An alternative Wald type test for two linear restrictions with applications to non-linear models

This paper proposes a new test procedure called the rel test to resolve the problem of small-sample local biasedness and non-monotonic power behavior of the Wald test for two linear restrictions caused by inaccuracy of the estimated covariance matrix of the estimator. This new test procedure, which does not need the covariance matrix of the estimator, involves finding the critical region based on contour points of the percentile confidence limit of a rel utilizing the bootstrap in order to obtain a test with the desired size and good power properties. Simulation results indicate that this new test procedure, the rel test, performs rather well both with respect to controlling size and having monotonic increasing power.     

On testing for an identity covariance matrix when the dimensionality equals or exceeds the sample size

This article explores the problem of testing the hypothesis that the covariance matrix is an identity matrix when the dimensionality is equal to the sample size or larger. Two new test statistics are proposed under comparable assumptions to those statistics in the literature. The asymptotic distribution of the proposed test statistics are found and are shown to be consistent in the general asymptotic framework. An extensive simulation study shows the newly proposed tests are comparable to, and in some cases more powerful than, the tests for an identity covariance matrix currently in the literature.     

Specification test for a linear regression model with ARCH process

ARCH models are used widely in analyzing economic and financial time series data. Many tests are available to detect the presence of ARCH; however, there is no acceptable procedure available for testing an estimated ARCH model.. In this paper we develop a test for a linear regression model with ARCH disturbances using the framework of the information matrix (IM) test. For the ARCH specification, the covariance matrix of the indicator vector is not block diagonal, and the IM test is turned out to be a test for variation in the fourth moment, i.e., a test for heterokurtosis. An illustrative example is provided to demonstrate the usefulness of the proposed test.     

Testing a block exchangeable covariance matrix

Testing hypotheses about the structure of a covariance matrix for doubly multivariate data is often considered in the literature. In this paper the Rao's score test (RST) is derived to test the block exchangeable covariance matrix or block compound symmetry (BCS) covariance structure under the assumption of multivariate normality. It is shown that the empirical distribution of the RST statistic under the null hypothesis is independent of the true values of the mean and the matrix components of a BCS structure. A significant advantage of the RST is that it can be performed for small samples, even smaller than the dimension of the data, where the likelihood ratio test (LRT) cannot be used, and it outperforms the standard LRT in a number of contexts. Simulation studies are performed for the sample size consideration, and for the estimation of the empirical quantiles of the null distribution of the test statistic. The RST procedure is illustrated on a real data set from the medical studies.     

On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

A test of the mean square error criterion for linear admissible estimators

A test for choosing between a linear admissible estimator and the least squares estimator (LSE) is developed. A characterization of linear admissible estimators useful for comparing estimators is presented and necessary and sufficient conditions for superiority of a linear admissible estimator over the LS estimetor is derived for the test. The test is based on the MSE matrix superiority, but also new resl?!ts concerning covariance matrix comparisons of linear estimators are derived. Further,shown that the test of Toro - Vizcarrondo and Wailace applies iioi only the restricted least squares estimators but also to certain estimators outside this class.     

An Improved Estimation in Regression Parameter Matrix in Multivariate Regression Model

We consider shrinkage and preliminary test estimation strategies for the matrix of regression parameters in multivariate multiple regression model in the presence of a natural linear constraint. We suggest a shrinkage and preliminary test estimation strategies for the parameter matrix. The goal of this article is to critically examine the relative performances of these estimators in the direction of the subspace and candidate subspace restricted type estimators. Our analytical and numerical results show that the proposed shrinkage and preliminary test estimators perform better than the benchmark estimator under candidate subspace and beyond. The methods are also applied on a real data set for illustrative purposes.     

A monte carlo study of the power of alternative tests under the generalized manova model

Monte Carlo simulations are performed for a broad range of conditions. These simulations indicate that the powers of alternative tests under the generalized MANOVA model for small samples differ significantly, if a large reduction of the number of polynomial parameters is applied. The results show that, if the response covariance matrix ∑ is known, the best alternative is to use ∑. If, however, ∑ is unknown, substitution of an identity matrix for ∑ is recommended. This alternative usually results in a test with more power than the test with the usual estimate of ∑ employing covariates or the test with an estimate of E obtained from another sample.     

Location‐invariant Multi‐sample U‐tests for Covariance Matrices with Large Dimension

For two or more multivariate distributions with common covariance matrix, test statistics for certain special structures of the common covariance matrix are presented when the dimension of the multivariate vectors may exceed the number of such vectors. The test statistics are constructed as functions of location‐invariant estimators defined as U‐statistics, and the corresponding asymptotic theory is used to derive the limiting distributions of the proposed tests. The properties of the test statistics are established under mild and practical assumptions, and the same are numerically demonstrated using simulation results with small or moderate sample sizes and large dimensions.     

Consistent model selection and data-driven smooth tests for longitudinal data in the estimating equations approach

Summary.  Model selection for marginal regression analysis of longitudinal data is challenging owing to the presence of correlation and the difficulty of specifying the full likelihood, particularly for correlated categorical data. The paper introduces a novel Bayesian information criterion type model selection procedure based on the quadratic inference function, which does not require the full likelihood or quasi-likelihood. With probability approaching 1, the criterion selects the most parsimonious correct model. Although a working correlation matrix is assumed, there is no need to estimate the nuisance parameters in the working correlation matrix; moreover, the model selection procedure is robust against the misspecification of the working correlation matrix. The criterion proposed can also be used to construct a data-driven Neyman smooth test for checking the goodness of fit of a postulated model. This test is especially useful and often yields much higher power in situations where the classical directional test behaves poorly. The finite sample performance of the model selection and model checking procedures is demonstrated through Monte Carlo studies and analysis of a clinical trial data set.     

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.     

Bayesian goodness-of-fit test for censored data

We propose a Bayesian computation and inference method for the Pearson-type chi-squared goodness-of-fit test with right-censored survival data. Our test statistic is derived from the classical Pearson chi-squared test using the differences between the observed and expected counts in the partitioned bins. In the Bayesian paradigm, we generate posterior samples of the model parameter using the Markov chain Monte Carlo procedure. By replacing the maximum likelihood estimator in the quadratic form with a random observation from the posterior distribution of the model parameter, we can easily construct a chi-squared test statistic. The degrees of freedom of the test equal the number of bins and thus is independent of the dimensionality of the underlying parameter vector. The test statistic recovers the conventional Pearson-type chi-squared structure. Moreover, the proposed algorithm circumvents the burden of evaluating the Fisher information matrix, its inverse and the rank of the variance–covariance matrix. We examine the proposed model diagnostic method using simulation studies and illustrate it with a real data set from a prostate cancer study.     

Asymptotic expansions for the distributions of statistics based on a correlation matrix

An asymptotic expansion is given for the distribution of the α-th largest latent root of a correlation matrix, when the observations are from a multivariate normal distribution. An asymptotic expansion for the distribution of a test statistic based on a correlation matrix, which is useful in dimensionality reduction in principal component analysis, is also given. These expansions hold when the corresponding latent root of the population correlation matrix is simple. The approach here is based on a perturbation method.     

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.     

Study of redundancy of vector variables in canonical correlations

Fujikoshi (1982) obtained the necessary and sufficient conditions for the increased number of variables in the two sets of vectors not affecting the original nonzero canonical correlations and used these to obtain the likelihood ratio test procedure. He assumed a nonsingular covariance matrix due to random variables. Here, we study the same problem when the covariance matrix is singular and establish some further results. In this study, we note that the unit canonical correlations have to be separated in some of the situations. These results are valid for complex random vector variables and in some situations, the test for redundancy is given for complex random variables.     

Testing for normally in censored regressions

This paper derives a Lagrange Multiplier test for normality in censored regressions. The test is derived against the generalized log-gamma distribution, in which normal is a special case. The resulting test statistic coincides to some extent with previously suggested score and conditional moment tests. Estimation of the variance is performed by using the matrix of second order derivatives in order to get an easy to use test statistic. Small sample performance of the test is studied and compared to other tests by Monte Carlo experiments.     

Nonparametric analysis of treatment effects with missing observations

This paper develops a test for comparing treatment effects when observations are missing at random for repeated measures data on independent subjects. It is assumed that missingness at any occasion follows a Bernoulli distribution. It is shown that the distribution of the vector of linear rank statistics depends on the unknown parameters of the probability law that governs missingness, which is absent in the existing conditional methods employing rank statistics. This dependence is through the variance–covariance matrix of the vector of linear ranks. The test statistic is a quadratic form in the linear rank statistics when the variance–covariance matrix is estimated. The limiting distribution of the test statistic is derived under the null hypothesis. Several methods of estimating the unknown components of the variance–covariance matrix are considered. The estimate that produces stable empirical Type I error rate while maintaining the highest power among the competing tests is recommended for implementation in practice. Simulation studies are also presented to show the advantage of the proposed test over other rank-based tests that do not account for the randomness in the missing data pattern. Our method is shown to have the highest power while also maintaining near-nominal Type I error rates. Our results clearly illustrate that even for an ignorable missingness mechanism, the randomness in the pattern of missingness cannot be ignored. A real data example is presented to highlight the effectiveness of the proposed method.     

Testing for a change lit the regression matrix of a multivariate linear model

A Bayesian test procedure Is developed to test; the null hypothesis of no change In the regression matrix of a multivariate lin¬ear model against the alternative hypothesis of exactly one change The resulting test is based on the marginal posterior distribution of the change point; To illustrate the test procedure a numerical example using a bivariate regression model is considered.     

A note on multivariate linear regression

In this note, a hypothesis test based on relevant statistical differences is proposed for multivariate linear regression models whose design matrix rank does not equal the number of regression variables. A statistical example is also provided to illustrate the proposed hypothesis test.     

A james-stein type detour of U-statistics

For a vector of estimable parameters, a modified version of the James-Stein rule (incorporating the idea of preliminary test estimators) is utilized in formulating some estimators based on U-statistics and their jackknifed estimator of dispersion matrix. Asymptotic admissibility properties of the classical U-statistics, their preliminary test version and the proposed estimators are studied.