Local likelihood density estimation on random fields

Local likelihood methods hold considerable promise in density estimation. They offer unmatched flexibility and adaptivity as the resulting density estimators inherit both of the best properties of nonparametric approaches and parametric inference. However, the adoption of local likelihood methods with dependent observations, in particular with random fields, is inhibited by lack of knowledge about their properties in the case. In the present paper we detail asymptotic properties of the local likelihood density estimators for stationary random fields in the usual smoothing context of the bandwidth, h, tending to zero as the sample size tends to infinity. The asymptotic analysis is substantially more complex than in ordinary kernel density estimation on random fields.