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We develop a weighted local likelihood estimate for the parameters thatgovern the local spatial dependency of a locally stationary random field. Theadvantage of this local likelihood estimate is that it smoothly downweights theinfluence of far away observations, works for irregular sampling locations, andwhen designed appropriately, can trade bias and variance for reducingestimation error. This paper starts with an exposition of our technique on theproblem of estimating an unknown positive function when multiplied by astationary random field. This example gives concrete evidence of the benefitsof our local likelihood as compared to na\"ive local likelihoods where thestationary model is assumed throughout a neighborhood. We then discuss thedifficult problem of estimating a bandwidth parameter that controls the amountof influence from distant observations. Finally we present a simulationexperiment for estimating the local smoothness of a local Mat\'ern random fieldwhen observing the field at random sampling locations in $[0,1]^2$.

Local likelihood estimation of local parameters for nonstationary random fields