The estimation of the total of an attribute defined over a continuous planar
domain is required in many applied settings, such as the estimation of canopy
coverage in the Monterano Nature Reserve in Italy. If the design-based approach
is considered, the scheme for the placement of the sample sites over the domain
is fundamental in order to implement the survey. In real situations, a commonly
adopted scheme is based on partitioning the domain into suitable strata, in
such a way that a single sample site is uniformly placed (i.e., selected with
uniform probability density) in each stratum and sample sites are independently
located. Under mild conditions on the function representing the target
attribute, it is shown that this scheme gives rise to an unbiased spatial total
estimator which is "superefficient" with respect to the estimator based on the
uniform placement of independent sample sites over the domain. In addition, the
large-sample normality of the estimator is proven and variance estimation
issues are discussed.Comment: Published in at http://dx.doi.org/10.1214/11-AOAS509 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
