Design-based spatial interpolation with data driven selection of the smoothing parameter

In the inverse distance weighting interpolation the interpolated, value is a weighted mean of the sampled values, with weights decreasing with the distances. The most widely adopted class of distance functions is the class of negative powers of order α and the appropriate choice of the smoothing parameter α is a crucial issue. In this paper, we give sufficient conditions for the design-based consistency of the inverse distance weighting interpolator when α is selected by cross-validation techniques, and a pseudo-population bootstrap approach is introduced to estimate the accuracy of the resulting interpolator. A simulation study is performed to empirically confirm the theoritical findings and to investigate the finite-sample properties of the interpolator obtained using leave-one-out cross-validation. Moreover, a comparison with the nearest neighbor interpolator, which is the limiting case for α = ∞, is performed. Finally, the estimation of the surface of the Shannon diversity index of tree diameter at breast height in the experimental watershed of Bonis forest (Southern Italy) is described.