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Is there a good published expository account, with mathematical details, of the various approaches that have been taken to the Behrens–Fisher problem?

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@Proc: Thanks for that clarification. I see what you are suggesting now. :) It may be hard to find sufficiently general overviews of sufficient depth in paper introductions. I'd be interested in your list, particularly any subset that seems most relevant to Michael's question. – cardinal Jul 25 '12 at 14:25
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@cardinal Comparison of tests 1. Classical approach 2. Classical, fiducial and frequentist 3. Empirical Bayesian 4. Bayesian 5. Bootstrap 6. Nonparametric 7. – user10525 Jul 25 '12 at 14:50
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Jeffreys priors 8. Reference priors 9. Matching priors 10. – user10525 Jul 25 '12 at 15:03
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@Proc: Those would make a wonderful, rather comprehensive answer. Y. Linnik was prolific in his studying of this problem and proved some of the significant negative results about this problem. A short overview of some of it can be found in Y. V. Linnik (1966), Latest investigations on Behrens-Fisher problem Sankyha, Series A, Vol. 28, No. 1, pp. 15-24. – cardinal Jul 25 '12 at 16:31
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There is also: S-H Kim and A. Cohen (1998), On the Behrens-Fisher problem: A review, J. Educ. and Behav. Stat., vol. 23, no. 4, pp. 356-377. I have only very briefly skimmed it, though, so I really can't vouch for it. – cardinal Jul 25 '12 at 16:44
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This article by L. J. Savage in 1976 was the motivation for a seminar we held for graduate students and professors at Stanford in 1977. I was a student then and gave my talk on the Behrens-Fisher problem. Faculty and visiting faculty participating included Seymour Geisser, Brad Efron and David Hinkley (and possibly other that I can't recollect). Paper from Annals of Statistics 1976 "On Rereading R. A. Fisher." The work and controversy on the Behren's-Fisher problem was one of many topics discussed through Savage's interpretation of Fisher's writings which I think included some heated debates. One with M. S. Barlett in particular. Savage points to the gems of wisdom more than this one flaw. This problem was the one that exposed the difference between fiducial inference and the Neyman-Pearson hypothesis testing approach. Prior to that Fisher recognized philosophical differences but thought that the two methods gave the same answers. But they do differ when nusiance parameters are involved (in the case of Behrens-Fisher the unknown population variance).

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1176343456

In the questioning period of my talk I discovered that Seymour Geisser was like an encyclopedia on this problem. you may find his book (published around the time of his death) Modes of Statistical Inference which is a rare book that discusses fiducial inference along with frequentist and Bayesian approaches. here is an amazon link for that. This link also conatins my customer review of the book which includes a lot of what I have said here about Seymour. Modes of Parametric Statistical Inference by Seymour Geisser and Wesley Johnson.

http://www.amazon.com/Parametric-Statistical-Inference-Probability-Statistics/dp/0471667269/ref=cm_cr_pr_product_top

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FYI-- using named hyperlinks is actually quite simple: while you're typing your answer click the little icon that looks like a chain link. It will pop up a window that will make the process self explanatory. – Macro Jul 24 '12 at 22:31
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Thanks. I will try next time. I am worn out from today. – Michael Chernick Jul 24 '12 at 22:35
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Chuanhai Liu recently developed an interesting framework of statistical inference, called 'Inferential Model'. Behrens-Fisher problem is one of the examples which can be quite elegantly tackled using this framework; if interested, take a look at Chapter 4.2 of the following paper.

http://www.stat.purdue.edu/~chuanhai/docs/immarg.pdf

It also contains some references to a number of key papers and review papers. I am not an expert on this topic, so I am not sure how comprehensive the reference is, but I guess it could be a good starting point!

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This looks like an interesting paper. The Welch papers are classics. They provided a frequentist approach. This is commonly used today with a t approximation which usually will have non-integer degrees of freedom. It is commonly used in packages such as SAS and you even can find it in the Excel t test of its data analysis packages. The work of Fisher is unfortunately not referenced. I don't think the fiducial approach is emphasized. The Segal 1938 paper is one that I am not familiar with but must represent the Fisherian approach at the time before fiducial inference went out of favor. – Michael Chernick Jul 25 '12 at 3:08
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Barnard is an authority and his recent paper that is referenced should be good. The papers by P.L. Hsu and Henry Scheffe are also frequestist approaches – Michael Chernick Jul 25 '12 at 3:09
It is interesting that the problem has had revived research interest in recent years and I believe this are ticle by Martin provides a good survey of recent work. – Michael Chernick Jul 25 '12 at 3:16

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