×

zbMATH — the first resource for mathematics

On the Nadaraya-Watson kernel regression estimator for irregularly spaced spatial data. (English) Zbl 1437.62143
Summary: We investigate the asymptotic normality of the Nadaraya-Watson kernel regression estimator for irregularly spaced data collected on a finite region of the lattice \(\mathbb{Z}^d\) where \(d\) is a positive integer. The results are stated for strongly mixing random fields in the sense of Rosenblatt (1956) and for weakly dependent random fields in the sense of W. B. Wu [Proc. Natl. Acad. Sci. USA 102, No. 40, 14150–14154 (2005; Zbl 1135.62075)]. Only minimal conditions on the bandwidth parameter and simple conditions on the dependence structure of the data are assumed.
MSC:
62G08 Nonparametric regression and quantile regression
62M30 Inference from spatial processes
62M40 Random fields; image analysis
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Biau, G.; Cadre, B., Nonparametric spatial prediction, Stat. Inference Stoch. Process., 7, 3, 327-349 (2004) · Zbl 1125.62317
[2] Carbon, M.; Francq, Ch.; Tran, L. T., Kernel regression estimation for random fields, J. Statist. Plann. Inference, 137, 3, 778-798 (2007) · Zbl 1104.62105
[3] Dabo-Niang, S.; Rachdi, M.; Yao, A., Kernel regression estimation for spatial functional random variables, Far East J. Theor. Stat., 37, 2, 77-113 (2011) · Zbl 1279.62090
[4] Dabo-Niang, S.; Yao, A.-F., Kernel regression estimation for continuous spatial processes, Math. Methods Statist., 16, 4, 298-317 (2007) · Zbl 1140.62071
[5] Dedecker, J., A central limit theorem for stationary random fields, Probab. Theory Related Fields, 110, 397-426 (1998) · Zbl 0902.60020
[6] Dedecker, J., Exponential inequalities and functional central limit theorems for random fields, ESAIM Probab. Stat., 5, 77-104 (2001) · Zbl 1003.60033
[7] El Machkouri, M., Nonparametric regression estimation for random fields in a fixed-design, Stat. Inference Stoch. Process., 10, 1, 29-47 (2007) · Zbl 1110.62052
[8] El Machkouri, M., Asymptotic normality for the Parzen-Rosenblatt density estimator for strongly mixing random fields, Stat. Inference Stoch. Process., 14, 1, 73-84 (2011) · Zbl 1274.62235
[9] El Machkouri, M., Kernel density estimation for stationary random fields, ALEA Lat. Am. J. Probab. Math. Stat., 11, 1, 259-279 (2014) · Zbl 1291.62083
[10] El Machkouri, M.; Stoica, R., Asymptotic normality of kernel estimates in a regression model for random fields, J. Nonparametr. Stat., 22, 8, 955-971 (2010) · Zbl 1203.62065
[11] El Machkouri, M.; Volný, D.; Wu, W. B., A central limit theorem for stationary random fields, Stochastic Process. Appl., 123, 1, 1-14 (2013) · Zbl 1308.60025
[12] Hallin, M.; Lu, Z.; Tran, L. T., Local linear spatial regression, Ann. Statist., 32, 6, 2469-2500 (2004) · Zbl 1069.62075
[13] Kulkarni, P. M., Estimation of parameters of a two-dimensional spatial autoregressive model with regression, Statist. Probab. Lett., 15, 2, 157-162 (1992) · Zbl 0766.62054
[14] Liggett, T. M., Interacting Particle Systems (1985), Springer-Verlag · Zbl 0559.60078
[15] Lindeberg, J. W., Eine neue Herleitung des Exponentialgezetzes in der Wahrscheinlichkeitsrechnung, Math. Z., 15, 211-225 (1922) · JFM 48.0602.04
[16] Lu, Z.; Chen, X., Spatial nonparametric regression estimation: Non-isotropic case, Acta Math. Appl. Sin. Engl. Ser., 18, 641-656 (2002) · Zbl 1019.62039
[17] Lu, Z.; Chen, X., Spatial kernel regression estimation: weak consistency, Statist. Probab. Lett., 68, 125-136 (2004) · Zbl 1058.62079
[18] Lu, Z.; Cheng, P., Distribution-free strong consistency for nonparametric kernel regression involving nonlinear time series, J. Statist. Plann. Inference, 65, 1, 67-86 (1997) · Zbl 0907.62053
[19] Masry, E.; Fan, J., Local polynomial estimation of regression functions for mixing processes, Scand. J. Stat., 24, 2, 165-179 (1997) · Zbl 0881.62047
[20] McLeish, D. L., A maximal inequality and dependent strong laws, Ann. Probab., 3, 5, 829-839 (1975) · Zbl 0353.60035
[21] Nadaraya, È. A., On non-parametric estimates of density functions and regression, Teor. Verojatnost. Primenen., 10, 199-203 (1965) · Zbl 0134.36302
[22] Parzen, E., On the estimation of a probability density and the mode, Ann. Math. Stat., 33, 1965-1976 (1962)
[23] Rio, E., Covariance inequalities for strongly mixing processes, Ann. Inst. H. Poincaré Probab. Statist., 29, 4, 587-597 (1993) · Zbl 0798.60027
[24] Robinson, P. M., Nonparametric estimators for time series, J. Time Series Anal., 4, 3, 185-207 (1983) · Zbl 0544.62082
[25] Rosenblatt, M., A central limit theorem and a strong mixing condition, Proc. Nat. Acad. Sci. USA, 42, 43-47 (1956) · Zbl 0070.13804
[26] Rosenblatt, M., Remarks on some nonparametric estimates of a density function, Ann. Math. Stat., 27, 832-837 (1956) · Zbl 0073.14602
[27] Roussas, G. G., Nonparametric estimation in mixing sequences of random variables, J. Statist. Plann. Inference, 18, 2, 135-149 (1988) · Zbl 0658.62048
[28] Stroock, D. W.; Zegarlinski, B., The logarithmic sobolev inequality for discrete spin systems on a lattice, Comm. Math. Phys., 149, 1, 175-193 (1992) · Zbl 0758.60070
[29] Watson, G. S., Smooth regression analysis, Sankhya A, 26, 359-372 (1964) · Zbl 0137.13002
[30] Wu, W. B., Nonlinear system theory: another look at dependence, Proc. Natl. Acad. Sci. USA, 102, 40, 14150-14154 (2005), (electronic) · Zbl 1135.62075
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.