The union-intersection principle applied to the test of dimensionality

In the first section Anderson-Rao-Fujikoshi's test statistics for testing the hypothesis of dimensionality are reviewed and then Olkin-Tomsky's generalized union-intersection principle is applied to show that a new class of test statistics for testing the hypothesis of dimensionality are derived which includes the likelihood ratio test statistics, the trace test statistics and a version of ROY'S maximum root test statistics.     

Multivariate analysis with an autoregressive covariance model

This paper addresses the problem of testing the multivariate linear hypothesis when the errors follow an antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as a nonstationary autoregressive model of general order. Three test statistics are derived that provide analogs to three commonly used MANOVA statistics: Wilks' Lambda, the Lawley-Hotelling Trace, and Pillai's Trace. Formulas are given for each of these statistics that show how they can be obtained From any statistical computing package that calculates the usual MANOVA statistics. These antedependent statistics would be appropriate in analyzing certain multivariate data sets in which repeated measurements are taken on the same subjects over a period of time.     

Properties of two tests for outliers in multivariate data

Mardia's multivariate kurtosis and the generalized distance have desirable properties as multivariate outlier tests. However, extensive critical values have not been published heretofore. A published approximation formula for critical values of the kurtosis is shown to inadequately control the type I error rate, with observed error rates often differing from their intended values by a factor of two or more. Critical values derived from simulations for both tests for up to 25 dimensions and 500 observations are presented. The power curves of both tests are discussed. The generalized distance is the more powerful test when exactly one outlier is present and the contaminant is substantially mean-shifted. However, as the number of outliers increases, the kurtosis becomes the more powerful test. The two tests are compared with respect to power and vulnerability to masking. Recommendations for the use of these tests and interpretation of results are given.     

EXACT DISTRIBUTION OF CERTAIN GENERAL TEST STATISTICS IN MULTIVARIATE ANALYSIS

In this paper, exact solution of Wilks' type-B integral equation has been obtained in its most general form as a series of weighted gamma distributions. This general result then gives the distributions of many test statistics in multivariate analysis. In particular the distributions of Wilks' Λ, the sphericity test criterion, and Bartlett's test statistic, are derived in easily computable form.     

A new functional statistic for multivariate normality

The test statistics of assessing multivariate normality based on Roy’s union-intersection principle (Roy, Some Aspects of Multivariate Analysis, Wiley, New York, 1953) are generalizations of univariate normality, and are formed as the optimal value of a nonlinear multivariate function. Due to the difficulty of solving multivariate optimization problems, researchers have proposed various approximations. However, this paper shows that the (nearly) global solution contrarily results in unsatisfactory power performance in Monte Carlo simulations. Thus, instead of searching for a true optimal solution, this study proposes a functional statistic constructed by the q% quantile of the objective function values. A comparative Monte Carlo analysis shows that the proposed method is superior to two highly recommended tests when detecting widely-selected alternatives that characterize the various properties of multivariate normality.     

Elliptically contoured model and factorization of wllks1 A

In a series of papers, Kshirsagar (1964, 1971) and McHenry and Kshirsagar (1977), factorize Wilks' A into a number of factors and find the independent null multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of independent or dependent variables. Essentially the factors of Wilks' A are the factors of certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distribution of these factors. The present paper generalizes Kshirsagar's (1971) normal theory to the elliptically contoured model, and shows that his results are null robust for the elliptically contoured model.     

Analysis of seasonal level shift (SLS) detection in SARIMA models

This study aims at exploring correct identification of seasonal outliers using most commonly applied test statistics. We evaluate the performance of seasonal level shift (SLS) by means of empirical level of significance, power of the test for sensitivity in detecting changes, and the vulnerability to masking of outliers by misspecification frequencies. We observe that the size of SLS affects the sampling distribution of η_SLS (test statistics for SLS detection) in case of SAR (1) and SMA (1) model. The empirical critical values for 1%, 5%, and 10% upper percentiles are higher than the usual cut off points and the empirical level of significance is inversely related to sample size and the model coefficients. The empirical power of the test statistics is not satisfactory at small sample size, and for large model coefficient. η_SLS gets confused with IO. The potential list of types of outliers should retain both IO and SLS as a part of outlier detection procedure for most efficient results. We apply the method suggested by Kaiser and Maravall with five possible types of outliers, that is, AO, IO, LS, TC, and SLS, to a number of quarterly and monthly time series data from Pakistan.     

CASE-DELETION DIAGNOSTICS FOR TEST STATISTICS IN MULTIVARIATE REGRESSION

Influence measures in multivariate regression analysis have been widely developed, especially through use of the case-deletion approach. However, there seem to be few accounts of the influence of observations on test statistics in hypothesis testing. This paper examines four common multivariate tests, namely the Wilks' ratio, Lawley-Hotelling trace, Pillai's trace and Roy's greatest root for testing a general linear hypothesis of the regression coefficients in multivariate regression. The influence of observations is measured using the case-deletion approach. The proposed diagnostic measures, except that of Roy's greatest root, can be expressed in terms of statistics without involving the actual deletion of observations. An illustrative example is given with satisfactory results.     

Detection of outliers in growth curve models

The growth curve model introduced by Potthoff and Roy (1964) is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. In this paper, we discuss procedures for detection of outliers in growth curve models for mean-slippage and dispersion-slippage outlier model. The distributions of the test statistics are discussed and the values of significant probabilities are given using Bonferronl's bounds. Some simulation results are also presented.     

A NOTE ON MULTIVARIATE LINEAR MODELS WITH NON-ADDITIVITY

Tukey's non-additivity test in an analysis of variance model is extended to a multivariate linear model with covariates. If non-additivity is found to exist, a Wilks's Lambda test for the dimensionality of the matrix of the non-additivity parameters is derived and the Lambda criterion is then factorized into two independent test criteria to test meaningful hypotheses concerning the multivariate model.     

Detection of outliers in familial data

Procedures for detection of outliers in familial data is given for mean-slippage and dispersion-slippage model of outliers for equal and unequal family sizes. The distributions of the test statistics are derived and Bonferroni's bounds for the values of significant probabilities are given.     

A note on finding critical values for a locally optimal multivariate test statistic

A previous paper provided a locally optimal test statistic for combining s independent test statistics of a common multivariate hypothesis. Difficulty in calculating the critical values limited the applicability of the test. In this paper, it is shown that approximate critical values can be easily calculated, with negligible loss of accuracy for most situations.     

Comparing several experimental groups with a control in the multivariate case

This paper presents a multivariate extension of Dunnett's test for comparing simultaneously k treatment group means with a single control group mean. A test based on Hotelling T²statistics is presented and approximate critical values are evaluated for the case of equal numbers of observations in each group, for the .05 and .01 levels of significance, for 1 to 5 variates, for 1 to 10 treatment groups, and for varying degrees of freedom. The accuracy of the procedure for generating approximate critical values is assessed via simulation studies conducted for selected cases and an example is presented using real data.     

The problem of outlier tests in sample surveys

Outliers can occur as readily in samples from the finite populations (e.g. in sample surveys) as in samples from infinite populations. However, in the vast literature on outliers there is almost no mention of outlier tests for data from sample surveys. We examine the behaviour of some standard outlier test statistics for infinite populations when these are applied to finite populations, examining their properties by extensive simulation studies. Some anomalous results are obtained Nsuggesting a fundamental difficulty in testing outliers for the finite population case.     

Testing equality of several exponential distributions

Roy's union-intersection principle is used to develop a test procedure to test the equality of scale parameters of several exponential distributions. Upper five and one percent values of the test statistic for two and three exponential distributions are tabulated and an illustrative simulated example is qiven.     

Small sample power of multivariate nonparametric tests for monotone trend

Bhattacharyya and Kioiz (1966) propose two multivariate nonparametric tests for monotone trend, one involving coordinate-wise Mann statistics and the other, coordinate-wise Spearman statistics. Dietz and Killeen (1981) propose a different test statistic based on coordinate-wise Mann statistics. The Pitman asymptotic relative efficiency of all three tests with respect to a normal theory competitor equals the cube root of the efficiency of a multivariate signed rank test with respect to Hotelling's T2. In this article, the small sample power of the nonparametric tests, the normal theory test, and a Bonferroni approach involving coordinate-wise univariate Mann or Spearman tests is examined in a simulation study. The Mann statistic of Dietz and Killeen and the Spearman statistic of Bhattacharyya and Klotz are found to perform well under both null and alternative hypotheses     

Tables of critical values for two group interim analyses

Tables of critical values are given, which can be used to execute interim analyses in clinical trials involving two groups when the joint distribution of the test statistics can be approximated by a multivariate normal distribution. Critical values are given for both the one and two interim analyses cases for a variety of partitions of α and correlation structures. Results of power calculations are presented, which reflect the effects of both the correlation structure and partitions of α.Several examples are given, which illustrate how to apply the tables to a variety of experiments     

Testing hypotheses about covariance matrices using bootstrap methods

The usual chi-squared approximation to test statistics based on normal theory for testing covariance structures of multivariate populations is very sensitive to the normality assumption. Two general bootstrap procedures are developed in this paper to obtain approximately valid critical values for these test statistics when the data are not normally distributed. The first is based on separate sampling from individual samples, and the second is based on sampling from pooled samples. Although the second method requires more assumptions, its small sample properties are better.     

Tests for noncorrelation of two multivariate ARMA time series

In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross-correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross-correlations asymptotically follows the same distribution as the corresponding vector of cross-correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi-square random variables. Two test procedures are described. The first one is based on the residual cross-correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite-sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi-square distribution. Finally, the procedures developed are applied to two different sets of economic data.     

Testing for serial correlation in the presence of dynamic heteroscedasticity

Standard serial correlation tests are derived assuming that the disturbances are homoscedastic, but this study shows that asympotic critical values are not accurate when this assumption is violated. Asymptotic critical values for the ARCH(2)-corrected LM, BP and BL tests are valid only when the underlying ARCH process is strictly stationary, whereas Wooldridge's robust LM test has good properties overall. These tests exhibit similar bahaviour even when the underlying process is GARCH (1,1). When the regressors include lagged dependent variables, the rejection frequencies under both the null and alternative hypotheses depend on the coefficientsof the lagged dependent variables and the other model parameters. They appear to be robust across various disturbance distributions under the null hypothesis.