Local linear spatial regression

Abstract

A local linear kernel estimator of the regression function x\mapsto g(x):=E[Y_i|X_i=x], x\in R^d, of a stationary (d+1)-dimensional spatial process {(Y_i,X_i),i\in Z^N} observed over a rectangular domain of the form I_n:={i=(i_1,...,i_N)\in Z^N| 1\leq i_k\leq n_k,k=1,...,N}, n=(n_1,...,n_N)\in Z^N, is proposed and investigated. Under mild regularity assumptions, asymptotic normality of the estimators of g(x) and its derivatives is established. Appropriate choices of the bandwidths are proposed. The spatial process is assumed to satisfy some very general mixing conditions, generalizing classical time-series strong mixing concepts. The size of the rectangular domain I_n is allowed to tend to infinity at different rates depending on the direction in Z^N.Comment: Published at http://dx.doi.org/10.1214/009053604000000850 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Asymptotic theory for nonparametric regression with spatial data

“…Estimates of the nonparametric regression function are typically obtained at several …xed points by some method of smoothing. In a spatial context, nonparametric regression has been discussed by, for example, Tran and Yakowitz (1993), Hallin, Lu and Tran (2004a). The most commonly used kind of smoothed nonparametric regression estimate in econometrics is still the Nadaraya-Watson kernel estimate.…”

Kernel spatial density estimation in infinite dimension space

“…Lu & Chen (2004), Hallin et al (2004b)). However, in some problems the assumption that this rate is the same in all directions is called isotropic divergence: that is n → +∞ and n j n k ≤ C for some constant 0 < C < ∞ and ∀ j, k ∈ {1, ..., N } (see ? or Hallin et al (2004b)). …”

“…As raised by Hallin et al (2004b), this gap leads to re-dened two important notions: asymptotic notion and, since we deal with index-dependent data, the notion of neighborhood and the related notion of dependent measure.…”

B-spline estimation for varying coefficient regression with spatial data

“…[11][12][13][14] developed the statistical properties of the kernel density estimator of random field. The results on the estimation of spatial regression function include [15,16], which investigated the properties of a Nadaraya-Waston kernel estimator for nonparametric function, [13], which established the asymptotic distribution of the local linear estimator for nonparametric function, [17], which developed the properties of the local linear kernel estimator for semiparametric spatial regression model, and [18], which established the asymptotic properties of the additive estimators. The nonparametric methods and theory about spatial data so far focus on local estimation methods, such as kernel and local linear methods.…”