New approximations for the variance in Cavalieri sampling

The theory of Cavalieri sampling − or systematic sampling along an axis − has received a recent impetus. The error variance may be represented by the sum of three components, namely the extension term, the ‘Zitterbewegung’, and higher order terms. The extension term can be estimated from the data, and it constitutes the standard variance approximation used so far. The Zitterbewegung oscillates about zero, and neither this nor higher order terms have hitherto been considered to predict the variance. The extension term is always a good approximation of the variance when the number of observations is very large, but not necessarily when this number is small. In this paper we propose a more general representation of the variance, and from it we construct a flexible extension term which approximates the variance satisfactorily for an arbitrary number of observations. Furthermore, we generalize the current connection between the smoothness properties of the measurement function (e.g. the section area function of an object when the target is the volume) and the corresponding properties of its covariogram to facilitate the computation of the new variance approximations; this enables us to interpret the behaviour of the variance from the ‘overall shape’ of the measurement function. Our approach applies mainly to measurement functions whose form is known analytically, but it helps also to understand the behaviour of the variance when the measurement function is known at sufficiently many points; in fact, we illustrate the concepts with both synthetic and real data.