Asymptotic expansions for two-stage rank tests

Stein's two-stage procedure produces a t-test which can realize a prescribed power against a given alternative, regardless of the unknown variance of the underlying normal distribution. This is achieved by determining the size of a second sample on the basis of a variance estimate derived from the first sample. In the paper we introduce a nonparametric competitor of this classical procedure by replacing the t-test by a rank test. For rank tests, the most precise information available are asymptotic expansions for their power to order n ^-1, where n is the sample size. Using results on combinations of rank tests for sub-samples, we obtain the same level of precision for the two-stage case. In this way we can determine the size of the additional sample to the natural order and moreover compare the nonparametric and the classical procedure in terms of expected additional numbers of observations required.     

On the k-sample Behrens-Fisher problem for high-dimensional data

For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T ² test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T ² test not powerful or not even well defined. To overcome this difficulty, we propose and study an L ²-norm based test. The asymptotic powers of the proposed L ²-norm based test and Hotelling’s T ² test are derived and theoretically compared. Methods for implementing the L ²-norm based test are described. Simulation studies are conducted to compare the L ²-norm based test and Hotelling’s T ² test when the latter can be well defined, and to compare the proposed implementation methods for the L ²-norm based test otherwise. The methodologies are motivated and illustrated by a real data example. The work was supported by the National University of Singapore Academic Research Grant (Grant No. R-155-000-085-112)     

Limiting size index distributions for ball-bin models with Zipf-type frequencies

We consider a random ball-bin model where balls are thrown randomly and sequentially into a set of bins. The frequency of choices of bins follows the Zipf-type (power-law) distribution; that is, the probability with which a ball enters the ith most popular bin is asymptotically proportional to 1/i ^α , α > 0. In this model, we derive the limiting size index distributions to which the empirical distributions of size indices converge almost surely, where the size index of degree k at time t represents the number of bins containing exactly k balls at t. While earlier studies have only treated the case where the power α of the Zipf-type distribution is greater than unity, we here consider the case of α ≤ 1 as well as α > 1. We first investigate the limiting size index distributions for the independent throw models and then extend the derived results to a case where bins are chosen dependently. Simulation experiments demonstrate not only that our analysis is valid but also that the derived limiting distributions well approximate the empirical size index distributions in a relatively short period.     

Asymptotic expansions for classical and generalized divided differences including applications

It is a well-known fact that the classical (i.e. polynomial) divided difference of orderm, when applied to a functiong, converges to themth-derivative of this function, if the evaluation points all collapse to a single one.In the first part of this paper we shall sharpen this result in the sense that we prove the existence of an asymptotic expansion with limitg ^(m) /m!. This result allows the application of extrapolation methods for the numerical differentiation of funtions.Moreover, in the second and main part of the paper we study generalized divided differences, which were introduced by Popoviciu [10] and further investigated for example by Karlin [2], Walz [15] and, mainly, Mühlbach [6–8]; we prove the existence of an asymptotic expansion also for these generalized divided differences, if the underlying function space is a Polya space. As a by-product, our results show that the generalized divided difference of orderm converges to the value of a certainmth order differential operator.     

On Some Tests of the Covariance Matrix Under General Conditions

We consider the problem of testing the hypothesis about the covariance matrix of random vectors under the assumptions that the underlying distributions are nonnormal and the sample size is moderate. The asymptotic expansions of the null distributions are obtained up to n ^−1/2. It is found that in most cases the null statistics are distributed as a mixture of independent chi-square random variables with degree of freedom one (up to n ^−1/2) and the coefficients of the mixtures are functions of the fourth cumulants of the original random variables. We also provide a general method to approximate such distributions based on a normalization transformation.     

Order Determination for Multivariate Autoregressive Processes Using Resampling Methods

LetX₁, …, X_nbe observations from a multivariate AR(p) model with unknown orderp. A resampling procedure is proposed for estimating the orderp. The classical criteria, such as AIC and BIC, estimate the orderpas the minimizer of the function[formula]wherenis the sample size,kis the order of the fitted model, Σ²_kis an estimate of the white noise covariance matrix, andC_nis a sequence of specified constants (for AIC,C_n=2m²/n, for Hannan and Quinn's modification of BIC,C_n=2m²(ln ln n)/n, wheremis the dimension of the data vector). A resampling scheme is proposed to estimate an improved penalty factorC_n. Conditional on the data, this procedure produces a consistent estimate ofp. Simulation results support the effectiveness of this procedure when compared with some of the traditional order selection criteria. Comments are also made on the use of Yule–Walker as opposed to conditional least squares estimations for order selection.     

A Bayesian Decision Theory Approach to Classification Problems

We address the classification problem where an item is declared to be from populationπ_jif certain of its characteristicsvare assumed to be sampled from the distribution with pdf f_j(vθ_j), wherej=1, 2, …, m. We first solve the two population classification problem and then extend the results to the generalmpopulation classification problem. Usually only the form of the pdf's is known. To use the classical classification rule the parameters,θ_j, must be replaced by their estimates. In this paper we allow the parameters of the underlying distributions to be generated from prior distributions. With this added structure, we obtain Bayes rules based on predictive distributions and these are completely determined. Using the first-order expansion of the predictive density, where the coefficients of powers ofn⁻¹remain uniformly bounded innwhen integrated, we obtain an asymptotic bound for the Bayes risk.     

Concentrated matrix Langevin distributions

This paper concerns the matrix Langevin distributions, exponential-type distributions defined on the two manifolds of our interest, the Stiefel manifold V_k,m and the manifold P_k,m−k of m×m orthogonal projection matrices idempotent of rank k which is equivalent to the Grassmann manifold G_k,m−k. Asymptotic theorems are derived when the concentration parameters of the distributions are large. We investigate the asymptotic behavior of distributions of some (matrix) statistics constructed based on the sample mean matrices in connection with testing hypotheses of the orientation parameters, and obtain asymptotic results in the estimation of large concentration parameters and in the classification of the matrix Langevin distributions.     

A two-stage least-squares finite element method for the stress-pressure-displacement elasticity equations

A new stress-pressure-displacement formulation for the planar elasticity equations is proposed by introducing the auxiliary variables, stresses, and pressure. The resulting first-order system involves a nonnegative parameter that measures the material compressibility for the elastic body. A two-stage least-squares finite element procedure is introduced for approximating the solution to this system with appropriate boundary conditions. It is shown that the two-stage least-squares scheme is stable and, with respect to the order of approximation for smooth exact solutions, the rates of convergence of the approximations for all the unknowns are optimal both in the H¹-norm and in the L²-norm. Numerical experiments with various values of the parameter are examined, which demonstrate the theoretical estimates. Among other things, computational results indicate that the behavior of convergence is uniform in the nonnegative parameter. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 297–315, 1998     

Inplace 2D matching in compressed images

The compressed matching problem is the problem of finding all occurrences of a pattern in a compressed text. In this paper we discuss the 2-dimensional compressed matching problem in Lempel–Ziv compressed images. Given a pattern P of (uncompressed) size m×m, and a text T of (uncompressed) size n×n, both in 2D-LZ compressed form, our algorithm finds all occurrences of P in T. The algorithm is strongly inplace, that is, the amount of extra space used is proportional to the best possible compression of a pattern of size m². The best compression that the 2D-LZ technique can obtain for a file of size m² is O(m). The time for performing the search is O(n²) and the preprocessing time is O(m³). Our algorithm is general in the sense that it can be used for any 2D compression which can be sequentially decompressed in small space.     

A Statistic for Testing the Null Hypothesis of Elliptical Symmetry

We present and study a procedure for testing the null hypothesis of multivariate elliptical symmetry. The procedure is based on the averages of some spherical harmonics over the projections of the scaled residual (1978, N. J. H. Small, Biometrika65, 657–658) of the d-dimensional data on the unit sphere of ^d. We find, under mild hypothesis, the limiting null distribution of the statistic presented, showing that, for an appropriate choice of the spherical harmonics included in the statistic, this distribution does not depend on the parameters that characterize the underlying elliptically symmetric law. We describe a bivariate simulation study that shows that the finite sample quantiles of our statistic converge fairly rapidly, with sample size, to the theoretical limiting quantiles and that our procedure enjoys good power against several alternatives.     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Continuity Properties of Distributions with Some Decomposability

Absolute continuity and smoothness of distributions in the nested subclasses ~L _m (B), m = 0, 1, 2,..., of the class of all B-decomposable distributions are studied. All invertible matrices are classified into two types in terms of P.V. numbers. The minimum integer m for which all full distributions in ~L _m (B) are absolutely continuous and the minimum integer m for which all absolutely continuous distributions in ~L _m (B) have the densities of class C ^r for 0 r are discussed according to the type of the matrix B related to P.V. numbers.     

Physics-based preconditioners for solving PDEs on highly heterogeneous media

Eigenvalues of smallest magnitude become a major bottleneck for iterative solvers especially when the underlying physical properties have severe contrasts. These contrasts are commonly found in many applications such as composite materials, geological rock properties, and thermal and electrical conductivity. The main objective of this work is to construct a method as algebraic as possible that could efficiently exploit the connectivity of highly heterogeneous media in the solution of diffusion operators. We propose an algebraic way of separating binary-like systems according to a given threshold into high- and low-conductivity regimes of coefficient size O (m) and O (1), respectively where m ≫ 1. The condition number of the linear system depends both on the mesh size and the coefficient size m. For our purposes, we address only the m dependence since the condition number of the linear system is mainly governed by the high-conductivity subblock. Thus, the proposed strategy is inspired by capturing the relevant physics governing the problem. Based on the algebraic construction, a two-stage preconditioning strategy is developed as follows: (1) a first stage that comprises approximation to the components of the solution associated to small eigenvalues and, (2) a second stage that deals with the remaining solution components with a deflation strategy (if ever needed). The deflation strategies are based on computing near invariant subspaces corresponding to smallest eigenvalues and deflating them by the use of recycled the Krylov subspaces. More detail on the proposed preconditioners can be found in [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Randomness-optimal oblivious sampling

We present the first efficient oblivious sampler that uses an optimal number of random bits, up to an arbitrary constant factor bigger than 1. Specifically, for any α>0, it uses (1+α)(m+log γ⁻¹) random bits to output d=poly(ϵ⁻¹, log γ⁻¹, m) sample points z₁,…,z_d∈{0, 1}^m such that for any function f: {0, 1}^m→[0, 1], Pr [|(1/d)∑_i=1^df(z_i)− E f|≤ϵ]≥1−γ. Our proof is based on an improved extractor construction. An extractor is a procedure which takes as input the output of a defective random source and a small number of truly random bits, and outputs a nearly random string. We present the first optimal extractor, up to constant factors, for defective random sources with constant entropy rate. We give applications to constructive leader election and reducing randomness in interactive proofs. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 , 345–367 (1997)     

Nonsingularity in the no-boundary Universe

In the no-boundary Universe of Hartle and Hawking, the path integral for the quantum state of the Universe must be summed only over nonsingular histories. If the quantum corrections to the Hamilton-lacobi equation in the interpretation of the wave packet is taken into account, then all classical trajectories should be nonsingular. The quantum behaviour of the classical singularity in theS ¹×S ^m model (m⩾2) is also clarified. It is argued that the Universe should evolve from the zero momentum state, instead from a zero volume state, to a 3-geometry state.     

On the asymptotic accuracy of the bootstrap under arbitrary resampling size

We study the order of convergence of the Kolmogorov-Smirnov distance for the bootstrap of the mean and the bootstrap of quantiles when an arbitrary bootstrap sample size is used. We see that for the bootstrap of the mean, the best order of the bootstrap sample is of the order ofn, wheren is the sample size. In the case of non-lattice distributions and the bootstrap of the sample mean; the bootstrap removes the effect of the skewness of the distribution only when the bootstrap sample equals the sample size. However, for the bootstrap of quantiles, the preferred order of the bootstrap sample isn ^2/3. For the bootstrap of quantiles, if the bootstrap sample is of ordern ² or bigger, the bootstrap is not consistent.     

Bootstrap method and empirical process

In this paper we consider the sampling properties of the bootstrap process, that is, the empirical process obtained from a random sample of size n (with replacement) of a fixed sample of size n of a continuous distribution. The cumulants of the bootstrap process are given up to the order n ^–1 and their unbiased estimation is discussed. Furthermore, it is shown that the bootstrap process has an asymptotic minimax property for some class of distributions up to the order n ^–1/2.     

A Computational Approach for Full Nonparametric Bayesian Inference Under Dirichlet Process Mixture Models

Widely used parametric generalized linear models are, unfortunately, a somewhat limited class of specifications. Nonparametric aspects are often introduced to enrich this class, resulting in semiparametric models. Focusing on single or k-sample problems, many classical nonparametric approaches are limited to hypothesis testing. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric specifications are often most appealing for smaller sample sizes. Bayesian nonparametric approaches avoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulation-based model fitting approaches which yield inference that is confined to posterior moments of linear functionals of the population distribution. This article provides a computational approach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a k-sample problem, and comparison of survival times from different populations under fairly heavy censoring.