Bill’s 80th birthday

estimation of a normal mean matrix

A few days ago, I noticed the paper Estimation under matrix quadratic loss and matrix superharmonicity by Takeru Matsuda and my friend Bill Strawderman had appeared in Biometrika. (Disclaimer: I was not involved in handling the submission!) This is a “classical” shrinkage estimation problem in that covariance matrix estimators are compared under under a quadratic loss, using Charles Stein’s technique of unbiased estimation of the risk is derived. The authors show that the Efron–Morris estimator is minimax. They also introduce superharmonicity for matrix-variate functions towards showing that generalized Bayes estimator with respect to a matrix superharmonic priors are minimax., including a generalization of Stein’s prior. Superharmonicity that relates to (much) earlier results by Ed George (1986), Mary-Ellen Bock (1988),  Dominique Fourdrinier, Bill Strawderman, and Marty Wells (1998). (All of whom I worked with in the 1980’s and 1990’s! in Rouen, Purdue, and Cornell). This paper also made me realise Dominique, Bill, and Marty had published a Springer book on Shrinkage estimators a few years ago and that I had missed it..!

efficiency and the Fréchet-Darmois-Cramèr-Rao bound

 

Following some entries on X validated, and after grading a mathematical statistics exam involving Cramèr-Rao, or Fréchet-Darmois-Cramèr-Rao to include both French contributors pictured above, I wonder as usual at the relevance of a concept of efficiency outside [and even inside] the restricted case of unbiased estimators. The general (frequentist) version is that the variance of an estimator δ of [any transform of] θ with bias b(θ) is

I(θ)⁻¹ (1+b'(θ))²

while a Bayesian version is the van Trees inequality on the integrated squared error loss

(E(I(θ))+I(π))⁻¹

where I(θ) and I(π) are the Fisher information and the prior entropy, respectively. But this opens a whole can of worms, in my opinion since

• establishing that a given estimator is efficient requires computing both the bias and the variance of that estimator, not an easy task when considering a Bayes estimator or even the James-Stein estimator. I actually do not know if any of the estimators dominating the standard Normal mean estimator has been shown to be efficient (although there exist results for closed form expressions of the James-Stein estimator quadratic risk, including one of mine the Canadian Journal of Statistics published verbatim in 1988). Or is there a result that a Bayes estimator associated with the quadratic loss is by default efficient in either the first or second sense?
• while the initial Fréchet-Darmois-Cramèr-Rao bound is restricted to unbiased estimators (i.e., b(θ)≡0) and unable to produce efficient estimators in all settings but for the natural parameter in the setting of exponential families, moving to the general case means there exists one efficiency notion for every bias function b(θ), which makes the notion quite weak, while not necessarily producing efficient estimators anyway, the major impediment to taking this notion seriously;
• moving from the variance to the squared error loss is not more “natural” than using any [other] convex combination of variance and squared bias, creating a whole new class of optimalities (a grocery of cans of worms!);
• I never got into the van Trees inequality so cannot say much, except that the comparison between various priors is delicate since the integrated risks are against different parameter measures.

Larry Brown (1940-2018)

Just learned a few minutes ago that my friend Larry Brown has passed away today, after fiercely fighting cancer till the end. My thoughts of shared loss and deep support first go to my friend Linda, his wife, and to their children. And to all their colleagues and friends at Wharton. I have know Larry for all of my career, from working on his papers during my PhD to being a temporary tenant in his Cornell University office in White Hall while he was mostly away in sabbatical during the academic year 1988-1989, and then periodically meeting with him in Cornell and then Wharton along the years. He and Linday were always unbelievably welcoming and I fondly remember many times at their place or in superb restaurants in Phillie and elsewhere.  And of course remembering just as fondly the many chats we had along these years about decision theory, admissibility, James-Stein estimation, and all aspects of mathematical statistics he loved and managed at an ethereal level of abstraction. His book on exponential families remains to this day one of the central books in my library, to which I kept referring on a regular basis… For certain, I will miss the friend and the scholar along the coming years, but keep returning to this book and have shared memories coming back to me as I will browse through its yellowed pages and typewriter style. Farewell, Larry, and thanks for everything!

Charles M. Stein [1920-2016]

I have just heard that Charles Stein, Professor at Stanford University, passed away last night. Although the following image is definitely over-used, I truly feel this is the departure of a giant of statistics.  He has been deeply influential on the fields of probability and mathematical statistics, primarily in decision theory and approximation techniques. On the first field, he led to considerable changes in the perception of optimality by exhibiting the Stein phenomenon, where the aggregation of several admissible estimators of unrelated quantities may (and will) become inadmissible for the joint estimation of those quantities! Although the result can be explained by mathematical and statistical reasoning, it was still dubbed a paradox due to its counter-intuitive nature. More foundationally, it led to expose the ill-posed nature of frequentist optimality criteria and certainly contributed to the Bayesian renewal of the 1980’s, before the MCMC revolution. (It definitely contributed to my own move, as I started working on the Stein phenomenon during my thesis, before realising the fundamentally Bayesian nature of the domination results.)

“…the Bayesian point of view is often accompanied by an insistence that people ought to agree to a certain doctrine even without really knowing what this doctrine is.” (Statistical Science, 1986)

The second major contribution of Charles Stein was the introduction of a new technique for normal approximation that is now called the Stein method. It relies on a differential operator and produces estimates of approximation error in Central Limit theorems, even in dependent settings. While I am much less familiar with this aspect of Charles Stein’s work, I believe the impact it has had on the field is much more profound and durable than the Stein effect in Normal mean estimation.

(During the Vietnam War, he was quite active in the anti-war movement and the above picture from 2003 shows that his opinions had not shifted over time!) A giant truly has gone.