Spatial kernel regression estimation: weak consistency.

In this paper, we introduce a kernel method to estimate a spatial conditional regression under mixing spatial processes. Some preliminary statistical properties including weak consistency and convergence rates are investigated. The sufficient conditions on mixing coefficients and the bandwidth are established to ensure distribution-free weak consistency, which requires no assumption on the regressor and allows the mixing coefficients decreasing to zero slowly. However, to achieve an optimal convergence rate, some requirements on the regressor and the decreasing rate of mixing coefficients tending to zero are needed.

[1]  R. Martin,et al.  Spatial Processes and Spatial Time Series Analysis , 1988 .

[2]  Noel A Cressie,et al.  Statistics for Spatial Data. , 1992 .

[3]  A. Krzyżak,et al.  Distribution-Free Pointwise Consistency of Kernel Regression Estimate , 1984 .

[4]  L. Györfi,et al.  Strong consistency and rates for recursive probability density estimators of stationary processes , 1987 .

[5]  P. Kulkarni.  Estimation of parameters of a two-dimensional spatial autoregressive model with regression , 1992 .

[6]  L. Tran.  Kernel density estimation on random fields , 1990 .

[7]  P. Robinson.  TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION , 1987 .

[8]  Y. Ogata.  Evaluation of spatial Bayesian models—Two computational methods☆ , 1996 .

[9]  D. Tjøstheim,et al.  Nonparametric Estimation and Identification of Nonlinear ARCH Time Series Strong Convergence and Asymptotic Normality: Strong Convergence and Asymptotic Normality , 1995, Econometric Theory.

[10]  Note on the uniform convergence of density estimates for mixing random variables , 1987 .

[11]  S. Yakowitz.  NEAREST‐NEIGHBOUR METHODS FOR TIME SERIES ANALYSIS , 1987 .

[12]  J. Besag.  Spatial Interaction and the Statistical Analysis of Lattice Systems , 1974 .

[13]  G. Roussas.  Nonparametric estimation in mixing sequences of random variables , 1988 .

[14]  S. Yakowitz,et al.  Nearest neighbor estimators for random fields , 1993 .

[15]  C. J. Stone,et al.  Nonparametric function estimation involving time series , 1992 .

[16]  P. Robinson.  NONPARAMETRIC ESTIMATORS FOR TIME SERIES , 1983 .

[17]  C. J. Stone,et al.  Optimal Rates of Convergence for Nonparametric Estimators , 1980 .

[18]  Marc Hallin,et al.  Density estimation for spatial linear processes , 2001 .

[19]  Graciela Boente,et al.  Consistency of a nonparametric estimate of a density function for dependent variables , 1988 .

[20]  P. Whittle.  ON STATIONARY PROCESSES IN THE PLANE , 1954 .

[21]  M. Hallin,et al.  Kernel density estimation on random fields: the L1 theory , 1996 .

[22]  C. C. Neaderhouser.  Convergence of block spins defined by a random field , 1980 .

[23]  L. Devroye.  On the Almost Everywhere Convergence of Nonparametric Regression Function Estimates , 1981 .

[24]  Wolfgang Härdle,et al.  Nonparametric Curve Estimation from Time Series , 1989 .

[25]  M. Rosenblatt.  Stationary sequences and random fields , 1985 .

[26]  Zudi Lu,et al.  Distribution-free strong consistency for nonparametric kernel regression involving nonlinear time series , 1997 .

[27]  Gutti Jogesh Babu,et al.  Spatial point processes in astronomy , 1996 .