Gao, Jiti and Lu, Zudi and Tjostheim, Dag
(2003):
*Semiparametric spatial regression: theory and practice.*

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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 four dimensional nonparametric regression. In this paper, a semi-parametric spatial regression approach is proposed to avoid this problem. An estimation procedure based on combining the so-called marginal integration technique with local linear kernel estimation is developed in the semi-parametric spatial regression setting. Asymptotic distributions are established under some mild conditions. The same convergence rates as in the one-dimensional regression case are established. An application of the methodology to the classical Mercer 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 |
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Original Title: | Semiparametric spatial regression: theory and practice |

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: | 11991 |

Depositing User: | jiti Gao |

Date Deposited: | 08 Dec 2008 08:38 |

Last Modified: | 06 Oct 2019 06:47 |

References: | Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems. J. Roy. Statist. Soc. Ser. B 36, 192-225. Bjerve, S. and Doksum, K. (1993). Correlation curves: measures of association as functions of covariate values. Ann. Statist. 21, 890-902. Bolthausen, E. (1982). On the central limit theorem for stationary random fields. Ann. Probab. 10, 1047-1050. Carbon, M., Hallin, M. and Tran, L.T. (1996). Kernel density estimation for random fields: the L_1 theory. J. Nonparametr. Statist. 6, 157-170. Chiles, J.-P. and Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty. John Wiley, New York. Cressie, N. A. C. (1993). Statistics for Spatial Data. John Wiley, New York. Fan, J., Hardle, W. and Mammen, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26, 943-971. Gao, J. (1998). Semiparametric regression smoothing of nonlinear time series. Scand. J. Statist. 25, 521-539. Gao, J. and King, M. L. (2005). Model estimation and specification testing in nonparametric and semiparametric regression. Unpublished report. Available from www.maths.uwa.edu.au/~jiti/jems.pdf. Gao, J., Lu, Z. and Tj{\o}stheim, D. (2004). Moment inequality for spatial processes. Unpublished technical report. Available from www.maths.uwa.edu.au/~jiti/glt04.pdf. Guyon, X. (1995). Random Fields on a Network. Springer-Verlag, New York. Guyon, X. and Richardson, S. (1984). Vitesse de convergence du th\'{e}oreme de la limite centrale pour des champs faiblement d\'{e}pendants. Z. Wahrsch. Verw. Gebiete 66, 297-314. {\sc Hallin, M., Lu, Z. and Tran, L. T.} (2001). Density estimation for spatial linear processes. Bernoulli 7, 657-668. {\sc Hallin, M., Lu, Z. and Tran, L. T.} (2004a). Density estimation for spatial processes: L_1 theory. J. Multi. Anal 88, 61-75. {\sc Hallin, M., Lu, Z. and Tran, L. T.} (2004b). Local linear spatial regression. Ann. Statist. 32, 2469-2500. {\sc H\"{a}rdle, W., Liang, H. and Gao, J.} (2000). Partially Linear Models. Springer--Verlag, New York. {\sc Hengartner, N. W. and Sperlich, S.} (2003). Rate optimal estimation with the integration method in the presence of many covariates. Available from www.maths.uwa.edu.au/~jiti/hs.pdf. {\sc Jones, M.C. and Koch, I.} (2003). Dependence maps: local dependence in practice. Statistics and Computing 13, 241-255. {\sc Lin, Z. and Lu, C.} (1996). Limit Theory for Mixing Dependent Random Variables. Kluwer Academic Publishers, London. {\sc Lu, Z. and Chen, X.} (2002). Spatial nonparametric regression estimation: Non-isotropic case. Acta Mathematicae Applicatae Sinica, English Series (Springer-Verlag) 18, 641-656. {\sc Lu, Z. and Chen, X.} (2004). Spatial kernel regression estimation: weak consistency. Statist. & Probab. Lett. 68, 125-136. {\sc Lu, Z., Lundervold, A., Tj{\o}stheim, D. and Yao, Q.} (2005). Exploring spatial nonlinearity through additive approximation. Discussion paper, Department of Statistics, London School of Economics, London, U.K. {\sc Mammen, E., Linton, O. and Nielsen, J. P.} (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions. Ann. Statist. 27, 1443-1490. {\sc McBratney, A. B. and Webster, R.} (1981). Detection of ridge and furrow pattern by spectral analysis of crop yield. Intern. Statist. Rev. 49, 45-52. {\sc Mercer, W. B. and Hall, A. D.} (1911). The experimental error field trials. J. Agricultural Science 4, 107-132. {\sc Napahetian, B. S.} (1987). An approach to limit theorems for dependent random variables. Probab. Theory Appl. 32, 589-594. {\sc Neaderhouser, C. C.} (1980). Convergence of blocks spins defined on random fields. J. Statist Phys. 22, 673-684. {\sc Nielsen, J. P. and Linton, O. B.} (1998). An optimization interpretation of integration and back-fitting estimators for separable nonparametric models. J. Roy. Statist. Soc. Ser. B 60, 217-222. {\sc Possolo, A.} (1991). Spatial Statistics and Imaging. Institute of Mathematical Statistics. Lecture notes-monograph series, New York. {\sc Rivoirard, J.} (1994). Introduction to Disjunctive Kriging and Non-linear Geostatistics. Clarendon Press, Oxford. {\sc Rosenblatt, M.} (1985). Stationary Sequences and Random Fields. Birkhauser, Boston. {\sc Sperlich, S., Tj{\o}stheim, D. and Yang, L.} (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18, 197-251. {\sc Stein, M. L.} (1999). Interpolation of Spatial Data. Springer-Verlag, New York. {\sc Takahata, H.} (1983). On the rates in the central limit theorem for weakly dependent random fields. Z. Wahrsch. Verw. Gebiete 64, 445-456. {\sc Tran, L. T.} (1990). Kernel density estimation on random fields. J. Multi. Anal. 34. 37-53. {\sc Tran, L. T. and Yakowitz, S.} (1993). Nearest neighbor estimators for random fields. J. Multi. Anal. 44, 23-46. {\sc Wackernagel, H.} (1998). Multivariate Geostatistics. Springer-Verlag, Berlin. {\sc Whittle, P.} (1954). On stationary processes in the plane. Biometrika 41, 434-449. {\sc Whittle, P.} (1963). Stochastic process in several dimensions. Bull. Int. Statist. Inst. 40, 974-985. {\sc Winkler, G.} (1995). Image Analysis, Random Fields and Dynamic Monte Carlo Methods. Springer-Verlag, Berlin. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/11991 |