Systematic sampling on the circle and on the sphere
Useful approximations have been developed along the years to predict the precision of systematic sampling for measurable functions of a bounded support in Rd. Recently, the theory of systematic sampling on R has received a thrust. In geometric sampling, design based unbiased estimators exist, however, which imply systematic sampling on the circle (S1) and the semicircle (H1); the planimeter estimator of an area, or the Buffon-Steinhaus estimator of curve length in the plane constitute popular examples. Over the last two decades, many other estimators of geometric measures have been obtained which imply systematic sampling also on the sphere (S2). In this paper we adapt the theory available for non-periodic functions of bounded support on R to periodic functions, and thereby to S1 and H1, and we obtain new estimators of the corresponding variance approximations. Further we consider - we believe for the first time - the problem of predicting the precision of systematic sampling in S2. The paper starts with a historical perspective, and ends with suggestions for further research.
Permanent link to this document: http://projecteuclid.org/euclid.aap/1013540235
Digital Object Identifier: doi:10.1239/aap/1013540235
Mathematical Reviews number (MathSciNet): MR1788086
Zentralblatt MATH identifier: 01545581