On the inadmissibility of Watterson’s estimator

We consider the estimation of the scaled mutation parameter θθ, which is one of the parameters of key interest in population genetics. We provide a general result showing when estimators of θθ can be improved using shrinkage when taking the mean squared error as the measure of performance. As a consequence, we show that Watterson’s estimator is inadmissible, and propose an alternative shrinkage-based estimator that is easy to calculate and has a smaller mean squared error than Watterson’s estimator for all possible parameter values 0<θ<∞0<θ<∞. This estimator is admissible in the class of all linear estimators. We then derive improved versions for other estimators of θθ, including the MLE. We also investigate how an improvement can be obtained both when combining information from several independent loci and when explicitly taking into account recombination. A simulation study provides information about the amount of improvement achieved by our alternative estimators.