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We consider the problem of estimating the trend for a spatial random process model expressed as Z(x) = μ(x) + ε(x) + δ(x), where the trend μ is a smooth random function, ε(x) is a mean zero, stationary random process, and {δ(x)} are assumed to be i.i.d. noise with zero mean. We propose a new model for stochastic trend in ℝ d by generalizing the notion of a structural model for trend in time series. We estimate the stochastic trend nonparametrically using a local linear regression method and derive the asymptotic mean squared error of the trend estimate under the proposed model for trend. Our results show that the asymptotic mean squared error for the stochastic trend is of the same order of magnitude as that of a deterministic trend of comparable complexity. This result suggests from the point of view of estimation under stationary noise, it is immaterial whether the trend is treated as deterministic or stochastic. Moreover, we show that the rate of convergence of the estimator is determined by the degree of decay of the correlation function of the stationary process ε(x) and this rate can be different from the usual rate of convergence found in the literature on nonparametric function estimation. We also propose a data dependent selection procedure for the bandwidth parameter which is based on a generalization of Mallow's C p criterion. We illustrate the methodology by simulation studies and by analyzing a data on surface temperature anomalies.

We consider the problem of estimating the trend for a spatial random process model expressed as Z ( x ) = μ ( x ) + ε ( x ) + δ ( x ), where the trend μ is a smooth random function, ε ( x ) is a mean zero, stationary random process, and { δ ( x )} are assumed to be i.i.d. noise with zero mean. We propose a new model for stochastic trend in ℝ d by generalizing the notion of a structural model for trend in time series. We estimate the stochastic trend nonparametrically using a local linear regression method and derive the asymptotic mean squared error of the trend estimate under the proposed model for trend. Our results show that the asymptotic mean squared error for the stochastic trend is of the same order of magnitude as that of a deterministic trend of comparable complexity. This result suggests from the point of view of estimation under stationary noise, it is immaterial whether the trend is treated as deterministic or stochastic. Moreover, we show that the rate of convergence of the estimator is determined by the degree of decay of the correlation function of the stationary process ε ( x ) and this rate can be different from the usual rate of convergence found in the literature on nonparametric function estimation. We also propose a data dependent selection procedure for the bandwidth parameter which is based on a generalization of Mallow's C p criterion. We illustrate the methodology by simulation studies and by analyzing a data on surface temperature anomalies.