Mixed model regression estimation of a spatial total in the continuous plane paradigm

In environmental studies, it is a common practice to follow the continuous-plane paradigm for estimating parameters of interest, like totals or means, of spatial variables defined over a surface whose points correspond to points of a connected and compact subset of R2. In the design-based approach, randomness comes from the random sampling of points in the domain, wherein to observe the values of the spatial variable. Adopting this type of approach, under a model-assisted framework, in this paper we present a class of estimators of the total of the spatial variable, which are based on linear mixed models. We introduce first the mixed model regression estimator of the total and its properties. Methods for choosing the smoothing parameter are also discussed. Then, we present the penalized splines regression estimator of the total, as a member of the class, and its asymptotic properties when the number of knots increases with the sample size within an infill asymptotic framework. This estimator, under mild conditions, is proved to be asymptotically unbiased, consistent, asymptotically normally distributed, and more efficient than the basic unbiased estimator. Furthermore, it can be super-efficient, that is, the variance order may be higher than the usual n−1, according to the degree of smoothness of the spatial variable. A simulation study shows the performance of the estimator as well as that of the proposed variance estimators.