A Note on the Likelihood Ratio Test for Equality of k Normal Populations

The authors derive the analytic expressions for the mean and variance of the log-likelihood ratio for testing equality of k (k ≥ 2) normal populations, and suggest a chi-square approximation and a gamma approximation to the exact null distribution. Numerical comparisons show that the two approximations and the original beta approximation of Neyman and Pearson (1931 Neyman , J. , Pearson , E. S. ( 1931 ). On the problem of k samples . In: Neyman , J. , Pearson , E. S. , eds. Joint Statistical Papers . Cambridge : Cambridge University Press , pp. 116 – 131 . [Google Scholar]) are all accurate, and the gamma approximation is the most accurate.     

A Note on the Poisson Likelihood Ratio Test Statistic for Kulldorff's Scan Methods

Methods based on scan statistics are widely used in health-related applications to detect clusters of disease. The most common methods are based on the Bernoulli and Poisson models. Kulldorff (1997 Kulldorff , M. ( 1997 ). A spatial scan statistic . Communications in Statistics—Theory and Methods 26 : 1481 – 1496 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) derived the likelihood ratio test statistic for his scan method for both of these models. His scan statistic is widely used with freely available software, SaTScan^? (see Kulldorff, 2005 Kulldorff , M. ( 2005 ). SaTScan: software for the spatial, temporal, and space-time scan statistics , version 5.1 [computer program]. Information Management Services 2005; Available: http://www.satscan.org/. Accessed July 2007 . [Google Scholar]). We provide an alternative derivation of the likelihood ratio test statistic in the Poisson case. Our derivation is simpler and more general in the sense that it applies when the incidences are not aggregated into subregional counts.     

A Note on the Modified Likelihood Ratio Test for Homogeneity in Beta Mixture Models

This article presents a note on the modified likelihood ratio test for homogeneity in beta mixture models. Under consistency of the penalized maximum likelihood estimators, the limiting distribution of the test statistic converges to the chi-bar-squared distributions. The statistic degenerates to zero with a weight due to the negative definiteness of a complicated random matrix. The probability that this matrix is negative definite is related to the parameter values under the homogeneity hypothesis. The dependency pattern enables the introduction of an upper bound on the asymptotic null distribution. Simulation study is investigated to verify the accuracy of the results.     

Bartlett's Test Applied to Variance Component Models

Exact methods for testing equality between variance components obtained from several cases of the same type of balanced orthogonal design are discussed. In particular, methods for successively testing equality of a number of components using Bartlett's tests are outlined for univariate and multivariate responses. Two clinical trial examples of repeated‐measures data are presented.     

A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models

This article develops a method for testing the goodness-of-fit of a given parametric autoregressive conditional duration model against unspecified nonparametric alternatives. The test statistics are functions of the residuals corresponding to the quasi maximum likelihood estimate of the given parametric model, and are easy to compute. The limiting distributions of the test statistics are not free from nuisance parameters. Hence, critical values cannot be tabulated for general use. A bootstrap procedure is proposed to implement the tests, and its asymptotic validity is established. The finite sample performances of the proposed tests and several other competing ones in the literature, were compared using a simulation study. The tests proposed in this article performed well consistently throughout, and they were either the best or close to the best. None of the tests performed uniformly the best. The tests are illustrated using an empirical example.     

Properties of a One-Sided Likelihood Ratio Test as Applied to Pool Screening

In the control of tropical disease, data are typically collected using pool screening design. When eradication programs have been in place for a period of time, researchers are interested in testing whether the disease prevalence is below a certain target level using one-sided likelihood ratio test (LRT). We will investigate the finite and large-sample properties of this test statistic. Based on simulations, if the number of pools is not large enough, the LRT has inflated type I error rate. Consequently, researchers will make the critical mistake of stopping the treatment when the target level has not been reached.     

A Note on the Innovation Distribution of a Gamma Distributed Autoregressive Process

A representation of the innovation random variable for a gamma distributed first-order autoregressive process was found by Lawrance (1982) in the form of a compound Poisson distribution, connected with a shot-noise process. In this note we simplify the representation of Lawrance by providing a direct representation in terms of density functions.     

Use of the Likelihood Ratio Test on the Inverse Gaussian Distribution

In the following article, the likelihood ratio test is determined for four tests of hypotheses involving the inverse Gaussian distribution. For three of the hypotheses, the test produces the same statistic as the uniformly most powerful unbiased test.     

Testing for Unit Root Against Stationarity Using the Likelihood Ratio Test

In a first-order autoregressive model with drift, we derive the likelihood ratio test for a unit root against the stationary alternative. We also derive the test in a state space model with trend. Finite sample and asymptotic critical values are obtained by Monte Carlo simulations. A simulation study investigates the power performance of the likelihood ratio test and we also examine how a bias correction of the test affects the results.     

Application of the Generalized Likelihood Ratio Test for Detecting Changes in the Mean of Multivariate GARCH Processes

We derive several multivariate control charts to monitor the mean vector of multi-variate GARCH processes under the presence of changes, by means of maximizing the generalized likelihood ratio. This presentation is rounded up by a comparative performance study based on extensive Monte Carlo simulations. An empirical illustration shows how the obtained results can be applied to real data.     

基于加权似然比检验的监控线性曲线的控制图

文章提出了一种基于加权似然比检验的阶段二监控线性曲线的控制图,称为WLRT图,并通过平均运行长度来衡量控制图的性能表现。模拟结果表明,WLRT图对于线性曲线的截距、斜率、标准差的变化及截距和斜率同时变化都具有很好的检测能力。通过与其他几种控制图的性能比较,得出WLRT图能较快地发现过程变化,而且设计简单、操作方便。     

Simulating Properties of the Likelihood Ratio Test for a Unit Root in an Explosive Second-Order Autoregression

This paper provides a means of accurately simulating explosive autoregressive processes and uses this method to analyze the distribution of the likelihood ratio test statistic for an explosive second-order autoregressive process of a unit root. While the standard Dickey–Fuller distribution is known to apply in this case, simulations of statistics in the explosive region are beset by the magnitude of the numbers involved, which cause numerical inaccuracies. This has previously constituted a bar on supporting asymptotic results by means of simulation, and analyzing the finite sample properties of tests in the explosive region.     

Simulating Properties of the Likelihood Ratio Test for a Unit Root in an Explosive Second-Order Autoregression

This paper provides a means of accurately simulating explosive autoregressive processes and uses this method to analyze the distribution of the likelihood ratio test statistic for an explosive second-order autoregressive process of a unit root. While the standard Dickey-Fuller distribution is known to apply in this case, simulations of statistics in the explosive region are beset by the magnitude of the numbers involved, which cause numerical inaccuracies. This has previously constituted a bar on supporting asymptotic results by means of simulation, and analyzing the finite sample properties of tests in the explosive region.     

A Multistage Chi-square Test for Measurement of the Degree of Association in Two-way Table

The classical chi-square test for independence cannot convey additional information and the degree of the association of two factors in two-way contingency table. Besides, measures of association by contingency coefficient need to be used with care, because the association measures depend on the number of rows r and the number of columns c. This article proposes a multistage chi-square test to measure the degree of the association between the two factors in two-way contingency table. We also give simulation and real examples to assess the performance of the proposed method. The results show that our proposed method can effectively investigate the degree of association of two factors in two-way contingency table.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

Considering the Concordant Observations in Likelihood Ratio Test for Paired Binary Data

In the formula of the likelihood ratio test on fourfold tables with matched pairs of binary data, only the two parts b and c, which represent changes, are considered; the retained parts a and d, which represent concordant observations, are not included. To develop the test by considering all the four parts and the mixture distribution of likelihood ratio chi-squares, a formula based on the entire sample is proposed. The revised formula is the same as the unrevised one when a + d is zero. The revised test is more valid than the revised McNemar's test in most cases.     

On the Distribution of Likelihood Ratio Test Statistics for Cointegration Rank

This paper analyses the likelihood ratio test for the hypothesis of reduced cointegration rank in a Gaussian vector autoregressive model. The usual asymptotic distribution typically gives rather large size distortions. This is explained by the fact that the asymptotic distribution of the likelihood ratio test statistic varies across the parameter space. A much improved distribution approximation can be obtained using local asymptotic theory. The idea is discussed for some low dimensional examples.     

判别分析统计检验体系的探讨

判别分析已越来越受到人们的重视并取得了重要的应用成果,但应用中存在着简单套用的情况,对判别分析的适用性、判别效果的显著性、判别变量的判别能力以及判别函数的判别能力的检验等问题重视不够。为了更好地应用判别分析,就应对判别分析进行统计检验并建立统计检验体系,统计检验体系应包括：判别分析适用性检验,判别效果显著性检验,判别变量的判别能力检验和判别函数的判别能力检验。     

An Approximation for the Test of the Equality of the Smallest Eigenvalues of a Covariance Matrix

A limiting distribution of the likelihood ratio statistic for the test of the equality of the q smallest eigenvalues of a covariance matrix is obtained. This distribution can be used as an alternative to the chi-squared distribution which is usually used with this test. It is shown that this new method yields reasonable significance levels for those situations in which the chi-squared approximation is inadequate.