Sankhya B

, Volume 80, Issue 2, pp 369–394 | Cite as

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

  • Jung Won Hyun
  • Prabir Burman
  • Debashis PaulEmail author


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 \(\mathbb {R}^{d}\) 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.

Keywords and phrases.

Spatial process stochastic trend local polynomial smoothing bandwidth selection Mallows’ Cp 

AMS (2000) subject classification.

Primary: 62G08 Secondary: 62G20 


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Funding Information

Research of the Prabir Burman is partially supported by the NSF grants DMS-9108295, DMS-0907622, DMS-1148643, and NSA grant H98230-04-1-0109. Research of Debashis Paul is partially supported by the NSF grant DMS-1407530 and the NIH grant 1R01EB021707.


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Copyright information

© Indian Statistical Institute 2018

Authors and Affiliations

  1. 1.Department of BiostatisticsSt. Jude Children’s Research HospitalMemphisUSA
  2. 2.Department of StatisticsUniversity of California at DavisDavisUSA

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