Nonparametric relative error regression for spatial random variables

Regular Article
First Online:
Received:
Revised:

Abstract

Let \(\displaystyle Z_{\mathbf {i}}=\left( X_{\mathbf {i}},\ Y_{\mathbf {i}}\right) _{\mathbf {i}\in \mathbb {N}^{N}\, N \ge 1}\), be a \( \mathbb {R}^d\times \mathbb {R}\)-valued measurable strictly stationary spatial process. We consider the problem of estimating the regression function of \(Y_{\mathbf {i}}\) given \(X_{\mathbf {i}}\). We construct an alternative kernel estimate of the regression function based on the minimization of the mean squared relative error. Under some general mixing assumptions, the almost complete consistency and the asymptotic normality of this estimator are obtained. Its finite-sample performance is compared with a standard kernel regression estimator via a Monte Carlo study and real data example.

Keywords

Kernel method Relative error Non-parametric estimation  Associated variable 

Mathematics Subject Classification

62G20 62G08 

References

  1. Bernhard FA, Stahlecker P (2003) Relative squared error prediction in the generalized linear regression model. Stat Pap 44:107–115MATHCrossRefGoogle Scholar
  2. Biau G, Cadre B (2004) Nonparametric spatial prediction. Stat Inference Stoch Process 7:327–349MATHMathSciNetCrossRefGoogle Scholar
  3. Bobbia M, Misiti M, Misiti Y, Poggi JM, Portier B (2015) Spatial outlier detection in the PM10 monitoring network of Normandy. Atmos Pollut Res 6:476–483CrossRefGoogle Scholar
  4. Carbon M, Tran LT, Wu B (1997) Kernel density estimation for random fields. Stat Probab Lett 36:115–125MATHMathSciNetCrossRefGoogle Scholar
  5. Carbon M, Francq C, Tran LT (2007) Kernel regression estimation for random fields. J Stat Plan Inference 137:778–798MATHMathSciNetCrossRefGoogle Scholar
  6. Cressie NA (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  7. Dabo-Niang S, Thiam B (2010) Robust quantile estimation and prediction for spatial processes. Stat Probab Lett 80:1447–1458MATHMathSciNetCrossRefGoogle Scholar
  8. Dabo-Niang S, Yao AF (2007) Kernel regression estimation for continuous spatial processes. Math Methods Stat 16:1–20MathSciNetCrossRefGoogle Scholar
  9. Dabo-Niang S, Ould-Abdi S, Ould-Abdi A, Diop A (2014) Consistency of a nonparametric conditional mode estimator for random fields. Stat Methods Appl 23:1–39MathSciNetCrossRefGoogle Scholar
  10. Dabo-Niang S, Yao A, Pischedda L, Cuny P, Gilbert F (2009) Spatial kernel mode estimation for functional random, with application to bioturbation problem. Stoch Environ Res Risk Assess 24:487–497CrossRefGoogle Scholar
  11. Diggle P, Ribeiro PJ (2007) Model-based geostatistics. Springer, New YorkMATHGoogle Scholar
  12. Doukhan P (1994) Mixing: properties and examples. Lecture Notes in Statistics, vol 85. Springer- Verlag, New YorkGoogle Scholar
  13. El Machkouri M, Stoica R (2010) Asymptotic normality of kernel estimates in a regression model for random fields. J Nonparametric Stat 22:955–971MATHCrossRefGoogle Scholar
  14. Filzmoser P, Ruiz-Gazen A, Thomas-Agnan C (2014) Identification of local multivariate outliers. Stat Pap 55:29–47MATHMathSciNetCrossRefGoogle Scholar
  15. Gheriballah A, Laksaci A, Rouane R (2010) Robust nonparametric estimation for spatial regression. J Stat Plan Inference 140:1656–1670MATHMathSciNetCrossRefGoogle Scholar
  16. Guyon X (1987) Estimation d’un champ par pseudo-vraisemblance conditionnelle: Etude asymptotique et application au cas Markovien. In: Proceedings of the sixth Franco-Belgian meeting of statisticiansGoogle Scholar
  17. Hallin M, Lu Z, Yu K (2009) Local linear spatial quantile regression. Bernoulli 15:659–686MATHMathSciNetCrossRefGoogle Scholar
  18. Jones MC, Park H, Shinb K, Vines SK, Jeong SO (2008) Relative error prediction via kernel regression smoothers. J Stat Plan Inference 138:2887–2898MATHCrossRefGoogle Scholar
  19. Li J, Tran LT (2009) Nonparametric estimation of conditional expectation. J Stat Plan Inference 139:164–175MATHMathSciNetCrossRefGoogle Scholar
  20. Liu X, Lu CT, Chen F (2010) Spatial outlier detection: random walk based approaches. In: Proceedings of the 18th ACM SIGSPATIAL international conference on advances in geographic information systems (ACM GIS), San Jose, CAGoogle Scholar
  21. Lu Z, Chen X (2004) Spatial kernel regression: weak consistency. Stat Probab Lett 68:125–136MATHCrossRefGoogle Scholar
  22. Martnez J, Saavedra J, Garca-Nieto PJ, Pieiro JI, Iglesias C, Taboada J, Sancho J, Pastor J (2014) Air quality parameters outliers detection using functional data analysis in the Langreo urban area (Northern Spain). Appl Math Comput 241:1–10MathSciNetCrossRefGoogle Scholar
  23. Narula SC, Wellington JF (1977) Prediction, linear regression and the minimum sum of relative errors. Technometrics 19:185–190MATHCrossRefGoogle Scholar
  24. Omidi M, Mohammadzadeh M (2015) A new method to build spatio-temporal covariance functions: analysis of ozone data. Stat Pap. doi:10.1007/s00362-015-0674-2
  25. Robinson PM (2011) Asymptotic theory for nonparametric regression with spatial data. J Econom 165:5–19CrossRefGoogle Scholar
  26. Shen VY, Yu T, Thebaut SM (1985) Identifying error-prone softwarean empirical study. IEEE Trans Softw Eng 11:317–324CrossRefGoogle Scholar
  27. Tran LT (1990) Kernel density estimation on random fields. J Multivar Anal 34:37–53MATHCrossRefGoogle Scholar
  28. Volker S (2014) Stochastic geometry, spatial statistics and random fields: models and algorithms. Lecture Notes in Mathematics, vol 2120. Springer, New YorkGoogle Scholar
  29. Xu R, Wang J (2008) \(L_1\)- estimation for spatial nonparametric regression. J Nonparametric Stat 20:523–537MATHCrossRefGoogle Scholar
  30. Yang Y, Ye F (2013) General relative error criterion and M-estimation. Front Math China 8:695–715MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Mohammed Attouch
    • 1
  • Ali Laksaci
    • 1
  • Nafissa Messabihi
    • 1
  1. 1.Agence Thmatique de Recherche en Sciences et Technologie, Laboratoire de statistique et processus stochastiquesUniv. Djillali LiabèsSidi Bel AbbèsAlgeria

Personalised recommendations