Abstract
The expectation of a local function on a stationary random field can be estimated from observations in a large window by the empirical estimator, that is, the average of the function over all shifts within the window. Under appropriate conditions, the estimator is consistent and asymptotically normal. Suppose that the field is a Gibbs field with known finite range of interactions but otherwise unknown potential. We show that the empirical estimator is efficient if and only if the function is (equivalent to) a sum of functions each of which depends only on the values of the field on a clique of sites.
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Greenwood, P.E., Wefelmeyer, W. Characterizing Efficient Empirical Estimators for Local Interaction Gibbs Fields. Statistical Inference for Stochastic Processes 2, 119–134 (1999). https://doi.org/10.1023/A:1009993904851
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DOI: https://doi.org/10.1023/A:1009993904851