a [counter]example of minimaxity

A chance question on X validated made me reconsider about the minimaxity over the weekend. Consider a Geometric G(p) variate X. What is the minimax estimator of p under squared error loss ? I thought it could be obtained via (Beta) conjugate priors, but following Dyubin (1978) the minimax estimator corresponds to a prior with point masses at ¼ and 1, resulting in a constant estimator equal to ¾ everywhere, except when X=0 where it is equal to 1. The actual question used a penalised qaudratic loss, dividing the squared error by p(1-p), which penalizes very strongly errors at p=0,1, and hence suggested an estimator equal to 1 when X=0 and to 0 otherwise. This proves to be the (unique) minimax estimator. With constant risk equal to 1. This reminded me of this fantastic 1984 paper by Georges Casella and Bill Strawderman on the estimation of the normal bounded mean, where the least favourable prior is supported by two atoms if the bound is small enough. Figure 1 in the Negative Binomial extension by Morozov and Syrova (2022) exploits the same principle. (Nothing Orwellian there!) If nothing else, a nice illustration for my Bayesian decision theory course!

Benford’s Law satisfies Stigler’s Law

Looking around for other entries on Benford’s Law, I found this nice entry that attributes Benford’s Law to the astronomer Simon Newcomb, instead of Benford (who rediscovered the distribution fifty years later). This is quite in line with Stigler’s Law of Eponymy, which states that (almost) no scientific law is named after its original discoverer. The post of Peter Coles also covers the connection between Benford’s Law and Jeffreys’ prior for scale parameters, which is discussed in Jim Berger’s Statistical Decision Theory and Bayesian Analysis.