ABSTRACT: In this article, we consider testing certain hypotheses concerning the covariance matrix Σ when the number of observations N=n+1 on the p-dimensional random vector x, distributed as normal, is less than p, n<p, and n/p goes to zero. Specifically, we consider testing Σ=σ²I_p, Σ=I_p, Σ=Λ, a diagonal matrix, and Σ=σ² [(1-ρ)I_p+ρ1_p1_p'], an intraclass correlation structure, where 1_p=(1,1,...,1)', is a p-row vector of ones, and Ip is the p×p identity matrix. The first two tests are the adapted versions of the likelihood ratio tests when n>p, p-fixed, and p/n goes to zero, to the case when n<p, n-fixed, and n/p goes to zero. The third test is the normalized version of the Fisher's z-transformation which is shown to be asymptotically normally distributed as n and p go to infinity (irrespective of the manner). A test for the fourth hypothesis is constructed using the spherecity test for a (p-1) dimensional vector but this test can only reject the hypothesis, that is, if the hypothesis is not rejected, it may not imply that the hypothesis is true. The first three tests are compared with some recently proposed tests.

ABSTRACT: This paper is concerned with the problem of selecting the best subset of variables in canonical correlation analysis. Our method uses Akaike's information criterion to a variable selection model based on redundancy of a subset of variables. First we derive its corrected AIC. It is noted that the variable selection model can be formulated in the term of a conditional independence structure. We also derive corrected AIC for some similar conditional independence structures.

ABSTRACT: To select the number of factors in the factor analysis, we consider the following testing procedure: (i) Testing n against n+1 for the number of factors by the likelihood ratio test statistic is conducted from n=1 as long as the last testing is rejected (ii) When the testing is not rejected, the number of factors in the null hypothesis is selected as the true number. But such a procedure needs a Monte Carlo simulation to evaluate its p-value, because the likelihood ratio test statistic does not follow the chi-square distribution asymptotically as pointed out by Hayashi et al. (2006). To avoid the simulation, we consider some locally conic parametrization to evaluate the asymptotic distribution of the likelihood ratio test statistic, or consider some modified likelihood ratio test statistic which follow the chi-square distribution asymptotically.

ABSTRACT: When data are not missing at random (NMAR), maximum likelihood (ML) procedure will not generate consistent parameter estimates unless the missing data mechanism is correctly modeled. Understanding NMAR mechanism in a data set would allow one to better use the ML methodology. A survey or questionnaire may contain many items, certain items may be responsible for NMAR values in other items. The paper develops statistical procedures to identify the responsible items. By comparing ML estimates (MLE), several statistics are developed to test whether the MLEs are changed when excluding items. The items that cause a significant change of the MLEs are responsible for the NMAR mechanism. Normal distribution is used for obtaining the MLEs, a sandwich-type covariance matrix is used to account for distribution violations. The class of nonnormal distributions within which the procedure is valid is provided. Both the saturated model and structural models are considered. Effect sizes are also defined and studied. The results indicate that more missing data in a sample does not necessarily imply more significant test statistics due to smaller effect sizes. Knowing the true population means and covariances or the parameter values in structural equation models may not make things easier either.

ABSTRACT: Ordinal data analysis using correlations based on the assumption that the ordinal data is generated from underlying continuous normal variates is widely used. To avoid the problem of multiple integration over the multivariate normal density function, currently most estimation approaches are based on limited information via separating the target likelihood function, which may lead to non-optimal results. Using a single objective function that involves only double integration, we introduce a pairwise likelihood approach for SEM with ordinal variables. A simulation study to evaluate the performance of the proposed method is described and summarized.