James R. Schott

Determining the Dimensionality in Sliced Inverse Regression

Abstract A general regression problem is one in which a response variable can be expressed as some function of one or more different linear combinations of a set of explanatory variables as well as a random error term. Sliced inverse regression is a method for determining these linear combinations. In this article we address the problem of determining how many linear combinations are involved. Procedures based on conditional means and conditional covariance matrices, as well as a procedure combining the two approaches, are considered. In each case we develop a test that has an asymptotic chi-squared distribution when the vector of explanatory variables is sampled from an elliptically symmetric distribution.

A test for the equality of covariance matrices when the dimension is large relative to the sample sizes

A simple statistic is proposed for testing the equality of the covariance matrices of several multivariate normal populations. The asymptotic null distribution of this statistic, as both the sample sizes and the number of variables go to infinity, is shown to be normal. Consequently, this test can be used when the number of variables is not small relative to the sample sizes and, in particular, even when the number of variables exceeds the sample sizes. The finite sample size performance of the normal approximation for this method is evaluated in a simulation study.

A multivariate one-way classification model with random effects

In this paper a multivariate generalization of the one-way random effects model is investigated, maximum likelihood estimators are obtained, and the likelihood ratio test is derived for an hypothesis on the rank of the covariance matrix of the random effect vectors. Properties of the likelihood ratio test are investigated. A sequential procedure for determining the rank of the covariance matrix of the random effect vectors is presented.

Some tests for the equality of covariance matrices

A Wald statistic, which is asymptotically equivalent to the likelihood ratio criterion, is obtained for the test of the equality of covariance matrices. A more general Wald statistic is constructed under the assumption of elliptical distributions, and the comparison of these two statistics sheds some light on the asymptotic performance of the likelihood ratio test. In particular, we find that the likelihood ratio test is liberal for nonnormal elliptical populations with positive kurtosis and conservative for nonnormal elliptical populations with negative kurtosis. Further, the likelihood ratio test cannot be adjusted by a scalar multiple so as to retain its asymptotic chi-squared distribution over the class of elliptical distributions. A Wald test, appropriate for more general populations, is also obtained.

Dimensionality reduction in quadratic discriminant analysis

Abstract One common objective of many multivariate techniques is to achieve a reduction in dimensionality while at the same time retain most of the relevant information contained in the original data set. This reduction not only provides a parsimonious description of the data but, in many cases, also increases the reliability of subsequent analyses of the data. In this paper we consider the problem of determining the minimum dimension necessary for quadratic discrimination in normal populations with heterogeneous covariance matrices. Some asymptotic chi-squared tests are obtained. Simulations are used to investigate the adequacy of the chi-squared approximations and to compare the misclassification probabilities of reduced-dimension quadratic discrimination with full-dimension quadratic discrimination.

Testing for elliptical symmetry in covariance-matrix-based analyses

Many normal-theory test procedures for covariance matrices remain valid outside the family of normal distributions if the matrix of fourth-order moments has structure similar to that of a normal distribution. In particular, for elliptical distributions this matrix of fourth-order moments is a scalar multiple of that for the normal, and for this reason many normal-theory statistics can be adjusted by a scalar multiple so as to retain their asymptotic distributional properties across elliptical distributions. For these analyses, a test for the validity of these scalar-adjusted normal-theory procedures can be viewed as a test on the structure of the matrix of fourth-order moments. In this paper, we develop a Wald statistic for conducting such a test.

A note on the critical values used in stepwise tests for multiplicative components of interaction

A result is presented concerning the null distribution of a statistic used to determine the number of multiplicative components in a fixed two-way model. This result suggests critical values which are compared with previously suggested critical values.

Kronecker product permutation matrices and their application to moment matrices of the normal distribution

In this paper, we consider the matrix which transforms a Kronecker product of vectors into the average of all vectors obtained by permuting the vectors involved in the Kronecker product. An explicit expression is given for this matrix, and some of its properties are derived. It is shown that this matrix is particularly useful in obtaining compact expressions for the moment matrices of the normal distribution. The utility of these expressions is illustrated through some examples.

A Test for a Specific Principal Component of a Correlation Matrix

Abstract In the application of principal components analysis it is common to replace an observed sample principal component vector by another vector closely resembling the sample vector but which is easier to use or interpret. A useful test of hypothesis in this case is one that specifies the true ith principal component. In this article we obtain an asymptotically chi-squared procedure suitable for testing such a hypothesis when the principal components analysis is performed on a correlation matrix. The procedure easily extends to a principal components analysis based on M estimates of scatter.

Estimating correlation matrices that have common eigenvectors

Abstract In this paper we develop a method for obtaining estimators of the correlation matrices from k groups when these correlation matrices have the same set of eigenvectors. These estimators are obtained by utilizing the spectral decomposition of a symmetric matrix; that is, we obtain an estimate, say P , of the matrix P containing the common normalized eigenvectors along with estimates of the eigenvalues for each of the k correlation matrices. It is shown that the rank of the Hadamard product, P P, is a crucial factor in the estimation of these correlation matrices. Consequently, our procedure begins with an initial estimate of P which is then used to obtain an estimate P such that P ⊙ P has its rank less than or equal to some specified value. Initial estimators of the eigenvalues of Ωi, the correlation matrix for the ith group, are then used to obtain refined estimators which, when put in the diagonal matrix Di as its diagonal elements, are such that P D i P ′ has correlation-matrix structure.