Nonparametric kernel estimation of an isotropic variogram

Asymptotic normality of the Nadaraya–Watson semivariogram estimators

“…The Nadaraya-Watson estimator of γ (s) in this setting is given byγ h (s) defined in (1.1), which has been proved to be asymptotically unbiased and consistent. Then, suppose that conditions (S1)-(S5), (S7) and (S9) are satisfied for δ = l = m = 1 and that the semivariogram γ admits three continuous derivatives in a neighborhood of s > 0; it has been proved in García-Soidán et al (2004) …”

“…In Cressie and Hawkins (1980) the square roots of the absolute differences, |Z(s i ) − Z(s j )|, are considered instead, and a more robust semivariogram estimator is suggested in Genton (1998), based on a highly robust estimator of scale. On the other hand, two kernel-type estimators are proposed in García-Soidán et al (2003, 2004, which result from adapting the local linear and the Nadaraya-Watson estimators to the context of spatial data, respectively.…”

“…Some properties ofγ h (s) has been studied in García-Soidán et al (2004); in particular, it has been proved that it is asymptotically unbiased as well as consistent, by proceeding as in . Thus, in this paper we will obtain the dominant terms of the bias and the variance ofγ h (s).…”

Assessing the effect of clustered and biased multi‐stage sampling

“…An example is the geoR library from R, described in Ribeiro and Diggle (2001), which offers a kernel estimator for exploratory purposes with the bandwidth being chosen by the user. In Garcia-Soidán et al (2004), it is suggested a transformed version of the kernel estimator, not restricted to exploratory aims but allowed to be used in kriging. They adapt the Nadaraya-Watson regression estimation to the context of spatial data and propose an asymptotically optimal bandwidth parameter.…”

Goodness-of-fit tests for the spatial spectral density

“…In geostatistics, the estimation of the variogram plays a key role since kriging prediction methods depend on the variogram or covariogram estimation. There exists an extensive literature devoted to the estimation problem (see Cressie 1993 for parametric models and García-Soidán et al (2004a), García-Soidán et al (2004b) for nonparametric estimations of the variogram. However, there are slightly few references devoted to the testing problem.…”