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!

bounded normal mean

A few days ago, one of my students, Jacopo Primavera (from La Sapienza, Roma) presented his “reading the classic” paper, namely the terrific bounded normal mean paper by my friends George Casella and Bill Strawderman (1981, Annals of Statistics). Even though I knew this paper quite well, having read (and studied) it myself many times, starting in 1987 in Purdue with Mary Ellen Bock, it was a pleasure to spend another hour on it, as I came up with new perspectives and new questions. Above are my scribbled notes on the back of the [Epson] beamer documentation. One such interesting question is whether or not it is possible to devise a computer code that would [approximately] produce the support of the least favourable prior for a given bound m (in a reasonable time). Another open question is to find the limiting bounds for which a 2 point, a 3 point, &tc., support prior is the least favourable prior. This was established in Casella and Strawderman for bounds less than 1.08 and for bounds between 1.4 and 1.6, but I am not aware of other results in that direction… Here are the slides used by Jacopo: