PROPERTIES OF THE PATTERNED VECH OPERATOR

The vec operator arranges the columns of a matrix one below the other. When the matrix is symmetric such elements are not distinct but an extraction of only the distinct elements on or below the diagonal forms the operation denoted by vech. For other types of patterned matrices a ‘patterned vech’ operator is defined. The transformations from vech to vec are not uniquely defined. Here we examine properties of linear transformations which overcome the lack of uniqueness and develop properties of such linear transformations.     

Maximum likelihood estimation of the linearly structured correlation matrix by a Jacobi-type iterative scheme

Mukherjee and Maiti [Q-procedure for solving likelihood equations in the analysis of covariance structures, Comput. Statist. Quart. 2 (1988), pp. 105–128] proposed an iterative scheme to derive the maximum likelihood estimates of the parameters involved in the population covariance matrix when it is linearly structured. The present investigation provides a Jacobi-type of iterative scheme, MSIII, when the underlying correlation matrix is linearly structured. Such scheme is shown to be quite competent and efficient compared to the prevalent Fisher-scoring (FS) and the Newton–Raphson iterative scheme (NR). An illustrative example is provided for a numerical comparison of the iterates of MSIII, FS and NR choosing the Toeplitz matrix as the population correlation matrix. Numerical behaviour of such schemes is studied in the context of ‘bad’ initial try-out vectors. Additionally a simulation experiment is performed to judge the superiority of MSIII over FS.