A unified approach to decision-theoretic properties of the MLEs for the mean directions of several Langevin distributions

The two-parameter Langevin distribution has been widely used for analyzing directional data. We address the problem of estimating the mean direction in its Cartesian and angular forms. The equivariant point estimation is introduced under different transformation groups. The maximum likelihood estimator (MLE) is shown to satisfy many decision theoretic properties such as admissibility, minimaxity, the best equivariance and risk-unbiasedness under various loss functions. Moreover, it is shown to be unique minimax when the concentration parameter is assumed to be known. These results extend and unify earlier results on the optimality of the MLE. These findings are also established for the problem of simultaneous estimation of mean directions of several independent Langevin populations. Further, estimation of a common mean direction of several independent Langevin populations is studied. A simulation study is carried out to analyze numerically the risk function of the MLE.