A tale of two variances

We begin by showing that the standard repeated-sampling interpretation of the variance of a parameter estimate in a finite-dimensional parametric model is ambiguous and open to misinterpretation. Three operational interpretations are given, all numerically different in general and all compatible with repeated sampling from the same population with a fixed parameter. One of these is compatible with the standard large-sample calculation based on the inverse Fisher information. The others are not. One interpretation coincides with what Fisher appears to have had in mind in his 1943 derivation of the log-series model for species abundances. The different interpretations help to resolve an apparent contradiction between the Fisherian variance and the inverse-information variance obtained from the Ewens model.