Ronny O. Vallejos

A common practice in applied statistics is to determine the sample size under independence. When the available data have an obvious correlation structure the problem is how to determine the decrease of sample size as a function of correlation. This problem is relevant when a pilot study has been carried out in a certain region and it is of interest to study a regionalized variable in the same area. This problem has been discussed in the literature (Richardson and Clifford 1989), and recently some attention has been devoted to the determination of geographical sample sizes (Griffith 2005, 2008).

In this talk several proposals for the effective sample size (ESS) are reviewed. Also a new quantity to measure the reduction of the spatial sample size is suggested. We do an exploration for patterned correlation matrices commonly used in spatial statistics. Monotonic properties of the effective sample size will be discussed with respect to dimension and the sample size. Some preliminary theoretical results will be also described for random locations on the plane. The estimation problem of the proposed effective sample size will be also addressed in the context of spatial modeling.

**References:**

1. Clifford, P., Richardson, S., Hémon, D., (1989). Assessing the signifcance of the correlation between two spatial processes. Biometrics 45, 123-134.

2. Griffith, D., (2005). Effective geographic sample size in the presence of spatial autocorrelation. Annals of the Association of American Geographers 95, 740-760.

3. Griffith, D., (2008). Geographic sampling of urban soils for contaminant mapping: how many samples and from where. Environ. Geochem. Health 30, 495-509.

**Keywords:** Effective Sample Size; Spatial Correlation; Patterned Correlation Matrices; CAR processes

**Biography:** Ronny Vallejos obtained his Ph.D. in Statistics at the University of Maryland Baltimore County, USA, in 2006. Currently he is an assistant professor in the department of Mathematics at Santa Maria Technical University in Chile.