## Pitman closeness renewal?

As noticed there a few months ago, the Pitman closeness criterion for comparing estimators (through the probability

Pθ(|δ-θ|<|δ’-θ|)

which should be larger than .5 for the first estimator to be deemed “better” or “Pitman closer”) has been “resuscitated” by Canadian researchers. In 1993, I wrote a JASA (discussion) paper along with Gene Hwang and Bill Strawderman pointing out the many inconsistencies of this criterion as a decision tool.  It was entitled “Is Pitman Closeness a Reasonable Criterion?” (The answer was in the question, right?!)

In an arXiv posting today, Jozani, Balakrishnan, and Davies propose new characterisations for comparing (in this sense) symmetrically distributed estimators. There is nothing wrong with this mathematical exercise, obviously. However, the approach still seems to suffer from the same decisional inconsistencies as in the past:

1. the results in the paper (see, e.g., Lemma 1 and 2) only apply to independent estimators, which is rather unrealistic (to the point of having the authors applying it to dependent estimators, the sample median X[n/2] versus a fixed index observation, e.g. X3, and again at the end of the paper in the comparison of several order statistics). Having independent estimators to compare is a rather rare situation as one tries to make the most of a given sample;
2. the setup is highly dependent on considering a single (one-dimensional) location parameter, the results do not apply to more general settings (except location-scale cases with scale parameters known to some extent, see Lemma 5) ;
3. some results (see Remark 4) allow to find a whole range of estimators dominating a given (again independent) estimator δ’, but they do not give a ranking of those estimators, except in the weak sense of having the above probability maximal in one of the estimators δ (Lemma 9). This is due to the independence constraint on the comparison. There is therefore no possibility (in this setting) of obtaining an estimator that is the “Pitman closest estimator of θ“, as claimed by the authors in the final section of their paper.

Once again, I have nothing against these derivations, which are mostly correct, but I simply argue here that they cannot constitute a competitor to standard decision theory.