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Local Linear Estimation for Spatial Random Processes with Stochastic Trend and Stationary Noise

Author

Listed:
  • Jung Won Hyun

    (St. Jude Children’s Research Hospital)

  • Prabir Burman

    (University of California at Davis)

  • Debashis Paul

    (University of California at Davis)

Abstract

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 $\mathbb {R}^{d}$ 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 Cp criterion. We illustrate the methodology by simulation studies and by analyzing a data on surface temperature anomalies.

Suggested Citation

  • Jung Won Hyun & Prabir Burman & Debashis Paul, 2018. "Local Linear Estimation for Spatial Random Processes with Stochastic Trend and Stationary Noise," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 80(2), pages 369-394, November.
  • Handle: RePEc:spr:sankhb:v:80:y:2018:i:2:d:10.1007_s13571-018-0155-4
    DOI: 10.1007/s13571-018-0155-4
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    References listed on IDEAS

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    1. Burman, Prabir, 1991. "Regression function estimation from dependent observations," Journal of Multivariate Analysis, Elsevier, vol. 36(2), pages 263-279, February.
    2. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178, Decembrie.
    3. Harvey,Andrew C., 1991. "Forecasting, Structural Time Series Models and the Kalman Filter," Cambridge Books, Cambridge University Press, number 9780521405737, June.
    4. Sally A. Wood, 2002. "Bayesian mixture of splines for spatially adaptive nonparametric regression," Biometrika, Biometrika Trust, vol. 89(3), pages 513-528, August.
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