Asymptotic normality of the Nadaraya–Watson semivariogram estimators

García-Soidán, Pilar

doi:10.1007/s11749-006-0016-8

Cited by 14 publications

(10 citation statements)

References 11 publications

(14 reference statements)

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“…Estimator is the indicator version of the kernel variogram analyzed in García‐Soidán () and involves a weighted average. The consistency of

{\overset{γ}{true}}_{I_{Y} MathClass-punc, h} (normalt MathClass-punc, x)

can be proved and, additionally, its bias and variance can be approximated.…”

Section: Resultsmentioning

confidence: 99%

Estimation of the spatial distribution through the kernel indicator variogram

García-Soidán

Menezes

2012

Environmetrics

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The distribution function of a spatial random process provides the probability that the variable involved does not exceed a given threshold. Estimation of this function can be addressed in a parametric way, although the available data do not always support the distribution model assumption. Additional options for distribution approximation are based on applying the indicator kriging technique or the sill estimation, which demand assuming stationary conditions from the random process and also estimation of the indicator variogram. In this paper, the latter distribution approximation approaches will be redesigned, by previously establishing the appropriate application setting, so that characterization of the dependence structure will be reduced to the estimation of one indicator variogram for each threshold. Furthermore, a kernel‐type estimator is suggested for the latter aim, as a nonparametric alternative to Matheron's indicator variogram. Numerical studies for simulated data have been developed to illustrate the better performance of the kernel estimator, when compared with Matheron's one, as well as the behavior of the procedures described for estimation of the distribution function. Finally, an application is included, where the nitrate concentration in groundwater in the Beja district (Portugal) is measured, so that pollution risk maps of the referred region will have been generated. Copyright © 2012 John Wiley & Sons, Ltd.

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“…Estimator is the indicator version of the kernel variogram analyzed in García‐Soidán () and involves a weighted average. The consistency of

{\overset{γ}{true}}_{I_{Y} MathClass-punc, h} (normalt MathClass-punc, x)

can be proved and, additionally, its bias and variance can be approximated.…”

Section: Resultsmentioning

confidence: 99%

Estimation of the spatial distribution through the kernel indicator variogram

García-Soidán

Menezes

2012

Environmetrics

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“…In practice, λ n can be determined by the diameter of a sampling region for use here (cf. García-Soidán [12], Hall and Patil [14], Maity and Sherman [23], Matsuda and Yajima [24]). Let Z = {0, ±1, ±2, .…”

Section: Preliminariesmentioning

confidence: 98%

“…To fit the variogram model in a way that allows nonparametric confidence intervals (CIs) to access the precision of the estimated parameters, without making assumptions about the joint distribution of the data or the distribution of spatial locations, one can apply the SFDEL method using estimating functions as in Example 3 motivated by least squares estimation. Alternatively, one can apply a kernel bandwidth estimator of the varigoram for which large sample distributional results are recently known (García-Soidán [12], Maity and Sherman [23]).…”

Section: 1mentioning

confidence: 99%

“…In contrast, using the lags above and the Nadaraya-Watson kernel estimator of the semivariogram γ(h) (based on the Epanechnikov kernel, cf. García-Soidán [12]), the range parameter estimates are 1.505, 1.230, 1.335 for bandwidths h = 0.5, 1, 1.5, where h = 0.5 arises from MSE optimal order considerations. This approach can also produce large-sample nonparametric CIs based on normal limits for λ n [γ(h) − γ(h)], having a covariance matrix C · V , C = [ f 2 ] −2 f 4 , involving the unknown density f of locations {s i /λ n } 105 i=1 on [0, 1] 2 .…”

Section: 1mentioning

confidence: 99%

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A frequency domain empirical likelihood method for irregularly spaced spatial data

Bandyopadhyay¹,

Lahiri²,

Nordman³

2015

Ann. Statist.

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This paper develops empirical likelihood methodology for irregularly spaced spatial data in the frequency domain. Unlike the frequency domain empirical likelihood (FDEL) methodology for time series (on a regular grid), the formulation of the spatial FDEL needs special care due to lack of the usual orthogonality properties of the discrete Fourier transform for irregularly spaced data and due to presence of nontrivial bias in the periodogram under different spatial asymptotic structures. A spatial FDEL is formulated in the paper taking into account the effects of these factors. The main results of the paper show that Wilks' phenomenon holds for a scaled version of the logarithm of the proposed empirical likelihood ratio statistic in the sense that it is asymptotically distribution-free and has a chisquared limit. As a result, the proposed spatial FDEL method can be used to build nonparametric, asymptotically correct confidence regions and tests for covariance parameters that are defined through spectral estimating equations, for irregularly spaced spatial data. In comparison to the more common studentization approach, a major advantage of our method is that it does not require explicit estimation of the standard error of an estimator, which is itself a very difficult problem as the asymptotic variances of many common estimators depend on intricate interactions among several population quantities, including the spectral density of the spatial process, the spatial sampling density and the spatial asymptotic structure. Results from a numerical study are also reported to illustrate the methodology and its finite sample properties.

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“…For example, simulations (not included) indicated a bandwidth of w = 0.65 maintains nominal size when n = 950, but leads to deflated test size and power when n = 400 on a smaller domain. García-Soidán et al (2004), García-Soidán (2007), and Kim and Park (2012) develop theoretically optimal bandwidths for nonparametric semivariogram estimation, but these works are not applicable here because they focus on the isotropic case and require an estimate of the second derivative of the variogram. We have found that the empirical bandwidth used by MS tends to produce nominal size (see Table 6).…”

Section: Recommendationsmentioning

confidence: 99%