Unbiased estimation of squared concentration in the Fisher-von Mises-Langevin distribution and the impossibility of unbiased concentration
This work resolves a fundamental theoretical question in directional statistics, providing a practical unbiased estimator for a key parameter, which is relevant for researchers working with directional data.
The authors prove that unbiased estimation of the concentration parameter in the Fisher-von Mises-Langevin distribution is impossible, and instead propose an alternative parameterization (squared concentration, termed intensity) for which unbiased estimation is possible, providing (almost) unbiased estimators via a partial sum U-statistic.
The estimation of concentration parameter in Fisher-von Mises-Langevin distribution is the directional statistics analogue of the estimation of the precision matrix for the Gaussian distribution. In this work we show that unbiased estimation of this parameter is impossible. With this realization in hand, we provide an alternative parameterization of the Fisher-von Mises-Langevin distribution in terms of the squared concentration, which we term the intensity. We fruther show that unbiased estimation of thereof is possible, and provide (almost) unbiased estimators thereof in terms of a partial sum U-statistic. We showcase our new estimator on synthetic data, New York taxi trip data, and on spherical word embeddings.