Paleontological completeness and stratigraphic ranges depend on extinction rate, origination rate, preservation rate, and the length of the interval of time over which observations can be made. I derive expressions for completeness and the distribution of durations and ranges as functions of these parameters, considering both continuous- and discrete-time models.
Previous work has shown that, if stratigraphic ranges can be followed indefinitely forward, and if extinction and preservation occur at stochastically constant rates, then extinction rate and preservability can be estimated from (1) discrete (binned) stratigraphic ranges even if data on occurrences within ranges are unknown, and (2) continuous ranges if the number of occurrences within each range is known. I show that, regardless of whether the window of observation is finite or infinite, extinction and preservation rates can also be estimated from (3) continuous ranges when the number of occurrences is not known, and (4) discrete ranges when the number of occurrences is not known. One previous estimation method for binned data involves a sample-size bias. This is circumvented by using maximum likelihood parameter estimation. It is worth exploiting data on occurrences within ranges when these are available, since they allow preservation rate to be estimated with less variance. The various methods yield comparable parameter estimates when applied to Cambro-Ordovician trilobite species and Cenozoic mammal species.
Stratigraphic gaps and variable preservation affect stratigraphic ranges predictably. In many cases, accurate parameter estimation is possible even in the face of these complications. The distribution of stratigraphic ranges can be used to estimate the sizes of gaps if their existence is known.
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