Gao, Jiti and Lu, Zudi and Tjostheim, Dag (2003): Estimation in semiparametric spatial regression. Published in: Annals of Statistics , Vol. 34, No. 3 (June 2006): pp. 13951435.

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Abstract
Nonparametric methods have been very popular in the last couple of decades in time series and regression, but no such development has taken place for spatial models. A rather obvious reason for this is the curse of dimensionality. For spatial data on a grid evaluating the conditional mean given its closest neighbors requires a fourdimensional nonparametric regression. In this paper a semiparametric spatial regression approach is proposed to avoid this problem. An estimation procedure based on combining the socalled marginal integration technique with local linear kernel estimation is developed in the semiparametric spatial regression setting. Asymptotic distributions are established under some mild conditions. The same convergence rates as in the onedimensional regression case are established. An application of the methodology to the classical Mercer and Hall wheat data set is given and indicates that one directional component appears to be nonlinear, which has gone unnoticed in earlier analyses.
Item Type:  MPRA Paper 

Original Title:  Estimation in semiparametric spatial regression 
Language:  English 
Keywords:  Additive approximation; asymptotic theory; conditional autoregression; local linear kernel estimate; marginal integration; semiparametric regression; spatial mixing process 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General 
Item ID:  11979 
Depositing User:  jiti Gao 
Date Deposited:  09. Dec 2008 00:18 
Last Modified:  17. Feb 2013 18:39 
References:  [1] BESAG, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192–236. [2] BJERVE,S.andDOKSUM, K. (1993). Correlation curves: Measures of association as functions of covariate values. Ann. Statist. 21 890–902. [3] BOLTHAUSEN, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047–1050. [4] CARBON,M.,HALLIN,M.andTRAN, L. T. (1996). Kernel density estimation for random fields: The L1 theory. J. Nonparametr. Statist. 6 157–170. [5] CHILÈS,J.P.andDELFINER, P. (1999). Geostatistics: Modeling Spatial Uncertainty. Wiley, New York. [6] CRESSIE, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York. [7] FAN,J.andGIJBELS, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London. [8] FAN,J.,HÄRDLE,W.andMAMMEN, E. (1998). Direct estimation of lowdimensional components in additive models. Ann. Statist. 26 943–971. [9] GAO, J. (1998). Semiparametric regression smoothing of nonlinear time series. Scand. J. Statist. 25 521–539. [10] GAO,J.andKING, M. L. (2005). Estimation and model specification testing in nonparametric and semiparametric regression models. Unpublished report. Available at www.maths.uwa. edu.au/~jiti/jems.pdf. [11] GAO,J.,LU,Z.andTJØSTHEIM, D. (2005). Semiparametric spatial regression: Theory and practice. Unpublished technical report. Available at www.maths.uwa.edu.au/~jiti/glt05. pdf. [12] GUYON, X. (1995). Random Fields on a Network. Modeling, Statistics and Applications. Springer, New York. [13] GUYON,X.andRICHARDSON, S. (1984). Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants. Z. Wahrsch. Verw. Gebiete 66 297–314. [14] HALLIN,M.,LU,Z.andTRAN, L. T. (2001). Density estimation for spatial linear processes. Bernoulli 7 657–668. [15] HALLIN,M.,LU,Z.andTRAN, L. T. (2004). Kernel density estimation for spatial processes: L1 theory. J. Multivariate Anal. 88 61–75. [16] HALLIN,M.,LU,Z.andTRAN, L. T. (2004). Local linear spatial regression. Ann. Statist. 32 2469–2500. [17] HÄRDLE,W.,LIANG,H.andGAO, J. (2000). Partially Linear Models. PhysicaVerlag, Heidelberg. [18] HENGARTNER,N.W.andSPERLICH, S. (2003). Rate optimal estimation with the integration method in the presence of many covariates. Available at www.maths.uwa.edu.au/~jiti/hs.pdf. [19] JONES,M.C.andKOCH, I. (2003). Dependence maps: Local dependence in practice. Statist. Comput. 13 241–255. [20] LIN,Z.andLU, C. (1996). Limit Theory for Mixing Dependent Random Variables.Kluwer, Dordrecht. [21] LU,Z.andCHEN, X. (2002). Spatial nonparametric regression estimation: Nonisotropic case. Acta Math. Appl. Sinica English Ser. 18 641–656. [22] LU,Z.andCHEN, X. (2004). Spatial kernel regression estimation: Weak consistency. Statist. Probab. Lett. 68 125–136. [23] LU,Z.,LUNDERVOLD,A.,TJØSTHEIM,D.andYAO, Q. (2005). Exploring spatial nonlinearity using additive approximation. Discussion paper, Dept. Statistics, London School of Economics, London. Available at www.maths.uwa.edu.au/~jiti/llty.pdf. [24] MAMMEN,E.,LINTON,O.andNIELSEN, J. P. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27 1443–1490. [25] MCBRATNEY,A.B.andWEBSTER, R. (1981). Detection of ridge and furrow pattern by spectral analysis of crop yield. Internat. Statist. Rev. 49 45–52. [26] MERCER,W.B.andHALL, A. D. (1911). The experimental error of field trials. J. Agricultural Science 4 107–132. [27] NIELSEN,J.P.andLINTON, O. B. (1998). An optimization interpretation of integration and backfitting estimators for separable nonparametric models. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 217–222. [28] POSSOLO, A., ed. (1991). Spatial Statistics and Imaging.IMS,Hayward,CA. [29] RIVOIRARD, J. (1994). Introduction to Disjunctive Kriging and NonLinear Geostatistics. Clarendon Press, Oxford. [30] ROSENBLATT, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston. [31] SPERLICH,S.,TJØSTHEIM,D.andYANG, L. (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18 197–251. [32] STEIN, M. L. (1999). Interpolation of Spatial Data. Some Theory for Kriging. Springer, New York. [33] TRAN, L. T. (1990). Kernel density estimation on random fields. J. Multivariate Anal. 34 37–53. [34] TRAN,L.T.andYAKOWITZ, S. (1993). Nearest neighbor estimators for random fields. J. Multivariate Anal. 44 23–46. [35] WACKERNAGEL, H. (1998). Multivariate Geostatistics: An Introduction With Applications, 2nd ed. Springer, Berlin. [36] WHITTLE, P. (1954). On stationary processes in the plane. Biometrika 41 434–449. MR0067450 [37] WHITTLE, P. (1963). Stochastic process in several dimensions. Bull. Inst. Internat. Statist. 40 974–994. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/11979 