Point estimation through Stein's method

Stein operators allow to characterise probability distributions via differential operators. We use these characterisations to obtain a new class of point estimators for marginal parameters of strictly stationary and ergodic processes. These so-called Stein estimators satisfy the desirable classical properties such as consistency and asymptotic normality. As a consequence of the usually simple form of the operator, we obtain explicit estimators in cases where standard methods such as (pseudo-) maximum likelihood estimation require a numerical procedure to calculate the estimate. In addition, with our approach, one can choose from a large class of test functions which allows to improve significantly on the moment estimator. For several probability laws, we can determine an estimator that shows an asymptotic behaviour close to efficiency in the i.i.d.\ case. Moreover, for i.i.d. observations, we retrieve data-dependent functions that result in asymptotically efficient estimators and give a sequence of explicit Stein estimators that converge to the maximum likelihood estimator.