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Optimal estimators of the position of a mass extinction when recovery potential is uniform

¹ Department of Mathematics and Statistics, Swarthmore College, Swarthmore, Pennsylvania 19081scwang{at}swarthmore.edu

Numerous methods have been developed to estimate the position of a mass extinction boundary while accounting for the incompleteness of the fossil record. Here we describe the point estimator and confidence interval for the extinction that are optimal under the assumption of uniform preservation and recovery potential, and independence among taxa. First, one should pool the data from all taxa into one combined "supersample." Next, one can then apply methods proposed by Strauss and Sadler (1989) for a single taxon. This gives the optimal point estimator in the sense that it has the smallest variance among all possible unbiased estimators. The corresponding confidence interval is optimal in the sense that it has the shortest average width among all possible intervals that are invariant to measurement scale. These optimality properties hold even among methods that have not yet been discovered. Using simulations, we show that the optimal estimators substantially improve upon the performance of other existing methods. Because the assumptions of uniform recovery and independence among taxa are strong ones, it is important to assess to what extent they are satisfied by the data. We demonstrate the use of probability plots for this purpose. Finally, we use simulations to explore the sensitivity of the optimal point estimator and confidence interval to nonuniformity and lack of independence, and we compare their performance under these conditions with existing methods. We find that nonuniformity strongly biases the point estimators for all methods studied, inflates their standard errors, and degrades the coverage probabilities of confidence intervals. Lack of independence has less effect on the accuracy of point estimates as long as recovery potential is uniform, but it, too, inflates the standard errors and degrades confidence interval coverage probabilities.