Estimation in a Two Variance Components Model When one Component is Known

We study a mixed linear model with two variance components. We suppose that one component is known. The objective of the paper is the estimation of the unknown component. The usual MINQE estimators seem to be unadapted to the problem. So we propose a new family of quadratic estimators, based on a natural class of estimators and the idea upon which the MINQE theory is built. All the estimators are compared on simulated data.     

Robustness of the Multiple Correlation Coefficient When Sampling From a Mixture of Two Multivariate Normal Populations

The density of the multiple correlation coefficient is derived by direct integration when the sample covariance matrix has a linear non-central distribution. Using the density, we deduce the null and non-null distribution of the multiple correlation coefficient when sampling from a mixture of two multivariate normal populations with the same covariance matrix. We also compute actual significance levels of the test of the hypothesis H_o : ρ_1·2…p = 0 versus H_a:ρ_1·2…p > 0, given the mixture model.     

A Test for the Equality of Means of Two Groups with Different Variances When the Sample Size and the Dimension are Large

We consider a class of test statistics including the Dempster trace criterion in the case of two groups without assuming equal covariance matrices. The test statistics in the class are valid when the dimension is larger than the sample size. We obtain asymptotic distributions of the test statistics in the class and use these distributions to derive the limiting power in each case. We obtain the most powerful test in the class with respect to this limiting power.     

A Methodology to Implement Box-Cox Transformation When No Covariate is Available

Box-Cox transformation is one of the most commonly used methodologies when data do not follow normal distribution. However, its use is restricted since it usually requires the availability of covariates. In this article, the use of a non-informative auxiliary variable is proposed for the implementation of Box-Cox transformation. Simulation studies are conducted to illustrate that the proposed approach is successful in attaining normality under different sample sizes and most of the distributions and in estimating transformation parameter for different sample sizes and mean-variance combinations. Methodology is illustrated on two real-life datasets.     

Evaluating the Significance Test When the Correlation Coefficient is Different from Zero in the Test of Hypothesis

Sample size and correlation coefficient of populations are the most important factors which influence the statistical significance of the sample correlation coefficient. It is observed that for evaluating the hypothesis when the observed value of the correlation coefficient's r is different from zero, Fisher's Z transformation may be incorrect for small samples especially when population correlation coefficient ρ has big values. In this study, a simulation program has been generated for to illustrate how the bias in the Fisher transformation of the correlation coefficient affects estimate precision when sample size is small and ρ has big value. By the simulation results, 90 and 95% confidence intervals of correlation coefficients have been created and tabled. As a result, it is suggested that especially when ρ is greater than 0.2 and sample sizes of 18 or less, Tables 1 and 2 can be used for the significance test in correlations.     

A Bootstrap Approach for Testing Marginal Independence Between Two Categorical Variables When Subjects Have Repeated Responses

To assess independence in two-way contingency tables, the Pearson chi-square test or Fisher’s exact test are typically used. These tests assume that each subject contributes at most one count to only one table cell (e.g., sex versus blood type). In other situations, each subject may have more than one count contributing to the table and these counts may occur in different cells of the table. One may wish to test independence, adjusting for the within-subject correlation. We provide a simple nonparametric bootstrap approach and assess its performance through simulation studies. The method is illustrated on subjects with multiple mental health presentations to Emergency Departments.     

Partial Ordering and Aging Properties of Order Statistics When the Sample Size is Random: A Brief Review

In many practical situations, order statistics arise naturally with random sample size. In this article, we review results on partial orderings and aging properties of such order statistics. The comparison of order statistics for different sample sizes is also discussed here.     

A Model Selection Criterion for Discriminant Analysis of Several Groups When the Dimension is Larger than the Total Sample Size

This article deals with a criterion for selection of variables for the multiple group discriminant analysis in high-dimensional data. The variable selection models considered for discriminant analysis in Fujikoshi (1985 Fujikoshi , Y. ( 1985 ). Selection of variables in discriminant analysis and canonical correlation analysis . In: Krishnaiah , P. R. , ed. Multivariate Analysis . Vol. VI. Amsterdam : North-Holland , pp. 219 – 236 . [Google Scholar], 2002 Fujikoshi , Y. ( 2002 ). Selection of variables for discriminant analysis in a high-dimensional case . Sankhya Ser. A 64 : 256 – 257 . [Google Scholar]) are the ones based on additional information due to Rao (1948 Rao , C. R. ( 1948 ). Tests of significance in multivariate analysis . Biometrika 35 : 58 – 79 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar], 1970 Rao , C. R. ( 1970 ). Inference on discriminant function coefficients . In: Bose , R. C. , ed. Essays in Probability and Statistics . Chapel Hill , NC : University of North Carolina Press , pp. 537 – 602 . [Google Scholar]). Our criterion is based on Akaike information criterion (AIC) for this model. The AIC has been successfully used in the literature in model selection when the dimension p is smaller than the sample size N. However, the case when p > N has not been considered in the literature, because MLE can not be estimated corresponding to singularity of the within-group covariance matrix. A popular method used to address the singularity problem in high-dimensional classification is the regularized method, which replaces the within-group sample covariance matrix with a ridge-type covariance estimate to stabilize the estimate. In this article, we propose AIC-type criterion by replacing MLE of the within-group covariance matrix with ridge-type estimator. This idea follows Srivastava and Kubokawa (2008 Srivastava , M. S. , Kubokawa , T. ( 2008 ). Akaike information criterion for selecting components of the mean vector in high dimensional data with fewer observations . J. Japan Statist. Soc. 38 : 259 – 283 . [Google Scholar]). Simulations revealed that our proposed criterion performs well.