Classical confidence intervals and Bayesian probability estimates for ends of local taxon ranges

The observed local range of a fossil taxon in a stratigraphic section is almost certainly a truncated version of the true local range. True endpoints are parameters that may be estimated using only the assumption that fossil finds are distributed randomly between them. If thickness is rescaled so that true endpoints lie at 0 and 1, the joint distribution of gap lengths between fossil finds is given by the Dirichlet distribution. Observed ends of the range are maximum likelihood estimators of true endpoints, but they are biased seriously. Extension of the observed range at each end by a distance equal to the average gap length yields unbiased point estimators. Classical statistics can generate confidence intervals for ends of the taxon range; but with Bayesian inference, the probability that true endpoints lie in a certain region can be stated. For a 95% confidence level (classical) or a 95% probability (Bayesian), the range extensions exceed the observed range if the range is established on less than six finds; if only two finds are used, such range extensions are an order of magnitude longer than the observed range. Evidently the standard biostratigraphic practice that identifies zonal boundaries as horizons rather than confidence intervals may not be justified at the resolution of typical fossiliferous sections.