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https://www.ias.ac.in/article/fulltext/pmsc/121/02/0229-0244

Area approximations; convex sets; increasing domain; infill sampling; lattice data; spatial processes; spatial covariance.

Investigations of spatial statistics, computed from lattice data in the plane, can lead to a special lattice point counting problem. The statistical goal is to expand the asymptotic expectation or large-sample bias of certain spatial covariance estimators, where this bias typically depends on the shape of a spatial sampling region. In particular, such bias expansions often require approximating a difference between two lattice point counts, where the counts correspond to a set of increasing domain (i.e., the sampling region) and an intersection of this set with a vector translate of itself. Non-trivially, the approximation error needs to be of smaller order than the spatial region’s perimeter length. For all convex regions in 2-dimensional Euclidean space and certain unions of convex sets, we show that a difference in areas can approximate a difference in lattice point counts to this required accuracy, even though area can poorly measure the lattice point count of any single set involved in the difference. When investigating large-sample properties of spatial estimators, this approximation result facilitates direct calculation of limiting bias, because, unlike counts, differences in areas are often tractable to compute even with non-rectangular regions. We illustrate the counting approximations with two statistical examples.

Daniel J Nordman¹ Soumendra N Lahiri²

1. Department of Statistics, Iowa State University, Ames, IA 50011, USA
2. Department of Statistics, Texas A&M University, College Station, TX 77843, USA

Published

May 2011

Manuscript received

14 May 2010

Manuscript revised

2 January 2011