On the bias of Mahalanobis distance due to limited sample size effect

The relationship between sample size and the bias of principal components of Mahalanobis distance is studied by computer simulation. The results shows that the bias of Mahalanobis distance in non-dominant components (the components corresponding to smaller eigenvalues of the covariance matrix) are larger than those in dominant components, and that the bias is smaller when the non-dominant eigenvalues are replaced by a larger value. The obtained relationship is helpful to know the sample size needed to estimate mean vectors and covariance matrices. For given sample size, the relationship suggests and determines the number of reliable eigenvectors which should be employed in modified Mahalanobis distance to compensate the bias