Stein's Method of Moments

Abstract: Stein operators allow to characterise probability distributions via differential operators. Based on these characterisations, we develop a new method of point estimation for marginal parameters of strictly stationary and ergodic processes, which we call \emph{Stein's Method of Moments} (SMOM). These SMOM 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. Moreover, for i.i.d.\ observations, we retrieve data-dependent functions that result in asymptotically efficient estimators and give a sequence of explicit SMOM estimators that converge to the maximum likelihood estimator. Our simulation study demonstrates that for a number of important univariate continuous probability distributions our SMOM estimators possess excellent small sample behaviour, often outperforming the maximum likelihood estimator and other widely-used methods in terms of lower bias and mean squared error. We also illustrate the pertinence of our approach on a real data set related to rainfall modelisation.