B. Lee this result provides conditions under which U statistics projections and V statistics are asymptotically equivalent. The use of this equivalence in nonparametric regression models is illustrated with several examples; the estimation of conditional variances, skewness, kurtosis and the construction of a nonparametric R-squared measure.     

二维指数信号模型中参数Bootstrap估计的强相合性

本文主要研究了在统计信号处理当中具有广泛应用的二维带白噪声指数信号模型中参数估计的Bootstrap逼近, 借助于回归模型中Bootstrap逼近的构造方法, 给出了二维指数信号模型参数的自助估计, 并证明了自助估计具有强相合性. 最后采用了Monte-Carlo法对所提的方法进行随机模拟, 模拟的结果表明当噪声不服从正态分布时, Bootstrap方法的估计效果优于最小二乘估计.     

Nuclearity of a nonnegative definite integral kernel on a separable metric space

The aim of this paper is to characterize the nuclearity of an integral operator, defined by a continuous non-negative definite square integrable kernel on a separable metric space, in terms of the integrability of the trace of the kernel function. Nuclearity here plays a role forU-statistics.     

How to get Central Limit Theorems for global errors of estimates

The asymptotic behavior of global errors of functional estimates plays a key role in hypothesis testing and confidence interval building. Whereas for pointwise errors asymptotic normality often easily follows from standard Central Limit Theorems, global errors asymptotics involve some additional techniques such as strong approximation, martingale theory and Poissonization. We review these techniques in the framework of density estimation from independent identically distributed random variables, i.e., the context for which they were introduced. This will avoid the mathematical difficulties associated with more complex statistical situations in which these tools have proved to be useful.     

Asymptotics of Studentized <Emphasis Type="Italic">U</Emphasis>-type processes for changepoint problems

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E|h(X ₁, X ₂)|² < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.