Two Early Contributions to the Ewens Saga

Abstract

The mixture model devised by Fisher, Corbet and Williams [Journal of Animal Ecology 12 (1943) 42–58] for species sampling and the sequential prediction approach pioneered by Good [Biometrika 40 (1953) 237–264] and Good and Toulmin [Biometrika 43 (1956) 45–63] are both closely related to the Ewens sampling formula. Fisher’s two-parameter joint distribution for the species counts includes the Ewens distribution as the conditional distribution given the sample size. The log-series model, as it is known in the ecological literature, is closely related to a Poisson process model devised by Arratia, Barbour and Tavaré [Ann. Appl. Probab. 2 (1992) 519–535]. Oddly, despite its advantages for statistical inference, Fisher does not mention the conditional distribution. Likewise, athough Good (1953) pioneered the sequential prediction approach, neither he nor Toulmin discovered the Ewens process in a form equivalent to the modern-day Chinese restaurant process.