Open Access
May 2023 Fisher’s measure of variability in repeated samples
Poly H. da Silva, Arash Jamshidpey, Peter McCullagh, Simon Tavaré
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Bernoulli 29(2): 1166-1194 (May 2023). DOI: 10.3150/22-BEJ1494


Fisher (1943) claimed that the expected value of the sample variance of the number of species found in large samples, each of n specimens taken from the same population, is asymptotically θlog2. This is at odds with the value θlogn obtained directly from the Ewens Sampling Formula (ESF), where θ specifies the rate at which new species are found. To resolve this apparent contradiction, we assume the species frequency spectrum in the population is determined by the ESF and that the samples are disjoint subsets drawn sequentially from this single population. We find an explicit formula for the required expected value for p samples of arbitrary size; in the limit of large equally-sized samples, it indeed has the value θlog2. We obtain limit theorems for the sample variance of p samples of size n under various limiting regimes as p,n or both tend to ∞. We discuss further the behavior of the number of species present in all samples, and revisit Fisher’s log-series distribution as the limiting distribution of the number of specimens observed in typical species in a future, large sample.


We thank Stephen Senn for bringing the Anscombe-Fisher correspondence to our attention. We thank two reviewers and the associate editor for comments that improved the paper. PHdS and ST were supported in part by NSF grant DMS-2030562.


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Poly H. da Silva. Arash Jamshidpey. Peter McCullagh. Simon Tavaré. "Fisher’s measure of variability in repeated samples." Bernoulli 29 (2) 1166 - 1194, May 2023.


Received: 1 October 2021; Published: May 2023
First available in Project Euclid: 19 February 2023

MathSciNet: MR4550219
zbMATH: 07666814
Digital Object Identifier: 10.3150/22-BEJ1494

Keywords: Chinese restaurant process , Ewens sampling formula , exchangeability , log-series model , Poisson approximation , sequential sampling

Vol.29 • No. 2 • May 2023
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