## Bernoulli

• Bernoulli
• Volume 13, Number 2 (2007), 447-472.

### Exploring spatial nonlinearity using additive approximation

#### Abstract

We propose to approximate the conditional expectation of a spatial random variable given its nearest-neighbour observations by an additive function. The setting is meaningful in practice and requires no unilateral ordering. It is capable of catching nonlinear features in spatial data and exploring local dependence structures. Our approach is different from both Markov field methods and disjunctive kriging. The asymptotic properties of the additive estimators have been established for $α$-mixing spatial processes by extending the theory of the backfitting procedure to the spatial case. This facilitates the confidence intervals for the component functions, although the asymptotic biases have to be estimated via (wild) bootstrap. Simulation results are reported. Applications to real data illustrate that the improvement in describing the data over the auto-normal scheme is significant when nonlinearity or non-Gaussianity is pronounced.

#### Article information

Source
Bernoulli, Volume 13, Number 2 (2007), 447-472.

Dates
First available in Project Euclid: 18 May 2007

https://projecteuclid.org/euclid.bj/1179498756

Digital Object Identifier
doi:10.3150/07-BEJ5093

Mathematical Reviews number (MathSciNet)
MR2331259

Zentralblatt MATH identifier
1127.62087

#### Citation

Lu, Zudi; Lundervold, Arvid; Tjøstheim, Dag; Yao, Qiwei. Exploring spatial nonlinearity using additive approximation. Bernoulli 13 (2007), no. 2, 447--472. doi:10.3150/07-BEJ5093. https://projecteuclid.org/euclid.bj/1179498756

#### References

• Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion)., J. Roy. Statist. Soc. Ser. B 36 192–236.
• Bolthausen, E. (1982). On the central limit theorem for stationary mixing random fields., Ann. Probab. 4 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. New York: Wiley.
• Cressie, N.A.C. (1993)., Statistics for Spatial Data. New York: Wiley.
• Doukhan, P. (1994)., Mixing: Properties and Examples. New York: Springer-Verlag.
• Diggle, P.J. (1985). A kernel method for smoothing point process data., Appl. Statist. 34 138–147.
• Diggle, P.J. and Marron, J.S. (1988). Equivalence of smoothing parameter selectors in density and intensity estimation., J. Amer. Statist. Assoc. 83 793–800.
• Fan, J., Härdle, W. and Mammen, E. (1998). Direct estimation of additive and linear components for high-dimensional data., Ann. Statist. 26 943–971.
• Gao, J., Lu, Z. and Tjøstheim, D. (2006). Estimation in semiparametric spatial regression., Ann. Statist. 34 1395–1435.
• 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. (2004a). Density estimation for spatial processes: the $L_1$ theory., J. Multivariate Anal. 88 61–75.
• Hallin, M., Lu, Z. and Tran, L.T. (2004b). Local linear spatial regression., Ann. Statist. 32 2469–2500.
• Linton, O. and Nielsen, J.P. (1995). A kernel method of estimating structured nonparametric regression based on marginal integration., Biometrika 82 93–101.
• Mammen, E., Linton, O. and Nielsen, J. (1999). The existence and asymptotic properties of a backfitting projection algorithm under weak conditions., Ann. Statist. 27 1443–1490.
• Mammen, E. and Park, B.U. (2005). Bandwidth selection for smooth backfitting in additive models., Ann. Statist. 33 1260–1294.
• Mammen, E. and Park, B.U. (2006). A simple smooth backfitting method for additive models., Ann. Statist. 34 2252–2271.
• Matheron, G. (1973). Le krigeage disjonctif. Technical Report N-360, Centre de Géostatistique, Fontainebleau, France.
• Matheron, G. (1976). A simple substitute for conditional expectation: The disjunctive kriging. In M. Guarascio, M. David and C. Huijbregts (eds), Advanced Geostatistic in the Mining Industry NATO Sci. Ser. C Math. Phys. Sci. 24, pp. 221–236. Dordrecht: Kluwer.
• Newey, W.K. (1994). Kernel estimation of partial means., Econometric Theory 10 233–253.
• Nielsen, J.P. and Sperlich, S. (2004). Smooth backfitting in practice., J. Roy. Statist. Soc. Ser. B 67 43–61.
• Perera, G. (2001). Random fields on $Z^d$: Limit theorems and irregular sets. In M. Moore (ed.), Spatial Statistics: Methodological Aspects and Applications, pp. 57–82. New York: Springer-Verlag.
• Rivoirard, J. (1994)., Introduction to Disjunctive Kriging and Non-linear Geostatistics. Oxford: Clarendon Press.
• Tjøstheim, D. and Auestad, B. (1994). Nonparametric identification of nonlinear time series: projections., J. Amer. Statist. Assoc. 89 1398–1409.
• Whittle, P. (1954). On stationary processes in the plane., Biometrika 41 434–449.
• Yao, Q. (2003). Exponential inequalities for spatial processes and uniform convergence rates for density estimation. In H. Zhang and J. Huang (eds), Development of Modern Statistics and Related Topics –- In Celebration of Professor Yaoting Zhang's 70th Birthday, pp. 118–128. Singapore: World Scientific.