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SUMMARY Classical stereological methods are based on the assumption of isotropy, either of the structure under study or of the sectioning probe relative to the structure. Thus, with "distribution-free stereology", anisotropic structures can be studied only via isotropic uniform random probes, which are often difficult to generate or inefficient in practice. Here we present a parametric approach to model the direction of a randomly chosen line or surface element, whereby length and surface densities can be estimated by probes at fixed and known directions. For the material studied, namely skeletal muscle capillaries, the Dimroth-Watson axial distribution provides a good fit. A maximum likelihood procedure for estimating simultaneously the concentration parameter of the Dimroth-Watson distribution and either the length or the surface density for the structure is described and illustrated with the capillary data. When a particular axial model is known to be satisfactory, a short-cut estimation method of practical value can be used.

Precise criteria have been published recently for obtaining unbiased ratio estimators of structural parameters, defined in an n-dimensional opaque specimen, from observations in lower-dimensional sections. In this paper, the possibility is shown of obtaining linear unbiased estimators of minimum variance whenever the data can be described by a linear regression model through the origin. Estimation from singleand two-stage sampling is discussed; the former is illustrated by an example.