New approximations for the variance in Cavalieri sampling

The theory of Cavalieri sampling - or systematic sampling along an axis - h as received a recent impetus. The error variance may be represented by the sum of three components, namely the extension term, the 'Zitterbewegung', a nd higher order terms. The extension term can be estimated from the data, a nd it constitutes the standard variance approximation used so far. The Zitt erbewegung oscillates about zero, and neither this nor higher order terms h ave hitherto been considered to predict the variance. The extension term is always a good approximation of the variance when the number of observation s is very large, but not necessarily when this number is small. In this pap er we propose a more general representation of the variance, and from it we construct a flexible extension term which approximates the variance satisf actorily for an arbitrary number of observations. Furthermore, we generaliz e the current connection between the smoothness properties of the measureme nt function (e.g. the section area function of an object when the target is the volume) and the corresponding properties of its covariogram to facilit ate the computation of the new variance approximations; this enables us to interpret the behaviour of the variance from the 'overall shape' of the mea surement function. Our approach applies mainly to measurement functions who se form is known analytically, but it helps also to understand the behaviou r of the variance when the measurement function is known at sufficiently ma ny points; in fact, we illustrate the concepts with both synthetic and real data.