生长曲线模型中的球性检验问题

本文给出了生长曲线模型中球性检验似然比准则在原假设相接近的两类备择假设下的非零渐近分布。     

Static Vacuum Solutions from Convergent Null Data Expansions at Space-Like Infinity

We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as ‘null data’. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1 : 1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations. Submitted: October 26, 2006. Accepted: October 29, 2006.     

Edgeworth expansion of Wilks’ lambda statistic

An asymptotic expansion of the null distribution of the Wilks’ lambda statistic is derived when some of the parameters are large. Cornish-Fisher expansions of the upper percent points are also obtained. A monotone transformation which reduces the third and the fourth order cumulants is also derived. In order to study the accuracy of the approximation formulas, some numerical experiments are done, with comparing to the classical expansions when only the sample size tends to infinity.     

The new test criterion for the homogeneity of parameters of several populations

Summary The new test criterion for testing the homogeneity of parameters of several populations is proposed and the test properties of it is discussed. The asymptotic expansions of the distributions of test criterion are discussed under (i) null hypothesis, (ii) fixed alternative hypothesis and (iii) local alternative hypothesis converging to the null hypothesis with appropriate rate of convergence as the sample size increases. As a particular case the asymptotic theory of a statistic for a homogeneity of variances of normal populations is also discussed and the exact moments of it under a null hypothesis can be used to obtain a percentage point by a Pearsonian curve fitting. This Institute of statistical Mathematics     

On some distribution problems in Manova and discriminant analysis

Asymptotic expansions, valid for large error degrees of freedom, are given for the multivariate noncentral F distribution and for the distribution of latent roots in MANOVA and discriminant analysis. The asymptotic results are expressed in terms of elementary functions which are easy to compute and the results of some numerical work are included. The Bartlett test of the null hypothesis that some of the noncentrality parameters in discriminant analysis are zero is also briefly discussed.     

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.     

First order spectral perturbation theory of square singular matrix polynomials

We develop first order eigenvalue expansions of one-parametric perturbations of square singular matrix polynomials. Although the eigenvalues of a singular matrix polynomial P(λ) are not continuous functions of the entries of the coefficients of the polynomial, we show that for most perturbations they are indeed continuous. Given an eigenvalue λ₀ of P(λ) we prove that, for generic perturbations M(λ) of degree at most the degree of P(λ), the eigenvalues of P(λ)+?M(λ) admit covergent series expansions near λ₀ and we describe the first order term of these expansions in terms of M(λ₀) and certain particular bases of the left and right null spaces of P(λ₀). In the important case of λ₀ being a semisimple eigenvalue of P(λ) any bases of the left and right null spaces of P(λ₀) can be used, and the first order term of the eigenvalue expansions takes a simple form. In this situation we also obtain the limit vector of the associated eigenvector expansions.     

Asymptotic expansion of the null distribution of test statistic for linear hypothesis in nonnormal linear model

This paper is concerned with the null distribution of test statistic T for testing a linear hypothesis in a linear model without assuming normal errors. The test statistic includes typical ANOVA test statistics. It is known that the null distribution of T converges to χ² when the sample size n is large under an adequate condition of the design matrix. We extend this result by obtaining an asymptotic expansion under general condition. Next, asymptotic expansions of one- and two-way test statistics are obtained by using this general one. Numerical accuracies are studied for some approximations of percent points and actual test sizes of T for two-way ANOVA test case based on the limiting distribution and an asymptotic expansion.     

On the Singularity of Random Bernoulli Matrices— Novel Integer Partitions and Lower Bound Expansions

We prove a lower bound expansion on the probability that a random ±1 matrix is singular, and conjecture that such expansions govern the actual probability of singularity. These expansions are based on naming the most likely, second most likely, and so on, ways that a Bernoulli matrix can be singular; the most likely way is to have a null vector of the form e _i ±e _j , which corresponds to the integer partition 11, with two parts of size 1. The second most likely way is to have a null vector of the form e _i ±e _j ±e _k ±e _? , which corresponds to the partition 1111. The fifth most likely way corresponds to the partition 21111. We define and characterize the “novel partitions” which show up in this series. As a family, novel partitions suffice to detect singularity, i.e., any singular Bernoulli matrix has a left null vector whose underlying integer partition is novel. And, with respect to this property, the family of novel partitions is minimal. We prove that the only novel partitions with six or fewer parts are 11, 1111, 21111, 111111, 221111, 311111, and 322111. We prove that there are fourteen novel partitions having seven parts. We formulate a conjecture about which partitions are “first place and runners up” in relation to the Erd?s-Littlewood-Offord bound. We prove some bounds on the interaction between left and right null vectors.     

Average projection type weighted Cramér-von Mises statistics for testing some distributions

This paper addresses the problem of testing goodness-of-fit for several important multivariate distributions: (Ⅰ) Uniform distribution on p-dimensional unit sphere; (Ⅱ) multivariate standard normal distribution; and (Ⅲ) multivariate normal distribution with unknown mean vector and covariance matrix. The average projection type weighted Cramér-yon Mises test statistic as well as estimated and weighted Cramér-von Mises statistics for testing distributions (Ⅰ), (Ⅱ) and (Ⅲ) are constructed via integrating projection direction on the unit sphere, and the asymptotic distributions and the expansions of those test statistics under the null hypothesis are also obtained. Furthermore, the approach of this paper can be applied to testing goodness-of-fit for elliptically contoured distributions.     

Smoothed nonparametric tests and approximations of <Emphasis Type="Italic">p</Emphasis>-values

We propose new smoothed sign and Wilcoxon’s signed rank tests that are based on kernel estimators of the underlying distribution function of the data. We discuss the approximations of the p-values and asymptotic properties of these tests. The new smoothed tests are equivalent to the ordinary sign and Wilcoxon’s tests in the sense of Pitman’s asymptotic relative efficiency, and the differences between the ordinary and new tests converge to zero in probability. Under the null hypothesis, the main terms of the asymptotic expectations and variances of the tests do not depend on the underlying distribution. Although the smoothed tests are not distribution-free, making use of the specific kernel enables us to obtain the Edgeworth expansions, being free of the underlying distribution.     

Average projection type weighted Cramér- von Mises statistics for testing some distributions

This paper addresses the problem of testing goodness-of-fit for several important multivariate distributions: (I) Uniform distribution on p-dimensional unit sphere; (II) multivariate standard normal distribution; and (III) multivariate normal distribution with unknown mean vector and covariance matrix. The average projection type weighted Cramér-von Mises test statistic as well as estimated and weighted Cramér-von Mises statistics for testing distributions (I), (II) and (III) are constructed via integrating projection direction on the unit sphere, and the asymptotic distributions and the expansions of those test statistics under the null hypothesis are also obtained. Furthermore, the approach of this paper can be applied to testing goodness-of-fit for elliptically contoured distributions.     

Multi-point Taylor expansions of analytic functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.

     

Two-Point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in two points.     

Arithmetic and metric properties of Oppenheim continued fraction expansions

We introduce a class of continued fraction expansions called Oppenheim continued fraction (OCF) expansions. Basic properties of these expansions are discussed and metric properties of the digits occurring in the OCF expansions are studied.     

Bootstrap inference for misspecified moment condition models

We study the standard-bootstrap, the centered-bootstrap, and the empirical-likelihood bootstrap tests of hypotheses used in conjunction with generalized method of moments inference in correctly specified and misspecified moment condition models. We show that, under correct specification, the standard-bootstrap estimator of the null distribution of the J-test converges in distribution to a random distribution, verifying its inconsistency, while the centered and the empirical-likelihood bootstrap estimators are consistent. We provide higher-order expansions of the size distortions of the analytic and the bootstrap tests. We show that the standard-bootstrap parameter-tests are consistent under misspecification, while the centered-bootstrap parameter-tests are inconsistent. We propose a general bootstrap methodology which is highly accurate under correct specification and consistent under misspecification. In a simulation study, we explore the finite sample behavior of the analytic and the bootstrap tests for a panel data model and we apply our methodology on a real-world data set.     

The geometric structure of the expected/observed likelihood expansions

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.This research was partially supported by the Italian National Research Council grant n.93.00824.CT10.     

Uniform Asymptotic Expansions of Symmetric Elliptic Integrals

Symmetric standard elliptic integrals are considered when two or more parameters are larger than the others. The distributional approach is used to derive seven expansions of these integrals in inverse powers of the asymptotic parameters. Some of these expansions also involve logarithmic terms in the asymptotic variables. These expansions are uniformly convergent when the asymptotic parameters are greater than the remaining ones. The coefficients of six of these expansions involve hypergeometric functions with less parameters than the original integrals. The coefficients of the seventh expansion again involve elliptic integrals, but with less parameters than the original integrals. The convergence speed of any of these expansions increases for an increasing difference between the asymptotic variables and the remaining ones. All the expansions are accompanied by an error bound at any order of the approximation. January 31, 2000. Date revised: May 18, 2000. Date accepted: August 4, 2000.     

Series expansions of analytic functions

Many of the classical polynomial expansions of analytic functions share a common property: the space of “expandable” functions is a Banach space isometrically isomorphic to the space of complex sequences with limit 0. Under the isometries, these polynomial expansions all correspond to essentially the same biorthogonal expansion in this sequence space. Sufficient conditions for such an isometry to exist are obtained, and convergence properties of the expansions are studied. The results obtained also apply to expansions other than polynomial expansions.