On Robust Mahalanobis Distance Issued from Fast Mcd and Mvv

In modern activities such as banking, homeland security, information transportation, telecommunication, etc., people work with large and high dimension data sets. But, the higher the dimension the higher the probability that outliers will be present in the data sets. The ability to detect outliers in high dimension multivariate data sets is a challenging task. In this circumstance, robust estimates of location and scale are needed. One of the primary problems encountered in robust estimation of location and scale is to ensure that the estimators are highly robust and computationally efficient. The most popular and widely used highly robust method to estimate such parameters is the so-called fast minimum covariance determinant (FMCD). Although it satisfies the desirable statistical properties such as high breakdown point, affine-equivariant, and bounded influence function, however, its computational efficiency, which is as important as its effectiveness, is lower when the data sets are of high dimension. It is a direct consequence of the use of Mahalanobis squared distance (MSD), which needs the inversion of covariance matrix, in data ordering process and the use of covariance determinant as the objective function. It is known that covariance matrix inversion and covariance determinant have the same high order of computational complexity, i.e., O(p) where p is the number of variables. In this paper we use vector variance, introduced by Djauhari (2007) as a measure of multivariate dispersion, and then define an alternative objective function to increase the computational efficiency. Through simulation experiments we compare the effectiveness and the computational efficiency of MVV algorithm, introduced by Herwindiati et al. (2007), with those of FMCD algorithm. The two algorithms have the same structures and only differ in their objective functions. If the objective function of FMCD is by minimizing the covariance determinant, which of MVV is by minimizing the vector variance. Simulation experiments will show that MVV is as effective as FMCD. More precisely, the two algorithms give the same results. Furthermore, MVV has higher computational efficiency than that of FMCD.