A note on multivariate location and scatter statistics for sparse data sets

For a pp-dimensional data set of size n≤p+1n≤p+1 in general position, it is shown that the only affine equivariant multivariate location statistic is the sample mean vector and that any affine equivariant scatter matrix must be proportional to the sample covariance matrix, with the proportionality constant not being dependent on the data.