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...     

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.     

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.     

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.     

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.     

Asymptotic behavior of a linear difference equation with continuous time

We consider a class of linear homogeneous difference equations with constant coefficients and commensurate delays. We prove an asymptotic formula for the solutions as t → ∞ under the assumption of the existence of a simple “dominant” real characteristic value. Research supported by FONDECYT Grant, Chile, No. 1.070.980.     

The strong law of large numbers for L-statistics with dependent data

We prove the strong law of large numbers for the linear combinations of functions of order statistics (L-statistics) based on weakly dependent random variables. We also establish the Glivenko-Cantelli theorem for ?-mixing sequences of identically distributed random variables.     

On bounded maximum width sequential confidence ellipsoids based on generalized U-statistics

For a vector of (estimable) functionals of several independent distributions, sequential confidence ellipsoids (of bounded maximum width) based on a class of generalized U-statistics are studied. A stopping rule along with a procedure for choosing the component sample sizes at each stage is developed, so that the proposed confidence ellipsoid has a confidence coefficient asymptotically (as the prescribed maximum width shrinks to zero) equal to a preassigned 1 - α (0 < α < 1), and the expected total sample size is minimized for the procedure. Asymptotic efficiency of the procedure is also studied. The case of von Mises' functionals is treated briefly at the end.     

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.     

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.     

Adaptive choice of trimming proportions

We consider Jaeckel's (1971,Ann. Math. Statist.,42, 1540–1552) proposal for choosing the trimming proportion of the trimmed mean in the more general context of choosing a trimming proportion for a trimmedL-estimator of location. We obtain higher order expansions which enable us to evaluate the effect of the estimated trimming proportion on the adaptive estimator. We find thatL-estimators with smooth weight functions are to be preferred to those with discontinuous weight functions (such as the trimmed mean) because the effect of the estimated trimming proportion on the estimator is of ordern ^–1 rather thann ^–3/4. In particular, we find that valid inferences can be based on a particular smooth trimmed mean with its asymptotic standard error and the Studentt distribution with degrees of freedom given by the Tukey and McLaughlin (1963,Sankhy Ser. A,25, 331–352) proposal.     

Asymptotic bounds for the expected L error of a multivariate kernel density estimator

The kernel estimator of a multivariate probability density function is studied. An asymptotic upper bound for the expected L¹ error of the estimator is derived. An asymptotic lower bound result and a formula for the exact asymptotic error are also given. The goodness of the smoothing parameter value derived by minimizing an explicit upper bound is examined in numerical simulations that consist of two different experiments. First, the L¹ error is estimated using numerical integration and, second, the effect of the choice of the smoothing parameter in discrimination tasks is studied.     

Distributions of class Lα

L_α (0 α 1) is a class of infinitely divisible distributions defined by restricting the measure in the Levy-Khinchin formula to a special form. When α = 1, L_α is just the classical class L. Several properties for L_α classes, which are similar to the most important properties for the class L, are established. Also, a conjecture of Wolfe about unimodality of some L_α distributions is disproved by giving a counterexample.     

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.     

Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients

We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in . If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t^(n−θ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order <m.     

Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time

We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM ^[x]/G/1 queue with batch arrivals.     

On sharp Burkholder–Rosenthal-type inequalities for infinite-degree U-statistics

In this paper, we present a method that allows one to obtain a number of sharp inequalities for expectations of functions of infinite-degree U-statistics. Using the approach, we prove, in particular, the following result: Let D be the class of functions f :R₊→R₊ such that the function f(x+z)−f(x) is concave in xR₊ for all zR₊. Then the following estimate holds: for all fD and all U-statistics ∑_1i1<<ilnY_i1,…,il(X_i1,…,X_il) with nonnegative kernels Y_i1,…,il :R^l→R₊, 1i_kn; i_r≠i_s, r≠s; k,r,s=1,…,l; l=0,…,m, in independent r.v.'s X₁,…,X_n. Similar inequality holds for sums of decoupled U-statistics. The class D is quite wide and includes all nonnegative twice differentiable functions f such that the function f″(x) is nonincreasing in x>0, and, in particular, the power functions f(x)=x^t, 1<t2; the power functions multiplied by logarithm f(x)= (x+x₀)^t ln(x+x₀), 1<t<2, x₀max(e^{(3t2−6t+2)/(t(t−1)(2−t))},1); and the entropy-type functions f(x)=(x+x₀)ln(x+x₀), x₀1. As an application of the results, we determine the best constants in Burkholder–Rosenthal-type inequalities for sums of U-statistics and prove new decoupling inequalities for those objects. The results obtained in the paper are, to our knowledge, the first known results on the best constants in sharp moment estimates for U-statistics of a general type.     

On decoupling,series expansions,and tail behavior of chaos processes

Some important aspects of chaos random variables such as decoupling, an almost sure representation (a Karhunen-Loeve expansion) and integrability are discussed here, the first being a tool for, and the third as a consequence of, the second. The main goal in this note is to learn about the structure of the limit laws ofU-processes.Research partially supported by National Science Foundation Grant No. DMS-9000132 and University of Connecticut Grant No. G12-913501.     

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.     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.