On the distribution of some test statistics connected with the multivariate linear functional relationship model

An alternate representation of the densities of some test statistics for the structural coefficients of the multivariate linear functional relationship model is proposed in this article. These statistics are distributed as the ratio of a linear combination of chi-square variÂtes over the root of a product of chi-square variÂtes. A computable representation of their densities has already been derived by Provost (1984) with the help of the technique of the inverse Mellin transform. The connection of the alternate representation to the densities of products of independent beta type-2 and of independent F-random variables is also discussed.     

On estimating the dimensionality in discriminant analysis

In discriminant analysis, the dimension of the hyperplane which population mean vectors span is called the dimensionality. The procedures commonly used to estimate this dimension involve testing a sequence of dimensionality hypotheses as well as model fitting approaches based on (consistent) Akaike's method, (modified) Mallows' method and Schwarz's method. The marginal log-likelihood (MLL) method is developed and the asymptotic distribution of the dimensionality estimated by this method for normal populations is derived. Furthermore a modified marginal log-likelihood (MMLL) method is also considered. The MLL method is not consistent for large samples and two modified criteria are proposed which attain asymptotic consistency. Some comments are made with regard to the robustness of this method to departures from normality. The operating characteristics of the various methods proposed are examined and compared.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

多元t分布的尾部性质及其在股票市场的应用

研究多元t分布的厚尾性和尾部相依性,并用股票市场的真实数据来估计尾部相依系数,从而说明在金融市场尤其是股票市场中,用多元t分布来模拟数据比用多元正态分布更合理。     

A BETA-OPTIMAL TEST OF THE EQUICORRELATION COEFFICIENT

This paper considers the problem of testing for nonzero values of the equicorrelation coefficient of a standard symmetric multivariate normal distribution. Recently, SenGupta (1987) proposed a locally best test. We construct a beta-optimal test and present selected one and five percent critical values. An empirical power comparison of SenGupta's test with two versions of the beta-optimal test and the power envelope shows the relative strengths of the three tests. It also allows us to assess and confirm Efron's (1975) rule of when to question the use of a locally best test, at least for this testing problem. On the basis of these results, we argue that the two beta-optimal tests can be considered as approximately uniformly most powerful tests, at least at the five percent significance level.     

Exact testing in multivariate regression

An F statistic due to Rao (1951,1973) tests uniform mixed linear restrictions in the multivariateregression model. In combination with a generalization of the Bera-Evans-Savin exact functional relationship between the W, LR, and LM statistics, Rao's F serves to unify a number of exact test procedures commonly applied in disparate empirical literatures. Examples in demand analysis and asset pricing are provided. The availability of exact tests of restrictions in certain nonlinear models when the model is linear under the null, originally explored by Milliken-Graybill (1970), is extended to multivariate regression. Generalized RESET, J-, and Hausman-Wu tests are resented. As an extension of Dufour (1989), bounds tests exist for nonlinear and inequality restrictions. Applications include conservative bound tests for symmetry or negativity of the substitution matrix in demand systems.     

CONDITIONAL INFERENCE ABOUT AN EQUICORRELATION COEFFICIENT

SenGupta (1987) proposed a locally most powerful test which is globally (one sided) unbiased, and an estimator of p, the equicorrelation coefficient of a standard symmetric multivariate normal (SSMN) distribution. Here we use the idea in Williams (1984) to illustrate the construction and use of ancillary statistics to make inference about p. The test and confidence intervals based on this construction are conditionally optimal.     

A randomized Baumgartner statistic for multivariate two-sample testing hypothesis

In this paper, multivariate two-sample testing problems were examined based on the Jure?ková–Kalina's ranks of distances. The multivariate two-sample rank test based on the modified Baumgartner statistic for the two-sided alternative was proposed. The proposed statistic was a randomized statistic. Simulations were used to investigate the power of the suggested statistic for various population distributions.     

A survey on multivariate chi-square distributions and their applications in testing multiple hypotheses

We are concerned with three different types of multivariate chi-square distributions. Their members play important roles as limiting distributions of vectors of test statistics in several applications of multiple hypotheses testing. We explain these applications and consider the computation of multiplicity-adjusted p-values under the respective global hypothesis. By means of numerical examples, we demonstrate how much gain in level exhaustion or, equivalently, power can be achieved with corresponding multivariate multiple tests compared with approaches which are only based on univariate marginal distributions and do not take the dependence structure among the test statistics into account. As a further contribution of independent value, we provide an overview of essentially all analytic formulas for computing multivariate chi-square probabilities of the considered types which are available up to present. These formulas were scattered in the previous literature and are presented here in a unified manner.     

Explaining inconsistencies between patients and collaterals: validity in outcome studies of coercive alcoholism treatment

This study aims to assess the consistency of replies to questionnaires mailed to patients and two kinds of collaterals, i.e., social workers and significant others, at a public treatment center with socially unstable and compulsory committed patients. It compares the quantity and kind of discrepancies between replies by patients and collaterals on outcome data concerning social situation and drinking habits. It aims to measure the amount of systematic bias among factors that may explain inconsistencies between reports, especially the factors compulsory commitment, worse outcome, frequency of contact and type of collateral. The responders generally agreed. Variables in which there was less agreement were explored in logistic regressions using ten explanatory variables. Significant relations did not exceed those expected by chance. Discrepancies were not systematic in size and kind. On a six-rank ordinal scale of alcohol use or abuse, however, patients tended to underestimate the extent of their abuse. Inconsistencies here concerned the degree but not the presence of abuse. No difference in consistency due to type of collateral was found. In conclusion, the consistency of the questionnaires was high and independent of the social situation of the patient, of compulsory commitment and of other background or treatment factors, as well as of treatment outcome and type of collateral.