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Statistica Sinica 18(2008),667-688



Ji Meng Loh and Michael L. Stein

Columbia University and University of Chicago

Abstract: Various methods, such as the moving block bootstrap, have been developed to resample dependent data. These contain features that attempt to retain the dependence structure in the data. Asymptotic results for these methods have been based on increasing the number of observations as the observation region increases, combined with restrictions on the range of dependence of the data. In this work, we consider resampling a one-dimensional Gaussian random field observed on a regular lattice, with increasing number of observations within a fixed observation region. We show a consistency result for resampling the variogram when the underlying process has a variogram of the form $\gamma(t) = \theta t + S(t),
\vert S(t)\vert \le Dt^2$. For a class of smoother processes, specifically, for a process with variogram of the form $\gamma (t)= \beta t^2 +
\theta t^3 + R(t), \vert R(t)\vert \le Dt^4$, we provide a similar result when resampling the second-order variogram. We performed a simulation study in one and two dimensions. We used the Matérn model for the covariance function with varying values of the smoothness parameter $\nu$. We find that the empirical coverage of confidence intervals of the variogram approaches the nominal 95% level as the number of observations increases for models with small $\nu$, but as $\nu$ is increased the empirical coverage decreases, with the coverage becoming significantly lower than the nominal level when $\nu\ge 1$. When we consider the second-order variogram, we find that the empirical coverage approaches the nominal level for larger values of $\nu$ than for the first-order variogram, with the empirical coverage noticeably lower than the nominal level only when $\nu$ is about 2.

Key words and phrases: Fixed domain asymptotics, Gaussian processes, spatial bootstrap, variogram.

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