×

Estimation in semiparametric spatial regression. (English) Zbl 1113.62048

Summary: 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 W. B. Mercer and A. D. Hall [J. Agricult. Sci. 4, 107–132 (1911)] wheat data set is given and indicates that one directional component appears to be nonlinear, which has gone unnoticed in earlier analyses.

MSC:

62G08 Nonparametric regression and quantile regression
62M30 Inference from spatial processes
60J25 Continuous-time Markov processes on general state spaces

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. JSTOR: · Zbl 0327.60067
[2] Bjerve, S. and Doksum, K. (1993). Correlation curves: Measures of association as functions of covariate values. Ann. Statist. 21 890–902. · Zbl 0817.62025 · doi:10.1214/aos/1176349156
[3] Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields. Ann. Probab. 10 1047–1050. · Zbl 0496.60020 · doi:10.1214/aop/1176993726
[4] 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. · Zbl 0872.62040 · doi:10.1080/10485259608832669
[5] Chilès, J.-P. and Delfiner, P. (1999). Geostatistics : Modeling Spatial Uncertainty . Wiley, New York. · Zbl 0922.62098
[6] Cressie, N. A. C. (1993). Statistics for Spatial Data . Wiley, New York. · Zbl 0825.62477
[7] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications . Chapman and Hall, London. · Zbl 0873.62037
[8] Fan, J., Härdle, W. and Mammen, E. (1998). Direct estimation of low-dimensional components in additive models. Ann. Statist. 26 943–971. · Zbl 1073.62527 · doi:10.1214/aos/1024691083
[9] Gao, J. (1998). Semiparametric regression smoothing of nonlinear time series. Scand. J. Statist. 25 521–539. · Zbl 0921.62109 · doi:10.1111/1467-9469.00118
[10] 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.
[11] 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.
[12] Guyon, X. (1995). Random Fields on a Network. Modeling, Statistics and Applications . Springer, New York. · Zbl 0839.60003
[13] 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. · Zbl 0531.60019 · doi:10.1007/BF00531535
[14] Hallin, M., Lu, Z. and Tran, L. T. (2001). Density estimation for spatial linear processes. Bernoulli 7 657–668. · Zbl 1005.62034 · doi:10.2307/3318731
[15] Hallin, M., Lu, Z. and Tran, L. T. (2004). Kernel density estimation for spatial processes: \(L_1\) theory. J. Multivariate Anal. 88 61–75. · Zbl 1032.62033 · doi:10.1016/S0047-259X(03)00060-5
[16] Hallin, M., Lu, Z. and Tran, L. T. (2004). Local linear spatial regression. Ann. Statist. 32 2469–2500. · Zbl 1069.62075 · doi:10.1214/009053604000000850
[17] Härdle, W., Liang, H. and Gao, J. (2000). Partially Linear Models . Physica-Verlag, Heidelberg. · Zbl 0968.62006
[18] 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. · Zbl 1070.62021
[19] Jones, M. C. and Koch, I. (2003). Dependence maps: Local dependence in practice. Statist. Comput. 13 241–255. · doi:10.1023/A:1024270700807
[20] Lin, Z. and Lu, C. (1996). Limit Theory for Mixing Dependent Random Variables . Kluwer, Dordrecht. · Zbl 0889.60001
[21] Lu, Z. and Chen, X. (2002). Spatial nonparametric regression estimation: Non-isotropic case. Acta Math. Appl. Sinica English Ser. 18 641–656. · Zbl 1019.62039 · doi:10.1007/s102550200067
[22] Lu, Z. and Chen, X. (2004). Spatial kernel regression estimation: Weak consistency. Statist. Probab. Lett. 68 125–136. · Zbl 1058.62079 · doi:10.1016/j.spl.2003.08.014
[23] 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. · Zbl 1127.62087
[24] 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. · Zbl 0986.62028
[25] 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.
[26] Mercer, W. B. and Hall, A. D. (1911). The experimental error of field trials. J. Agricultural Science 4 107–132.
[27] 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. JSTOR: · Zbl 0909.62040 · doi:10.1111/1467-9868.00120
[28] Possolo, A., ed. (1991). Spatial Statistics and Imaging . IMS, Hayward, CA. · Zbl 0760.00005
[29] Rivoirard, J. (1994). Introduction to Disjunctive Kriging and Non-Linear Geostatistics . Clarendon Press, Oxford. · Zbl 0811.65063
[30] Rosenblatt, M. (1985). Stationary Sequences and Random Fields . Birkhäuser, Boston. · Zbl 0597.62095
[31] Sperlich, S., Tjøstheim, D. and Yang, L. (2002). Nonparametric estimation and testing of interaction in additive models. Econometric Theory 18 197–251. JSTOR: · Zbl 1109.62310 · doi:10.1017/S0266466602182016
[32] Stein, M. L. (1999). Interpolation of Spatial Data. Some Theory for Kriging . Springer, New York. · Zbl 0924.62100
[33] Tran, L. T. (1990). Kernel density estimation on random fields. J. Multivariate Anal. 34 37–53. · Zbl 0709.62085 · doi:10.1016/0047-259X(90)90059-Q
[34] Tran, L. T. and Yakowitz, S. (1993). Nearest neighbor estimators for random fields. J. Multivariate Anal. 44 23–46. · Zbl 0764.62076 · doi:10.1006/jmva.1993.1002
[35] Wackernagel, H. (1998). Multivariate Geostatistics: An Introduction With Applications , 2nd ed. Springer, Berlin. · Zbl 0912.62131
[36] Whittle, P. (1954). On stationary processes in the plane. Biometrika 41 434–449. JSTOR: · Zbl 0058.35601 · doi:10.1093/biomet/41.3-4.434
[37] Whittle, P. (1963). Stochastic process in several dimensions. Bull. Inst. Internat. Statist. 40 974–994. · Zbl 0129.10603
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.