In this article, we consider the least squares approach for estimating parameters of a spatial variogram and establish consistency and asymptotic normality of these estimators under general conditions. Large sample distributions are also established under a spatial regression model where the sampling design possibly has an infill component. These results allow us to investigate efficiencies of different least squares estimators in large samples. We provide two necessary and sufficient conditions for these estimators to be asymptotically efficient. In particular, we show that when the number of lags used to define the estimators is chosen to be equal to the number of variogram parameters to be estimated, the ordinary least squares estimator, the weighted least squares and the generalized least squares estimators are all asymptotically efficient. We also carry out a small simulation study to investigate the implications of these results in finite samples.

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