The likelihood ratio criterion and the asymptotic expansion of its distribution

Summary Asymptotic expansion of the distribution of the likelihood ratio criterion (LRC) for testing a composite hypothesis is derived under null hypothesis and a correction factor ρ which makes the term of order 1/n in the asymptotic expansion of the distribution of it vanish is obtained. The problem is extended to the case of a general composite hypothesis and of Pitman's local alternatives. The asymptotic distribution of LRC for a simple hypothesis is studied under a fixed alternative. The Institute of Statistical Mathematics     

序约束下多元正态均值的检验问题

设有k组均值有简单半序约束,协方差阵未知的p维正态分布．Sasabuchi等在2003年研究了均值是否相等的检验问题．考虑到似然比检验统计量的临界点难以获得,以致于它不容易实施,Sasabuchi提出了一个检验方法．称为Sasabuchi检验．Sasabuchi检验的一个不足之处在于,它并不优于经典的MANOVA检验．作者提出了一个新的检验方法,它比Sasabuchi检验有一致优的势,而且形式更为简单．通过模拟发现这个检验方法还优势于MANOVA．最后导出了这个检验统计量的渐近零分布．     

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。     

Preliminary test estimators in intraclass correlation model under unequal family sizes

The intraclass correlation model is well known in the literature of multivariate analysis and it is mainly used in studying familial data. This model is considered in this paper and the interest is focused on the estimation of the intraclass correlation on the basis of familial data from families which are randomly selected from two or more independent populations. The size of the families is considered unequal and the variances of the populations are considered unequal, too. In this statistical framework some preliminary test estimators are presented in a unified way and their asymptotic distribution is obtained. A decision-theoretic approach is developed to compare the estimators by using the asymptotic distributional quadratic risk under the null hypothesis of equality of the intraclass correlations and under contiguous alternative hypotheses, as well. Some interesting relationships are obtained between the estimators considered.     

上界型拟合优度检验

对简单零假设情况,构造出一类上界型拟合优度检验.取不同的参数λ和不同的权函数,这类检验不仅包含许多已存在的检验,如Kolmogorov-Smirov检验,Berk-Jones检验等,而且还给出一些新的检验.众所周知,对不同的问题,"最优"的检验是不同的,有必要对这类检验的性质进行讨论.该文对任意给定的λ和较一般的权函数q(·),在较弱的条件下,导出了相应上界型检验统计量在零假设下的渐近分布,研究了它们的局部渐近功效;在若干固定备择假设下,对该类检验的功效进行了模拟研究.模拟结果表明,在不同的备择假设下,功效较优的检验是不同的,不存在对所有情况一致最优的检验.     

A frequency-domain test for long range dependence

A new frequency-domain test statistic is introduced to test for short memory versus long memory. We provide its asymptotic distribution under the null hypothesis and show that it is consistent under any long memory alternative. Some simulation studies show that this test is more robust than various standard tests in terms of empirical size when the normality of observed process is lost.     

Asymptotics of the goodness-of-fit test for a partial linear model with randomly censored data

In this paper, we discuss the problem of testing the hypothesis that the underlying regression is a partial linear model. A test statistic, which is based on the quadratic form of a cusum process of residuals, is proposed. The asymptotic distributions of the test statistic under null hypothesis and the local alternative hypothesis are given. The number simulation shows that the test is available.     

浅谈假设检验中原假设和备择假设的建立

建立原假设和备择假设是进行假设检验的第一步.如何设立假设检验中的原假设和备择假设,是学生和初学者普遍困扰的问题.而根据奈曼—皮尔逊准则,原假设和备择假设的地位不对等.本文对如何建立假设检验中的原假设和备择假设进行了讨论,给出了几个建立原假设和备择假设的原则.     

A Partial Empirical Likelihood Based Score Test Under a Semiparametric Finite Mixture Model

We propose a score statistic to test the null hypothesis that the two-component density functions are equal under a semiparametric finite mixture model. The proposed score test is based on a partial empirical likelihood function under an I-sample semiparametric model. The proposed score statistic has an asymptotic chi-squared distribution under the null hypothesis and an asymptotic noncentral chi-squared distribution under local alternatives to the null hypothesis. Moreover, we show that the proposed score test is asymptotically equivalent to a partial empirical likelihood ratio test and a Wald test. We present some results on a simulation study.     

Testing equality between several populations covariance operators

In many situations, when dealing with several populations, equality of the covariance operators is assumed. An important issue is to study whether this assumption holds before making other inferences. In this paper, we develop a test for comparing covariance operators of several functional data samples. The proposed test is based on the Hilbert–Schmidt norm of the difference between estimated covariance operators. In particular, when dealing with two populations, the test statistic is just the squared norm of the difference between the two covariance operators estimators. The asymptotic behaviour of the test statistic under both the null hypothesis and local alternatives is obtained. The computation of the quantiles of the null asymptotic distribution is not feasible in practice. To overcome this problem, a bootstrap procedure is considered. The performance of the test statistic for small sample sizes is illustrated through a Monte Carlo study and on a real data set.     

删截分布未知的多维随机删截数据的PP拟合优度检验

文[5]在假定删截分布已知的条件下,用投影寻踪（PP）技巧讨论了多维随机删截数据的PP拟合优度检验问题．本文讨论截尾分布未知时,多维随机删截数据的拟合优度检验问题,得到了检验统计量在零假设成立时的渐近分布,并讨论了其Bootstrap逼近．     

On consistent testing for serial correlation of unknown form in vector time series models

Multivariate autoregressive models with exogenous variables (VARX) are often used in econometric applications. Many properties of the basic statistics for this class of models rely on the assumption of independent errors. Using results of Hong (Econometrica 64 (1996) 837), we propose a new test statistic for checking the hypothesis of non-correlation or independence in the Gaussian case. The test statistic is obtained by comparing the spectral density of the errors under the null hypothesis of independence with a kernel-based spectral density estimator. The asymptotic distribution of the statistic is derived under the null hypothesis. This test generalizes the portmanteau test of Hosking (J. Amer. Statist. Assoc. 75 (1980) 602). The consistency of the test is established for a general class of static regression models with autocorrelated errors. Its asymptotic slope is derived and the asymptotic relative efficiency within the class of possible kernels is also investigated. Finally, the level and power of the resulting tests are also studied by simulation.     

高维随机截尾数据的PP拟合优度检验

关于一维删截数据的拟合优度检验，已有相当多的文献，但高维截尾数据的拟合优度检验尚不多见．本文用PP技巧讨论了高维截尾数据的拟合优度检验，得到了检验统计量的渐近分布，并讨论了其Bootstrap逼近及逼近的相容性和检验的渐近功效．     

A simple test of exponentiality against NWBUE family of life distributions

In this paper we propose a simple procedure to test the null hypothesis of exponentiality against the alternative that it belongs to the new worse then better than used in expectation (NWBUE) family. The test is shown to be consistent and the asymptotic distribution of the test statistic has been obtained. The performance of the test against various classes of alternatives has been studied by means of simulation. Copyright © 2005 John Wiley & Sons, Ltd.     

Nonparametric test of restricted interchangeability

Exact and large sample distributions of the rank order test under the null hypothesis of restricted interchangeability are obtained. Under given regularity conditions and under Pitman's shift in location alternative, the asymptotic relative efficiency of this nonparametric test in comparison with Votaw's (1948, Ann. Math. Statist., 19, 447–473) likelihood ratio test is given.     

Generalized Cramér-von Mises tests of goodness of fit for doubly censored data

We generalize Cramér-von Mises statistics to test the goodness of fit of a lifetime distribution when the data are doubly censored. We derive the limiting distributions of our test statistics under the null hypothesis and the alternative hypothesis, respectively. We also give a strong consistent estimator for the asymptotic covariance of the self-consistent estimator for the survival function with doubly censored data. Thereby, a method, called the Fredholm Integral Equation method, is proposed to estimate the null distribution of test statistics. In this work, the perturbation theory for linear operators plays an important role, and some numerical examples are included.The author's research was supported by a Faculty Fellowship of University of Nebraska-Lincoln.     

HAC estimation and strong linearity testing in weak ARMA models

In the framework of ARMA models, we consider testing the reliability of the standard asymptotic covariance matrix (ACM) of the least-squares estimator. The standard formula for this ACM is derived under the assumption that the errors are independent and identically distributed, and is in general invalid when the errors are only uncorrelated. The test statistic is based on the difference between a conventional estimator of the ACM of the least-squares estimator of the ARMA coefficients and its robust HAC-type version. The asymptotic distribution of the HAC estimator is established under the null hypothesis of independence, and under a large class of alternatives. The asymptotic distribution of the proposed statistic is shown to be a standard χ² under the null, and a noncentral χ² under the alternatives. The choice of the HAC estimator is discussed through asymptotic power comparisons. The finite sample properties of the test are analyzed via Monte Carlo simulation.     

ON SOME TESTS FOR ELLIPTICAL SYMMETRY

In this paper, some test statistics of Kolmogorov type and Cramer-von Mises type based on projection pursuit technique are proposed for testing the sphericity problem of a high-dimensional distribution. The limiting distributions of the test statistics are derived under the null hypothesis and any fixed alternative. The asymptotic properties of Bootstrap approximation are investigated. Furthermore, for computational reasons, an approximation for the statistics, based on number theoretic method, is suggested.     

Approximation by randomly weighting method in censored regression model

Censored regression (“Tobit”) models have been in common use, and their linear hypothesis testings have been widely studied. However, the critical values of these tests are usually related to quantities of an unknown error distribution and estimators of nuisance parameters. In this paper, we propose a randomly weighting test statistic and take its conditional distribution as an approximation to null distribution of the test statistic. It is shown that, under both the null and local alternative hypotheses, conditionally asymptotic distribution of the randomly weighting test statistic is the same as the null distribution of the test statistic. Therefore, the critical values of the test statistic can be obtained by randomly weighting method without estimating the nuisance parameters. At the same time, we also achieve the weak consistency and asymptotic normality of the randomly weighting least absolute deviation estimate in censored regression model. Simulation studies illustrate that the performance of our proposed resampling test method is better than that of central chi-square distribution under the null hypothesis. This work was supported by National Natural Science Foundation of China (Grant No. 10471136), PhD Program Foundation of the Ministry of Education of China, and Special Foundations of the Chinese Academy of Sciences and University of Science and Technology of China     

Goodness of fit test for ergodic diffusions by tick time sample scheme

We consider a nonparametric goodness of fit test problem for the drift coefficient of one-dimensional ergodic diffusions, where the diffusion coefficient is a nuisance function which is estimated in some sense. Using a theory for the continuous observation case, we construct a test based on the data observed discretely in space, that is, the so-called tick time sampled data. It is proved that the asymptotic distribution of our test under the null hypothesis is the supremum of the standard Brownian motion, and thus our test is asymptotically distribution free. It is also shown that the test is consistent under any fixed alternative.