Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chisquared distribution as is the case with those derived from the ordinary covariance matrix. For example, Rousseeuw’s minimum covariance determinant (MCD) is a robust estimator with a high breakdown. However, even in quite large ...

We define the minimum covariance determinant functionals for multivariate location and scatter through trimming functions and establish their existence at any multivariate distribution. We provide a precise characterization including a separating ellipsoid property and prove that the functionals are continuous. Moreover we establish asymptotic normality for both the location and covariance esti...

The Minimum Covariance Determinant (MCD) scatter estimator is a highly robust estimator for the dispersion matrix of a multivariate, elliptically symmetric distribution. It is relatively fast to compute and intuitively appealing. In this note we derive its innuence function and compute the asymptotic variances of its elements. A comparison with the one step reweighted MCD and with S-estimators ...

We propose a simple and semi-parametric estimator for the tail index of a regular varying elliptical random vector. Since, for univariate random variables, our estimator boils down to the Hill estimator and it inherits the simplicity and asymptotic properties, we name it after Bruce M. Hill. The estimator is based on the distance between an elliptical probability contour and the outer – or exce...

Outlier detection is an integral component of statistical modelling and estimation. For highdimensional data, classical methods based on the Mahalanobis distance are usually not applicable. We propose an outlier detection procedure that replaces the classical minimum covariance determinant estimator with a high-breakdown minimum diagonal product estimator. The cut-off value is obtained from the...

The minimum regularized covariance determinant method (MRCD) is a robust estimator for multivariate location and scatter, which detects outliers by fitting matrix to the data. Its regularization ensures that well-conditioned in any dimension. MRCD assumes non-outlying observations are roughly elliptically distributed, but many datasets not of form. Moreover, computation time increases substanti...

The least trimmed squares estimator and the minimum covariance determinant estimator [5] are frequently used robust estimators of regression and of location and scatter. Consistency factors can be computed for both methods to make the estimators consistent at the normal model. However, for small data sets these factors do not make the estimator unbiased. Based on simulation studies we therefore...

Our aim is to construct a factor analysis method that can resist the effect of outliers. We start with a highly robust initial covariance estimator, after which the factors can be obtained from maximum likelihood or from principal factor analysis (PFA). We find that PFA based on the minimum covariance determinant scatter matrix works well. We also derive the influence function of the PFA method...