Suppose one knows that a distribution is essentially a multivariate Gaussian, but that some of the points are contaminated. The theory of M-estimation shows that if you start with a possibly crude estimate of the shape/covariance of the distribution, reweight each point by its Mahalanobis distance, recomputed the shape/covariance from these weighted samples, etc - you will converge in many cases to the true covariace of your inliers.
This works because for a normal distribution, the Mahalanobis distance, $x^T\Sigma x$ is the measure of how far a data point is from the center of the distribution.
Does this generalize to arbitrary distributions? I suppose one could find the possibly messy transformations of the pdf of a given distribution, for example gamma, to the standard normal, and translate the notion of distance in the normal setting to the distribution of interest. But this seems generally unwieldy. Has an idea along these lines been done anywhere using some notion of statistical distance? That is a general notion of how far away a point is from a given distribution?