on Pitman closeness

I 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Pertinent to Symmetric Populations”, given by Mohammad Jozania, at the University of Manitoba next week, and it rescinded my former (if negative) interest in Pitman nearness (or closeness).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,

\text{Pr}_\theta(|\hat\theta_1(X)-\theta|<|\hat\theta_2(X)-\theta|)

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 counter-intuitive 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.)

5 Responses to “on Pitman closeness”

  1. William Volterman Says:

    Having done some work with Pitman Closeness myself, I have of course read your paper. It was quite an interesting read.

    And while I agree with many points raised, I don’t think that any definitive statements regarding Pitman Closeness can be made based on your article. In fact I believe there were strong points made on both sides. (Though it is my opinion that all sides had their minds made up definitively before reading the others arguments anyways.)

    You will find many papers now on Pitman Closeness, primarily with ordered data. (For where transitivity often follows.)

    That said, I’m sure you’ve already made up your mind, so I will not trouble you further, but it couldn’t hurt to look up a few new papers on PC.

  2. Hi

    I just notified by a friend of mine that you have made a comment on my recent talk that I gave in our weekly seminars.

    Thanks for your comments.

    My main research area is in decision theory and I have recently begun to do research in the area of ranked set sampling as well.
    As you may know, ranked set sampling (RSS) is an efficient method for data collecting which can be used to obtain better estimators of the population mean when there are some cheap and easy to use auxiliary information that can be used for ranking the sampling units before the actual measurement. There are many variations of RSS and recently(??) in a paper at the Biometrical Journal it is shown that the median ranked set sampling works very good in estimating the median for symmetric populations. After one of our weekly seminars given by a colleague of mine who is working mostly on PC, I became interested to see what would be the best ranked set type sampling design for estimating the median for symmetric populations using the PC as the criterion of performance. It turns out that here also the median RSS or its randomized version would be the best RSS design to follow.

    In addition, let d1 nd d2 be two independent and symmetrically distributed estimators about the parameter \theta. Among other results, we show that a sufficient condition for d1 to be PC to \theta than d2 is that the distribution og d1 to be more peaked about \theta than d2.

    Anyway, the seminar that I gave talks about both pros and cons of the PC as a criterion. In this work, we also obtain some results in the same spirit as the one you have in the JASA paper which can be used to show some of the deficiencies of PC. The whole point is that as an statistician when we are making inference we should not base all of our inference solely using one criterion and we should try to see the performance of the estimators using other criteria of performance.

    I have seen your JASA paper, I also read its comments and discussions. It seems to me that, those who are not agreeing with you about not using PC are having their own reasoning as well. It is obvious that PC suffers from some problems (and disabilities) but I think this does not mean we should ignore it totally. PC is born (whether we like or not) with all of its pros and cons and it is now part of our life as statisticians. Imagine what will happen if we decide not to use any criteria which suffers from some deficiencies and disabilities. Like people (no matter black or white, etc.) I think different criteria and methods have the right to exist in the statistics world.

    By the way, I can send you a copy of my talk if you are interested and thanks again for your comments.

    Regards
    Mohammad

    • Sure, I’d be interested in the talk (and the papers if any are available). I (and others) however remain unconvinced by the pros of Pitman closeness, but don’t take it personally!

  3. Hi,

    But what kind of criterions would you recomend for comparison of estimators, since you disagree with the use of Pitman closeness criterion?
    In your book “The Bayesian choice” you mentioned that integrated risk can served as measure for ordering of estimators, but it requires integration over sample space (and steps into the land of frequentists) and not very friendly when it comes to its values calculation in complex problem.

    I have no access to your JASA paper, maybe the answer is already there.

    I would really appreaciate for your thoughts about it.

    Tomas

    • To me, the comparison of estimators only makes sense in a decision-theoretic perspective, namely via a loss function. If your perspective is Bayesian, you use the posterior expected loss (integrating the error over the parameter space). If not, you use the frequentist risk (integrating over the observation space). The main point in my JASA paper wrt this aspect is that Pitman closeness does not fit within decision theory.

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