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

Abstract

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.

Keywords:

• linearly structured correlation matrix
• maximum likelihood estimate of a vector parameter
• vec operator
• matrix and vector norms
• Jacobian matrix
• convergence of an iterative scheme
• Toeplitz matrix

AMS Subject Classification :

• 15A18
• 15A60
• 62H25

Notes

Related R code is available to the readers from the author on request.