Central limit theorem and the bootstrap for -statistics of strongly mixing data

The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley [R.C. Bradley, Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983),69–81] to prove a new generalized covariance inequality similar to Yoshihara’s [K. Yoshihara, Limiting behavior of U-statistics for stationary, absolutely regular processes, Z. Wahrsch. Verw. Gebiete 35 (1976), 237–252]. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see [P.J. Bickel, D.A. Freedman, Some asymptotic theory for the bootstrap, Ann. Statist. 9 (1981), 1196–1217]. For mixing data, Politis and Romano [D.N. Politis, J.P. Romano, A circular block resampling procedure for stationary data, in: R. Lepage, L. Billard (Eds.), Exploring the Limits of Bootstrap, Wiley, New York, 1992, pp. 263–270] proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean’s distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results.     

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.     

Consistency of general bootstrap methods for degenerate -type and -type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of U-type and V-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.     

Universally consistent conditionalU-statistics for absolutely regular processes and its applications for hidden Markov models

A general class of conditionalU-statistics was introduced by W. Stute as a generalization of the Nadaraya-Watson estimates of a regression function. It was shown that such statistics are universally consistent. Also, universal consistentcies of the window andk _n -nearest neighbor estimators (as two special cases of the conditionalU-statistics) were proved. In this paper, we extend these results from the independent case to dependent case. The result is applied to verify the Bayes risk consistency of the corresponding discrimination rules. Research supported by the Office of Naval Research Contract N00014-91-J-1020.     

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.     

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.     

Resampling student'st-type statistics

The present paper establishes conditional and unconditional central limit theorems for various resampling procedures for thet-statistic. The results work under fairly general conditions and the underlying random variables need not to be independent. Specific examples are then them(n) (double) bootstrap out ofk(n) observations, the Bayesian bootstrap and two-samplet-type permutation statistics. In case whenm(n)/k(n) is bounded away from zero and infinity necessary and sufficient conditions for the conditional central limit law of the bootstrapt-statistics are established. For high resampling intensity whenm(n)/k(n) tends to infinity the following general result is obtained. Without further other assumptions the bootstrap makes the resampledt-statistic automatically normal. The results are based on a general conditional limit theorem for weighted resampling statistics which is of own interest.     

Almost sure central limit theorems for functionals of absolutely regular processes with application to U-statistics

We prove an almost sure central limit theorem for functionals of absolutely regular processes and extend this result to U-statistics.     

Bootstrapping generalizedU-processes andV-processes and their applications in projection pursuit

In this paper, we study bootstrap approximation for generalizedU-processes (GUP) indexed by a class of functions. Under mild conditions we obtain that the asymptotic distributions of bootstrapping generalizedU-processes (BGUP) are the same as those of GUP almost surely. As a result, the asymptotic properties of bootstrap approximation for PP generalizedU-processes (BPPGUP) are obtained. In addition we have derived bootstrap approximation for generalizedV-processes (GVP). Thus, we can use BGUP or bootstrapping GVP (BGVP) to simulate GUP and GVP.This project is supported by the National Natural Science Foundation of China and the Science Foundation of Educational Committee of Guizhou.     

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.     

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.     

Approximations for multivariate U-statistics

Edgeworth approximations for multivariate U-statistics hold up to the order o(n^−1/2) under moment conditions and the assumption that the projection of the U-statistic to sums of i.i.d. random vectors is strongly nonlattice.     

A class of asymptotically normal degenerate quasi <Emphasis Type="Italic">U</Emphasis>-statistics

Some quasi U-statistics, unlike other variants of U-statistics, arising in distance based tests for homogeneity of groups, have first-order stationary kernels of degree 2, and yet they enjoy asymptotic normality under suitable hypotheses of invariance. Central limit theorems for a more general class of quasi U-statistics with possibly higher order stationarity (and degree) are formulated with the aid of appropriate martingale (array) characterizations as well as permutational invariance structures.     

A note on the almost sure limit theorem for U-statistic

Summary In this note we prove an almost sure limit theorem for the products of U-statistics.     

Weak and strong representations for trimmed U-statistics

Summary Weak and strong representations are proved for two classes of trimmed U-statistics, generalizing the trimmed mean. Applications of the strong representation theorems include laws of iterated logarithm and invariance principles for trimmed U-statistics.     

A Functional Law of the Iterated Logarithm for U-Statistic Type Processes

We prove a functional law of the iterated logarithm for U-statistics type processes. The result is used to determine the almost sure set of limit points for change-point estimators.     

The law of the iterated logarithm and Marcinkiewicz law of large numbers forB-valuedU-statistics

Suppose thatB is a separable Banach space and (S,l,P) a probability space.H is a measurable symmetric kernel function fromS ^m intoB. In this paper we shall further study some limit theorems forB-valuedU-statisticsU _m ⁿ H based onP andH. Special attention is paid upon the Marcinkiewicz type law of large numbers and the law of the iterated logarithm. Our results can be regarded as extensions of corresponding results for sums of independentB-valued random variables toU-statistics.Research supported by National Natural Science Foundation of China and Zhejiang Province.     

Weak invariance principle for finite-populationU-statistics

Under the weakest possible conditions, we establish the weak invariance principle for finite-populationU-statistics in this paper. It is worth while to point out that, for the sampling without replacement, the sequence of random delements inC[0, 1], associated with the sample partial sums or theU-statistics, converges in law to the standard Brown bridge, but not to the Brown motion as in the usual case of replacement sampling.     

A Glivenko-Cantelli Theorem and Strong Laws for L-Statistics

We prove a weighted Glivenko-Cantelli theorem and apply it to study the rate of convergence in the strong law for L-statistics.     

Asymptotically optimal sequential estimation of regular functionals of several distributions based on generalized U-statistics

For the several sample problem, a vector of estimable parameters is considered. For a fixed total sample size, a multistage (sequential) procedure based on generalized U-statistics is developed for choosing a partition of this sample size into individual sample size for which the generalized variance of the estimator of the parameter vector is asymptotically minimized.