JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.A flexible method is introduced to model the structure of a covariance matrix C and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix A, the matrix exponential transformation, C = exp (A), is a positive definite matrix. Because there is no constraint on the possible values of the upper triangular elements on A, any possible structure of interest can be imposed on them. The method presented here is not intended to replace the existing special models available for a covariance matrix, but rather to provide a broad range of further structures that supplements existing methodology. Maximum likelihood estimation procedures are used to estimate the parameters, and the large-sample asymptotic properties are obtained. A simulation study and two real-life examples are given to illustrate the method introduced. KEY WORDS: Covariance matrix; Golden-Thompson inequality; Matrix exponential transformation; Maximum likelihood estimation; Volterra integral equation. Tom Y. M. Chiu is Statistician, SPSS, Inc., Chicago, IL 60611. Tom Leonard is Associate Professor and Kam-Wah Tsui is Professor, Department of Statistics, University of Wisconsin, Madison, WI 53706. The authors are grateful to , two anonymous referees, and an associate editor for many helpful suggestions. Thanks are also due to Sook-Fwe Yap for computing advice and to Karen Pridham and the University of Wisconsin School of Nursing for providing the medical data set. The initial stages of this research were supported under the Army Research Office (ARO) Contract DAAG29-80-C-0041. of C. Because As = T[log(D)]5T' for any nonnegative integer s, substituting for As in (1) readily confirms that C = TDT'. Hence the matrix logarithmic transformation takes only logs of the eigenvalues of C, while leaving the eigenvectors unchanged. Several properties of the transformations (1) and (2) are now described when C is p x p positive definite matrix. a. Invariance under rotations: Let W denote any p x p orthonormal matrix. Then (2) implies that log(WCW') = WAW' = W[log C]W'. (3) b. The generalized variance: The determinant ICI satisfies log | C = tr(A),where tr(*) denotes the trace of a matrix. c. The inverse C-1 satisfies C' = exp{-A}.(5) d. It is not generally true that if Ct is some t x t positive definite submatrix of C consisting of t rows of C and the corresponding t columns, then
