best unbiased estimators

A question that came out on X validated today kept me busy for most of the day! It relates to an earlier question on the best unbiased nature of a maximum likelihood estimator, to which I pointed out the simple case of the Normal variance when the estimate is not unbiased (but improves the mean square error). Here, the question is whether or not the maximum likelihood estimator of a location parameter, when corrected from its bias, is the best unbiased estimator (in the sense of the minimal variance). The question is quite interesting in that it links to the mathematical statistics of the 1950’s, of Charles Stein, Erich Lehmann, Henry Scheffé, and Debabrata Basu. For instance, if there exists a complete sufficient statistic for the problem, then there exists a best unbiased estimator of the location parameter, by virtue of the Lehmann-Scheffé theorem (it is also a consequence of Basu’s theorem). And the existence is pretty limited in that outside the two exponential families with location parameter, there is no other distribution meeting this condition, I believe. However, even if there is no complete sufficient statistic, there may still exist best unbiased estimators, as shown by Bondesson. But Lehmann and Scheffé in their magisterial 1950 Sankhya paper exhibit a counter-example, namely the U(θ-1,θ-1) distribution:

since no non-constant function of θ allows for a best unbiased estimator.

Looking in particular at the location parameter of a Cauchy distribution, I realised that the Pitman best equivariant estimator is unbiased as well [for all location problems] and hence dominates the (equivariant) maximum likelihood estimator which is unbiased in this symmetric case. However, as detailed in a nice paper of Gabriela Freue on this problem, I further discovered that there is no uniformly minimal variance estimator and no uniformly minimal variance unbiased estimator! (And that the Pitman estimator enjoys a closed form expression, as opposed to the maximum likelihood estimator.) This sounds a bit paradoxical but simply means that there exists different unbiased estimators which variance functions are not ordered and hence not comparable. Between them and with the variance of the Pitman estimator.

Jubilee at the University of Calcutta

The main reason for my trip to India was taking part in the celebrations of the 75th anniversary of the Department of Statistics at the University of Calcutta and of the 100th anniversary of the birth of P.K. Bose (whom I did not know before visiting Kolkata). The Department of Statistics was created in 1941 by Mahalanobis, the very first statistics department in Asia. (Mahalanobis was also instrumental in creating the ISI in 1932. And Sankhyā in 1933.)  Fisher visited Calcutta very often and was very supportive of Mahalanobis’ efforts: in the corridor, the above picture of Fisher is displayed, with him surrounded by faculties and graduates from the Department when he came in 1941.

Although I missed the first two days of the conference (!), I enjoyed very much the exchanges I had with graduate students there, about my talk on folded MCMC and other MCMC and Bayesian issues. (With The Bayesian Choice being an easy conversational bridge-way between us as it is their Bayesian textbook.) The setting reminded me of the ISBA conference in Varanasi four years ago, with the graduate students being strongly involved and providing heavy support in the organisation, as well as eager to discuss academic and non-academic issue. (Plus offering us one evening an amazing cultural show of songs and dances.)

at CIRM

Thanks to a very early start from Paris, and despite horrendous traffic jams in Marseilles, I managed to reach CIRM with ten minutes to spare before my course. After my one-hour class, I was suddenly made aware of the (simplistic) idea that the slice sampling uniforms are simply auxiliary, meaning they can be used in many different ways.

I noticed Natesh Pillai just arXived an extension of his earlier Cauchy paper with XL. He proves that the result on the Cauchy distribution of any convex combination of normal ratios still holds when the pair of vectors is distributed from a product of elliptically symmetric functions. Some of Natesh’s remarks reminded me of the 1970 Sankhyã paper by Kelker on spherically symmetric variables. Especially because of Kelker’s characterisation of elliptically symmetric functions as scale mixtures of normals, which makes perfect sense since the scale cancels.

As I skimmed through my slides yesterday, fearing everyone knew about the MCMC basics, I decided to present today the Rao-Blackwellisation slides I gave in Warwick a few months ago.

From Series B to Series A

The respite from editorial duties has been rather brief after all since, a bit longer than a month after (reluctantly) leaving the co-editorship of the Journal of the Royal Society, Series B, I have received a proposal this morning to join the editorial board of Sankhya, Series A! A proposal that I swiftly accepted, therefore joining the editorial board of Sankhya, Series A for the second time.