Design-Based Maps for Finite Populations of Spatial Units

The estimation of the values of a survey variable in finite populations of spatial units is considered for making maps when samples of spatial units are selected by probabilistic sampling schemes. The single values are estimated by means of an inverse distance weighting predictor. The design-based asymptotic properties of the resulting maps, referred to as the design-based maps, are considered when the study area remains fixed and the sizes of the spatial units tend to zero. Conditions ensuring design-based asymptotic unbiasedness and consistency are derived. They essentially require the existence of a pointwise or uniformly continuous density function of the survey variable onto the study area, some regularities in the size and shape of the units, and the use of spatially balanced designs to select units. The continuity assumption can be relaxed into a Riemann integrability assumption when estimation is performed at a sufficiently small spatial grain and the estimates are subsequently aggregated at a greater grain. A computationally simple mean squared error estimator is proposed. A simulation study is performed to assess the theoretical results. An application to estimate the map of wine cultivations in Tuscany (Central Italy) is considered. Supplementary materials for this article are available online.