Estimation of covariance matrices using influence of eigenvectors and eigenvalues

This paper presents a new approach to the estimation of a dispersion matrix from a set of observed data. The approach provides a robust and resistant estimation. Its uniqueness lies in the fact that it is the only technique based upon influence functions of both the eigenvalues and eigenvectors of the estimated matrix. The approach is an iterative one, where at each stage the influence functions of the current estimate are used as weights to adjust the influence that each point has in the estimation process. The motivation behind the use of influence functions is that they measure the amount of change in parameter estimates at a point x. It is important to note that outlying points x do not necessarily unduly influence parameter estimates, while conversely, non-outlying points with large influence may change parameter estimates by nature of their orientation. This paper reviews influence functions of matrices and their associated eigenvalues and eigenvectors. Distributional properties of the sample influence functions are investigated. Several graphical methods for the identification of influential points are presented and discussed. A new covariance estimation procedure called ESI for eigen-structure influence is introduced and its performance is investigated using simulation techniques. The results are compared with the results of several other methods available in the literature. A specific application to biplots and other lower rank applications is discussed.