Some results on the existence, uniqueness, and computation of the M-estimates of multivariate location and scatter

The M-estimates of multivariate location and scatter are a class of robust alternatives to the sample mean vector and sample covariance matrix. They also have applications to bounded influence regression. There are a number of problems, though, which need to be resolved before they can become widely applicable. This paper addresses the problems of existence, uniqueness and computation of the M-estimates for finite sample sizes.The main contribution of this paper is the introduction of a new algorithm for calculating the scatter component. This algorithm is essentially a fixed point algorithm with a scale adjustment made at each iteration. The algorithm is shown to converge “monotonically” and under less restrictive conditions than those demanded by previous algorithms. The existence and uniqueness of the M-estimates are also established under less restrictive conditions than those previously known. These relaxed conditions are most important for the more robust M-estimates and also when using more conserv...