## The Annals of Statistics

### Estimation in semiparametric spatial regression

#### 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 semiparametric 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 semiparametric 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 and Hall wheat data set is given and indicates that one directional component appears to be nonlinear, which has gone unnoticed in earlier analyses.

#### Article information

Source
Ann. Statist., Volume 34, Number 3 (2006), 1395-1435.

Dates
First available in Project Euclid: 10 July 2006

https://projecteuclid.org/euclid.aos/1152540753

Digital Object Identifier
doi:10.1214/009053606000000317

Mathematical Reviews number (MathSciNet)
MR2278362

Zentralblatt MATH identifier
1113.62048

#### Citation

Gao, Jiti; Lu, Zudi; Tjøstheim, Dag. Estimation in semiparametric spatial regression. Ann. Statist. 34 (2006), no. 3, 1395--1435. doi:10.1214/009053606000000317. https://projecteuclid.org/euclid.aos/1152540753

#### References

• Besag, J. E. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion). J. Roy. Statist. Soc. Ser. B 36 192--236.
• 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 mixing 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.
• Chilès, J.-P. and Delfiner, P. (1999). Geostatistics: Modeling Spatial Uncertainty. Wiley, New York.
• Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley, New York.
• Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London.
• Fan, J., Härdle, 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). Estimation and model specification testing in nonparametric and semiparametric regression models. Unpublished report. Available at www.maths.uwa.edu.au/~jiti/jems.pdf.
• Gao, J., Lu, Z. and Tjøstheim, D. (2005). Semiparametric spatial regression: Theory and practice. Unpublished technical report. Available at www.maths.uwa.edu.au/~jiti/glt05.pdf.
• Guyon, X. (1995). Random Fields on a Network. Modeling, Statistics and Applications. Springer, New York.
• Guyon, X. and Richardson, S. (1984). Vitesse de convergence du théorème de la limite centrale pour des champs faiblement dépendants. Z. Wahrsch. Verw. Gebiete 66 297--314.
• Hallin, M., Lu, Z. and Tran, L. T. (2001). Density estimation for spatial linear processes. Bernoulli 7 657--668.
• Hallin, M., Lu, Z. and Tran, L. T. (2004). Kernel density estimation for spatial processes: $L_1$ theory. J. Multivariate Anal. 88 61--75.
• Hallin, M., Lu, Z. and Tran, L. T. (2004). Local linear spatial regression. Ann. Statist. 32 2469--2500.
• Härdle, W., Liang, H. and Gao, J. (2000). Partially Linear Models. Physica-Verlag, Heidelberg.
• Hengartner, N. W. and Sperlich, 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.
• Jones, M. C. and Koch, I. (2003). Dependence maps: Local dependence in practice. Statist. Comput. 13 241--255.
• Lin, Z. and Lu, C. (1996). Limit Theory for Mixing Dependent Random Variables. Kluwer, Dordrecht.
• Lu, Z. and Chen, X. (2002). Spatial nonparametric regression estimation: Non-isotropic case. Acta Math. Appl. Sinica English Ser. 18 641--656.
• Lu, Z. and Chen, X. (2004). Spatial kernel regression estimation: Weak consistency. Statist. Probab. Lett. 68 125--136.
• Lu, Z., Lundervold, A., Tjøstheim, D. and Yao, 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.
• 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.
• McBratney, A. B. and Webster, R. (1981). Detection of ridge and furrow pattern by spectral analysis of crop yield. Internat. Statist. Rev. 49 45--52.
• Mercer, W. B. and Hall, A. D. (1911). The experimental error of field trials. J. Agricultural Science 4 107--132.
• Nielsen, J. P. and Linton, O. B. (1998). An optimization interpretation of integration and back-fitting estimators for separable nonparametric models. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 217--222.
• Possolo, A., ed. (1991). Spatial Statistics and Imaging. IMS, Hayward, CA.
• Rivoirard, J. (1994). Introduction to Disjunctive Kriging and Non-Linear Geostatistics. Clarendon Press, Oxford.
• Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston.
• Sperlich, S., Tjøstheim, D. and Yang, L. (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18 197--251.
• Stein, M. L. (1999). Interpolation of Spatial Data. Some Theory for Kriging. Springer, New York.
• Tran, L. T. (1990). Kernel density estimation on random fields. J. Multivariate Anal. 34 37--53.
• Tran, L. T. and Yakowitz, S. (1993). Nearest neighbor estimators for random fields. J. Multivariate Anal. 44 23--46.
• Wackernagel, H. (1998). Multivariate Geostatistics: An Introduction With Applications, 2nd ed. Springer, Berlin.
• Whittle, P. (1954). On stationary processes in the plane. Biometrika 41 434--449.
• Whittle, P. (1963). Stochastic process in several dimensions. Bull. Inst. Internat. Statist. 40 974--994.