Mixtures of Global and Local Edgeworth Expansions and Their Applications

Edgeworth expansions which are local in one coordinate and global in the rest of the coordinates are obtained for sums of independent but not identically distributed random vectors. Expansions for conditional probabilities are deduced from these. Both lattice and continuous conditioning variables are considered. The results are then applied to derive Edgeworth expansions for bootstrap distributions, for Bayesian bootstrap distribution, and for the distributions of statistics based on samples from finite populations. This results in a unified theory of Edgeworth expansions for resampling procedures. The Bayesian bootstrap is shown to be second order correct for smooth positive “priors,” whenever the third cumulant of the “prior” is equal to the third power of its standard deviation. Similar results are established for weighted bootstrap when the weights are constructed from random variables with a lattice distribution.     

Singh's theorem in the lattice case

The asymptotic behavior of the parametric bootstrap estimator of the sampling distribution of a maximum likelihood estimator is investigated in a simple lattice case, integer valued random variables whose distributions form an exponential family. The expected value of the bootstrap estimator is compared with an Edgeworth expansion, less the continuity correction.     

EDGEWORTH EXPANSION AND BOOTSTRAP APPROXIMATION FOR THE STUDENTIZED MLE FROM RANDOMLY CENSORED EXPONENTIAL SAMPLES

In this paper,the author studies the asymptotic accuracies of the one-term Edgeworth expansions and the bootstrap approximation for the studentized MLE from randomly censored exponential population.It is shown that the Edgeworth expansions and the bootstrap approximation are asymptotically close to the exact distribution of the studentized MLE with a rate.     

BOOTSTRAP CRITICAL POINT FOR CIRCULAR MEAN DIRECTION AND ITS APPLICATIONS

ＢＯＯＴＳＴＲＡＰＣＲＩＴＩＣＡＬＰＯＩＮＴＦＯＲＣＩＲＣＵＬＡＲＭＥＡＮＤＩＲＥＣＴＩＯＮＡＮＤＩＴＳＡＰＰＬＩＣＡＴＩＯＮＳＷＵＣＨＡＯＢＩＡＯＡＮＤＤＥＮＧＷＥＩＣＡＩＡｂｓｔｒａｃｔ．Ｆｉｒｓｔｌｙ，ｔｈｉｓｐａｐｅｒｒｅｖｉｅｗｓＨａｌ’...     

On the Edgeworth expansion and the <Emphasis Type="Italic">M</Emphasis> out of <Emphasis Type="Italic">N</Emphasis> bootstrap accuracy for a Studentized trimmed mean

We investigate the second order accuracy of the M out of N bootstrap for a Studentized trimmed mean using the Edgeworth expansion derived in a previous paper. Some simulations, which support our theoretical results, are also given. The effect of extrapolation in conjunction with the M out of N bootstrap for Studentized trimmed means is briefly discussed. As an auxiliary result we obtain a Bahadur’s type representation for an M out of N bootstrap quantile. Our results supplement previous work on (Studentized) trimmed means by Hall and Padmanabhan [13], Bickel and Sakov [7], and Gribkova and Helmers [11].      

Edgeworth expansion for the survival function estimator in the Koziol-Green model

In the Koziol-Green or proportional hazards random censorship model, the asymptotic accuracy of the estimated one-term Edgeworth expansion and the smoothed bootstrap approximation for the Studentized Abdushukurov-Cheng-Lin estimator is investigated. It is shown that both the Edgeworth expansion estimate and the bootstrap approximation are asymptotically closer to the exact distribution of the Studentized Abdushukurov-Cheng-Lin estimator than the normal approximation.     

On a mapping approach to investigating the bootstrap accuracy

Summary. A simple mapping approach is proposed to study the bootstrap accuracy in a rather general setting. It is demonstrated that the bootstrap accuracy can be obtained through this method for a broad class of statistics to which the commonly used Edgeworth expansion approach may not be successfully applied. We then consider some examples to illustrate how this approach may be used to find the bootstrap accuracy and show the advantage of the bootstrap approximation over the Gaussian approximation. For the multivariate Kolmogorov–Smirnov statistic, we show the error of bootstrap approximation is as small as that of the Gaussian approximation. For the multivariate kernel type density estimate, we obtain an order of the bootstrap error which is smaller than the order of the error of the Gaussian approximation given in Rio (1994). We also consider an application of the bootstrap accuracy for empirical process to that for the copula process. Received: 23 June 1995 / In revised form: 18 June 1996     

Corrections to the Central Limit Theorem for Heavy-tailed Probability Densities

Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT.     

On Edgeworth expansions for dependency-neighborhoods chain structures and Stein's method

Let W be the sum of dependent random variables, and h(x) be a function. This paper provides an Edgeworth expansion of an arbitrary ``length' for %E{h(W)} in terms of certain characteristics of dependency, and of the smoothness of h and/or the distribution of W. The core of the class of dependency structures for which these characteristics are meaningful is the local dependency, but in fact, the class is essentially wider. The remainder is estimated in terms of Lyapunov's ratios. The proof is based on a Stein's method.Supported in part by NSF grant DMS-98-03623Supported in part by the Russian Foundation of Basic Research, grant # 00-01-00194, and by NSF grant DMS-98-03623Mathematics Subject Classification (2000):Primary 62E20; Secondary 60E05     

A Method for the Computer Calculation of Edgeworth Expansions for Smooth Function Models

Abstract

In this article I describe, in detail, a method for the computer calculation of Edgeworth expansions for a smooth function model accurate in the O(n ^–1) term. For such models, these expansions are an important tool for the analysis of normalizing transformations, the correction of an approximately normally distributed quantity for skewness, and the comparison of bootstrap inference procedures. The method is straightforward and is efficient in a sense described in the article. The implementation of the method in general is clear from its implementation in the Mathematica program (available through StatLib) for the particular case of the studentized mean.     

An edgeworth expansion for finite-population L-statistics

We consider the one-term Edgeworth expansion for finite-population L-statistics. We provide an explicit formula for the Edgeworth correction term and give sufficient conditions for the validity of the expansion that are expressed in terms of the weight function defining the statistics and moment conditions.     

Asymptotic theory for estimators under random censorship

Summary The product limit estimator of an unknown distributionF is represented as aU-statistic plus an error of the ordero(1/n). Using this, the maximum likelihood estimator of the specific risk rate in the time interval [0,M], is shown to admit a two term Edgeworth expansion. This risk rate for a specific cause of death is defined as the ratio of the probability of death, due to that particular cause, in the time interval [0,M], to the mean life time of an individual up to that time pointM. Similar expansions for the bootstrapped statistics are used to show that the bootstrap distribution, of the studentized estimator of the risk rate, approximates the sampling distribution better than the corresponding normal distribution.Research supported in part by NSA Grant MDA 904-90-H-1001 and by NSF Grant DMS-9007717     

Comparing Empirical Likelihood and Bootstrap Hypothesis Tests

A comparison between empirical likelihood and bootstrap tests for a mean parameter against a series of local alternative hypotheses is made by developing Edgeworth expansions for the power functions of the two tests. For univariate and bivariate cases, practical rules are proposed for choosing the more powerful test.     

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.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

Correction and addendum to “embedding problems with local conditions”

A correction is provided to a result of the author’s concerning embedding problems with p-group kernel. It is also shown that every such embedding problem for a Laurent series field in characteristic p has a proper solution. Supported in part by NSF Grant DMS-0500118. 2000 Math. Subj. Classif.: 14G32, 12G10, 14H30, 12E30.     

Bias and variance reduction techniques for bootstrap information criteria

We discuss the problem of constructing information criteria by applying the bootstrap methods. Various bias and variance reduction methods are presented for improving the bootstrap bias correction term in computing the bootstrap information criterion. The properties of these methods are investigated both in theoretical and numerical aspects, for which we use a statistical functional approach. It is shown that the bootstrap method automatically achieves the second-order bias correction if the bias of the first-order bias correction term is properly removed. We also show that the variance associated with bootstrapping can be considerably reduced for various model estimation procedures without any analytical argument. Monte Carlo experiments are conducted to investigate the performance of the bootstrap bias and variance reduction techniques.     

Q-Learning Algorithms with Random Truncation Bounds and Applications to Effective Parallel Computing

Motivated by an important problem of load balancing in parallel computing, this paper examines a modified algorithm to enhance Q-learning methods, especially in asynchronous recursive procedures for self-adaptive load distribution at run-time. Unlike the existing projection method that utilizes a fixed region, our algorithm employs a sequence of growing truncation bounds to ensure the boundedness of the iterates. Convergence and rates of convergence of the proposed algorithm are established. This class of algorithms has broad applications in signal processing, learning, financial engineering, and other related fields. G. Yin’s research was supported in part by the National Science Foundation under Grants DMS-0603287 and DMS-0624849 and in part by the National Security Agency under Grant MSPF-068-029. C.Z. Xu’s research was supported in part by the National Science Foundation under Grants CCF-0611750, DMS-0624849, CNS-0702488, and CRI-0708232. L.Y. Wang’s research was supported in part by the National Science Foundation under Grants ECS-0329597 and DMS-0624849 and by the Michigan Economic Development Council.     

Separating Cycles in Doubly Toroidal Embeddings

 We show that every 4-representative graph embedding in the double torus contains a noncontractible cycle that separates the surface into two pieces. As a special case, every triangulation of the double torus in which every noncontractible cycle has length at least 4 has a noncontractible cycle that separates the surface into two pieces. Received: May 22, 2001 Final version received: August 22, 2002 RID="*" ID="*" Supported by NSF Grants Numbers DMS-9622780 and DMS-0070613 RID="†" ID="†" Supported by NSF Grants Numbers DMS-9622780 and DMS-0070430     

Edgeworth expansion in partial linear models

In this paper, under some fairly general conditions, a first-order Edgeworth expansion for the standardized statistic of β in partial linear models is given, then a non-residual type of consistent estimation for the error variance is constructed, and finally an Edgeworth expansion for the corresponding studentized version is presented.