Herman Rubin

Herman Rubin, Professor of Statistics and Mathematics at Purdue University, passed away in West Lafayette, IN, on April 23, 2018; he was 91. Herman Rubin was among the last remaining great polymaths of the twentieth century. To all who knew him, or had heard about him, Herman was an inexplicable outlier in numerous ways. His unique ability to understand a new problem and arrive at the answer almost instantly baffled even the smartest mathematician. He never forgot a fact, theorem or proof that he had seen. He would solve a complete stranger’s problem without expecting co-authorship or anything else in return. He would fight for someone who opposed him at every step. He would always stand by his principles. Herman Rubin’s death marks the nearness of the end of a unique era following the second world war that saw the simultaneous emergence of a distinctive group of supremely talented statisticians who would shape the foundations of the subject for decades to come.

Herman Rubin was born October 27, 1926 in Chicago, Illinois. He obtained his PhD in Mathematics from the University of Chicago in 1948, at the age of 21; he was a student of Paul Halmos. He would serve the faculties of Stanford University, the University of Oregon, Michigan State University and Purdue University. After a stint at the Cowles Commission, he formed a productive intellectual affinity with Ted Anderson, Charles Stein and Ingram Olkin; at this time, he also became professionally close to David Blackwell and Meyer Girshick. With Ted Anderson, he wrote two phenomenal papers on fundamental multivariate analysis that worked out the fixed sample as well as asymptotic distribution theory of MLEs in factor analysis models and structural equation models. These results have entered into all standard multivariate analysis and econometrics texts, and have remained there for more than half a century. Herman’s most famous and classic contribution to inference is the widely used and fundamental idea of monotone likelihood ratio families. Anyone who has taken a course on testing hypotheses knows the absolute fundamentality of the idea and the results in the 1956 paper with Samuel Karlin. It was this work that led to Sam Karlin’s hugely influential TP2 and variation diminishing families, with shadows of Isaac Schoenberg and Bill Studden lurking in the background.

Following this period of a series of fundamental papers in statistics, we see a shift. He makes novel entries into various aspects of probability and asymptotics. With J. Sethuraman, he does theory of moderate deviations. With Herman Chernoff, he attacks the (then completely novel) problem of estimating locations of singularities, and how, precisely, the asymptotics are completely new. With Prakasa Rao as his student, he gets into the problem of cube root asymptotics for monotone densities. With C.R. Rao, he gives the classic Rao–Rubin characterization theorem. And, to many, the jewel in the crown is the invention of the idea of a Stratonovich integral.

Herman really did enjoy particular problems, as long as they were not mundane. Classic examples are his papers with Rick Vitale, quite shocking, that sets of independent events characterize an underlying probability measure, degeneracies aside; the pretty work with Jeesen Chen and Burgess Davis on how non-uniform can a uniform sample look to the eye; with Tom Sellke on roots of smooth characteristic functions; the Bayesian formulation of quality control with Meyer Girshick; the hilariously bizarre, but hard, problem of estimating a rational mean; his papers with Andrew Rukhin on the positive normal mean; the work on the notorious Binomial N problem; and Bayesian robustness of frequentist non-parametric tests… among others. Herman never considered who would cite or read a result; if he wanted to solve a problem, he did.

Herman was probably one of the lifelong Bayesians, but axiomatically so. He really did take most of the Savagian theory and axioms literally; he expanded on them, though later. An expansion was published in Statistics and Decisions (I believe, with extremely active help from Jim Berger). He would not budge an inch from his conviction that the loss and the prior are inseparable. He would refuse to discuss what is an appropriate loss function; he would insist you ask the client. He would nevertheless want to see the full risk function of a procedure and would study Bayes through the lens of Bayes risk—and even exclusively Bayes risk, namely the double integral. On asymptotic behaviors of procedures, he did not appear to care for second-order terms. He showed his concern time and again for only calculating a limit. A glorious example of this is his work with J. Sethuraman on efficiency defined through Bayes risks; this was so novel that it entered into the classic asymptotic text of Robert Serfling. He came back to it many years later in joint work with Kai-Sheng Song in an Annals of Statistics article.

In some ways, Herman was a paradox. He would publicly say only Bayes procedures should be used—but oppose the use of a single prior with all his teeth. He would be technically interested in the robustness of traditional frequentist procedures, although he would portray them as coming out of wrong formulations. (Well known examples are his oft-cited papers with Joe Gastwirth on the performance of the t-test under dependence.) He did not have a personal desire to burn the midnight oil writing a comprehensive review of some area; but he would be an invaluable asset to the one who was. An example is his review of infinitely divisible distributions with Arup Bose (and this writer). An all-time classic is his text on Equivalents of the Axiom of Choice, jointly written with his wife Jean E. Rubin, Professor of Mathematics at Purdue. Jim Berger thanks Herman profusely in the preface of his classic Springer book on decision theory and Bayesian analysis; Charles Stein acknowledges Herman (and Herbert Robbins) in his first shrinkage paper.

There was a fairly long period in the history of statistics, when nearly every paper written in his home department had Herman’s contributions to it. He never asked for, or received, credit for them. He epitomized the term scholar in its literal sense. With the passing of Herman Rubin, a shining beacon of knowledge and wisdom just moved. Herman was a consummate master of simulation, characteristic functions, and infinitely divisible distributions. He kept to himself a mountain of facts and results on these and other topics that never saw the light of the day. There was never a person who did not respect Herman Rubin’s brain; even Paul Erdős did. Herman never stopped thinking of good problems, and loved discussing them. He personified Albert Einstein’s quote, “Intellectual growth should commence at birth and cease only at death”.

The IMS Lecture Notes–Monograph Series published a collection of research articles in Herman Rubin’s honor in 2004; numerous leaders of the profession wrote an original article for this collection. As well as being IMS and AMS Fellow, he was also a Fellow of the American Association for the Advancement of Science, and a member of Sigma Xi.

Herman had sophisticated taste in music and literature. He was often seen in classical concerts and operas. He helped mathematical causes financially. Herman was probably one of the very few people anywhere who could work out the NY Times crossword puzzle on any day in about an hour.

Herman is survived by his son Arthur, and daughter Leonore. His wife Jean died in 2002.

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Written by Anirban DasGupta, Purdue University