The accuracy of one‐dimensional systematic sampling

Sampling designs with equally spaced measurements along a temporal or spatial axis (one‐dimensional systematic sampling) are widely used in stereology (volume estimation from serial sections, linear integration), for Monte Carlo integration, in forest research (line sampling), in time‐series analysis and in demographic studies. An unbiased estimate of the integral of a function f ( x ) in an interval [0, h ] is obtained from a discrete number of values of f ( x ) at arguments separated by a constant distance d , provided that one of the arguments is selected with uniform probability within an interval of length d. In this paper the accuracy of one‐dimensional random systematic sampling is explored. The relative mean square error of the estimate, CE 2 (Â), is determined for six geometrical models of f ( x ) directly by integration (which leads to explicit results) or indirectly via a Fourier transformation of f ( x ) (which leads to results in form of series expansions). Some CE 2 s thus obtained depend only on n =[ m ] and p = m ‐[ m ], the integer and decimal part of the real number m = h/d. These non‐elementary expressions have recently been denoted as np ‐functions (Mattfeldt, 1987). In practical studies lack of independent replications in systematic sampling often precludes the unbiased estimation of CE ( Â ). Short‐cut estimators were based on treatment of the systematic sample as a simple random sample, on the empirical autocorrelation, on various differences between the elements, on interpenetrating subsamples, and on the theory of regionalized variables. More involved methods which usually require a computer rely on Monte Carlo simulation, regression of log CE on log m , Fourier analysis, the Euler‐MacLaurin theorem, and methods of time‐series analysis. In an empirical study the absolute volumes of ten rat hearts were estimated from sets of systematic slices. Acceptable estimates of CE ( Â ) were obtained with the short‐cut estimators based on the theory of regionalized variables, by Monte Carlo methods, and with regression of log CE on log m.