Spatial auto-correlation and auto-regressive models estimation from sample survey data.

Maximum likelihood estimation of the model parameters for a spatial population based on data collected from a survey sample is usually straightforward when sampling and non-response are both non-informative, since the model can then usually be fitted using the available sample data, and no allowance is necessary for the fact that only a part of the population has been observed. Although for many regression models this naive strategy yields consistent estimates, this is not the case for some models, such as spatial auto-regressive models. In this paper, we show that for a broad class of such models, a maximum marginal likelihood approach that uses both sample and population data leads to more efficient estimates since it uses spatial information from sampled as well as non-sampled units. Extensive simulation experiments based on two well-known data sets are used to assess the impact of the spatial sampling design, the auto-correlation parameter and the sample size on the performance of this approach. When compared to some widely used methods that use only sample data, the results from these experiments show that the maximum marginal likelihood approach is much more precise.

A spatially balanced design with probability function proportional to the within sample distance.

The units observed in a biological, agricultural, and environmental survey are often randomly selected from a finite population whose main feature is to be geo-referenced thus its spatial distribution should be used as essential information in designing the sample. In particular our interest is focused on probability samples that are well spread over the population in every dimension which in recent literature are defined as spatially balanced samples. To approach the problem we used the within sample distance as the summary index of the spatial distribution of a random selection criterion. Moreover numerical comparisons are made between the relative efficiency, measured with respect to the simple random sampling, of the suggested design and some other classical solutions as the Generalized Random Tessellation Stratified (GRTS) design used by the US Environmental Protection Agency (EPA) and other balanced or spatially balanced selection procedures as the Spatially Correlated Poisson Sampling (SCPS), the balanced sampling (CUBE), and the Local Pivotal method (LPM). These experiments on real and simulated data show that the design based on the within sample distance selects samples with a better spatial balance thus gives estimates with a lower sampling error than those obtained by using the other methods. The suggested method is very flexible to the introduction of stratification and coordination of samples and, even if in its nature it is computationally intensive, it is shown to be a suitable solution even when dealing with high sampling rates and large population frames where the main problem arises from the size of the distance matrix.