An efficient estimator for the expectation $\int f\,\d P$ is constructed, where $P$ is a Gibbs random field, and $f$ is a local statistic, i.\,e. a functional depending on a finite number of coordinates. The estimator coincides with the empirical estimator under the conditions stated in Greenwood and Wefelmeyer \cite{greenwood_wefelmeyer_1999}, and covers the known special cases, namely the von~Mises statistic for the i.i.d. underlying fields and the case of one-dimensional Markov chains.
Gibbs random field, efficient estimator, empirical estimator
62F12, 62M40