On the non-standard distribution of empirical likelihood estimators with spatial data

This note highlights some unusual and unexpected behavior in point estimation using empirical likelihood (EL). In particular, frequency domain formulations of EL, based on the periodogram and estimating functions, have been proposed in the literature for time and spatial processes. However, in contrast to the time series case and most applications of EL, the maximum EL parameter estimator exhibits surprisingly non-standard asymptotic properties for irregularly located spatial data. In fact, a consistent normal limit cannot be guaranteed, as is typical for EL. Despite this, log-ratio EL statistics maintain standard chi-square limits with such spatial data.