M-estimation with incomplete and dependent multivariate data

We extend the theory of M-estimation to incomplete and dependent multivariate data. ML-estimation can still be considered a special case of M-estimation in this context. We notice that the unobserved data must be missing completely at random but not only missing at random, which is a typical assumption of ML-estimation, to guarantee the consistency of an M-estimator. Further, we show that the weight functions for scatter must satisfy a critical scaling condition, which is implicitly fulfilled both by the Gaussian and by Tyler’s weight function. We generalize this principal result by introducing the class of power weight functions, which contains the two aforementioned weight functions as limiting cases. A simulation study confirms our theoretical findings. If the data are heavy tailed or contaminated, the M-estimators turn out to be favorable compared to the ML-estimators that are based on the normal-distribution assumption.