Although it may sound like an excessive notion of optimality, one can hope at obtaining an estimator δ of a unidimensional parameter θ that is always closer to θ that any other parameter. In distribution if not almost surely, meaning the cdf of (δθ) is steeper than for other estimators enjoying the same cdf at zero (for instance ½ to make them all medianunbiased). When I saw this question on X validated, I thought of the Cauchy location example, where there is no uniformly optimal estimator, albeit a large collection of unbiased ones. But a simulation experiment shows that the MLE does better than the competition. At least than three (above) four of them (since I tried the Pitman estimator via Christian Henning’s smoothmest R package). The differences to the MLE empirical cd make it clearer below (with tomato for a score correction, gold for the Pitman estimator, sienna for the 38% trimmed mean, and blue for the median):I wonder at a general theory along these lines. There is a vague similarity with Pitman nearness or closeness but without the paradoxes induced by this criterion. More in the spirit of stochastic dominance, which may be achievable for location invariant and mean unbiased estimators…
Archive for Pitman closeness
more concentration, everywhere
Posted in R, Statistics with tags best equivariant estimator, Cauchy distribution, cross validated, ecdf, Pitman best equivariant estimator, Pitman closeness, Pitman nearness, R, smoothmest R package, stochastic dominance, uniform optimality on January 25, 2019 by xi'anPitman medal for Kerrie Mengersen
Posted in pictures, Statistics, Travel, University life with tags Amazon, Australia, best equivariant estimator, Comptes Rendus de l'Académie des Sciences, EJG Pitman, exponential families, George Darmois, invariance, jaguars, medal, Pitman closeness, prizes, QUT, Statistical Society of Australia. on December 20, 2016 by xi'anMy friend and coauthor of many years, Kerrie Mengersen, just received the 2016 Pitman Medal, which is the prize of the Statistical Society of Australia. Congratulations to Kerrie for a welldeserved reward of her massive contributions to Australian, Bayesian, computational, modelling statistics, and to data science as a whole. (In case you wonder about the picture above, she has not yet lost the medal, but is instead looking for jaguars in the Amazon.)
This medal is named after EJG Pitman, Australian probabilist and statistician, whose name is attached to an estimator, a lemma, a measure of efficiency, a test, and a measure of comparison between estimators. His estimator is the best equivariant (or invariant) estimator, which can be expressed as a Bayes estimator under the relevant right Haar measure, despite having no Bayesian motivation to start with. His lemma is the PitmanKoopmanDarmois lemma, which states that outside exponential families, sufficient is essentially useless (except for exotic distributions like the Uniform distributions). Darmois published the result first in 1935, but in French in the Comptes Rendus de l’Académie des Sciences. And the measure of comparison is Pitman nearness or closeness, on which I wrote a paper with my friends Gene Hwang and Bill Strawderman, paper that we thought was the final paper on the measure as it was pointing out several majors deficiencies with this concept. But the literature continued to grow after that..!
Pitman closeness renewal?
Posted in Statistics, University life with tags decision theory, JASA, median, Pitman closeness, Pitman nearness on July 26, 2012 by xi'anAs noticed there a few months ago, the Pitman closeness criterion for comparing estimators (through the probability
P_{θ}(δθ<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:

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. X_{3,} 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;
 the setup is highly dependent on considering a single (onedimensional) location parameter, the results do not apply to more general settings (except locationscale cases with scale parameters known to some extent, see Lemma 5) ;
 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.
on Pitman closeness
Posted in Statistics, University life with tags decision theory, Pitman closeness, Pitman nearness, transitivity on November 15, 2011 by xi'anI came by happenstance upon this talk, “Some Pitman Closeness Properties Pertinent to Symmetric Populations”, given by Mohammad Jozania, at the University of Manitoba next week, and it rescinded my former (if negative) interest in Pitman nearness (or closeness). This criterion, which originated in a 1937 paper of E.J.G. Pitman, compares two estimators in the light of the probability of one being closer (to the “truth”) than the other,
and there was a brief interest in the method at the end of the 1980’s, culminating with Keating and Mason’s book on the topic.
In a 1993 JASA paper I wrote with Gene Hwang and Bill Strawderman, entitled “Is Pitman Closeness a Reasonable Criterion?“, we demonstrated that, in many respects, this criterion was not appropriate for comparing estimators. For instance, the comparison was not transitive, two estimators with the same marginal distribution could sometimes be ranked, a Bayes estimator could not be properly derived, some counterintuitive orderings could be exhibited, &tc… This was an exciting (and fun) paper to write as it was only made of (counter)examples. (Hence our answer to the above question was definitive no.) Judging from the abstract to the talk,
In this talk, we focus on Pitman closeness probabilities when the estimators are symmetrically distributed about the unknown parameter θ. We first consider two symmetric estimators θ¹ and θ² and obtain necessary and sufficient conditions for θ¹ to be Pitman closer to the common median θ than θ². We then establish some properties in the context of estimation under Pitman closeness criterion. We define a Pitman closeness probability which measures the frequency with which an individual order statistic is Pitman closer to θ than some symmetric estimator. We show that, for symmetric populations, the sample median is Pitman closer to the population median than any other symmetrically distributed estimator of θ. Finally, we discuss the use of Pitman closeness probabilities in the determination of an optimal ranked set sampling scheme (denoted by RSS) for the estimation of the population median when the underlying distribution is symmetric. We show that the best RSS scheme from symmetric populations in the sense of Pitman closeness is the median and randomized median RSS for the cases of odd and even sample sizes, respectively.
it sounds like the authors have relaunched research in this area, hence that our 1993 definitive conclusion against the use of the criterion was not definitive for everyone… (I could not find a trace of the corresponding paper through google, but I would be interested in reading the recent research on the topic! Even though the result about the “optimality” of the sample median reminds me of earlier results, with the related drawback that this optimality is incompatible with the sufficiency principle.)