## Jack Wolfowitz: Applying Mathematical Statistics to Practical Problems

##### This is an excerpt from Jack Wolfowitz’s memoriam, which was written by Lionel Weiss and published in *The American Statistician* in 1981.

Jack (Jacob) Wolfowitz was born in Warsaw, Poland, on March 19, 1910, and came to the United States with his family in 1920. He graduated from the College of the City of New York in 1931 and worked toward his PhD while supporting himself as a high-school teacher. He earned his PhD in mathematics from New York University in 1942.

In 1934, Wolfowitz married Lillian Dundes, who he remained committed to in marriage for the rest of his life.

In 1938, Abraham Wald came to Columbia University; it was then that Wolfowitz and he started their remarkable collaboration. Their first joint paper, “Confidence Limits for Continuous Distributions,” was published in the *Annals of Mathematical Statistics * in 1939. Wolfowitz revered Wald, and after Wald was killed in a plane crash in 1950, a large photograph of Wald occupied a prominent place on Wolfowitz’s office wall.

During World War II, Wald and Wolfowitz worked together on war-related research at the Statistical Research Group at Columbia University. In 1945, Wolfowitz became an associate professor at The University of North Carolina at Chapel Hill. In 1946, he joined the faculty at Columbia University, where he stayed until 1951. In 1951, he joined the department of mathematics at Cornell University. In 1970, he joined the department of mathematics at the University of Illinois at Urbana-Champaign. When he retired from the last position in 1978, he became distinguished professor at the University of South Florida at Tampa, a position he held until he died on July 16, 1981.

As a teacher, Wolfowitz was unmatched in his ability to explain the intuition underlying the most complicated research results. His own drive toward intuitive understanding led him to many interesting results. An outstanding example of this was his investigation of why the maximum likelihood estimator works as well as it does in so many cases, an investigation that started in 1949 with “On Wald’s Proof of the Consistency of the Maximum Likelihood Estimate,” published in the *Annals of Mathematical Statistics, *and continued for most of the rest of his life.

It is interesting that during this investigation, Wolfowitz and Jack Kiefer found what seems to be the first example of a case that might arise in practice in which the method of maximum likelihood breaks down: It is the program of estimating µ and σ based on n iid observations, each with

which is discussed in “Consistency of the Maximum Likelihood Estimator in the Presence of Infinitely Many Incidental Parameters,” published in 1956 in *Annals of Mathematical Statistics.*

Wolfowitz’s research touched every important area of mathematical statistics and extended outside of mathematical statistics into such fields as probability theory, inventory theory, and coding theory. In this last field, he is considered as eminent a researcher as he is in mathematical statistics. A good account of his research, as well as a complete bibliography, can be found in* Jacob Wolfowitz: Selected Papers, * edited by Kiefer and published by Springer in 1980.

Most of Wolfowitz’s publications are highly mathematical, but much of his research has direct practical application. An example is the sequence of papers on the inventory problem, written with Aryeh (Arie) Dvoretzky and Kiefer, which appeared in *Econometrica* in April 1952, July 1952, and October 1953. These were pioneering papers in what is now a whole field of study; operations research departments give full courses in inventory theory.

Wolfowitz also felt strongly that the ultimate justification for research in mathematical statistics is its applicability to practical problems. In this connection, his 1967 paper, “Remarks on the Theory of Testing Hypotheses,” published in the *New York Statistician*, is particularly interesting. A few excerpts will illustrate the outlook expressed in this paper:

“The practical statisticians who now accept useless theory should rebel and not do so any more.”

“What about the mathematical statisticians, especially the talented and mathematically productive among them? Most of the papers they write on testing hypotheses are, at their best, mathematically ingenious and difficult, of no general mathematical interest, and no lasting statistical importance.”

It is significant that Wolfowitz chose this nontechnical article to be included in the volume of selected papers.

Wolfowitz’s research contains many asymptotic results. His attitude toward asymptotic theory was that it was merely a guide to the choice of appropriate statistical decision rules for moderate sample sizes, necessary because computations that are impractical for finite sample sizes often become relatively easy as sample sizes increase. Large-scale computer simulation is just becoming economically feasible, which will make it possible to evaluate statistical decision rules for small and moderate sample sizes. I think Wolfowitz would welcome such Monte Carlo investigations.

His honors include an honorary doctorate from the Technion in 1975, election to the National Academy of Sciences and American Academy of Arts and Sciences, and becoming a Fellow of the ASA, Econometric Society, International Statistical Institute, and Institute of Mathematical Statistics. He served a term as president of the Institute of Mathematical Statistics and was chosen as Rietz Lecturer and Wald Lecturer. In 1979, he was the Shannon Lecturer of the Institute of Electrical and Electronic Engineers. He gave invited addresses at numerous scientific meetings.

Wolfowitz read widely, traveled often, and kept himself informed about current events in all of the large nations and many of the smaller ones. A large university attracts visitors from all over the world, and Wolfowitz liked to get the points of view of such visitors. He was active in organizing protests against Soviet repression of minorities, intellectuals, and dissidents.

Wolfowitz enjoyed long walks in the countryside and welcomed companions on these walks. If all the walkers were professional colleagues, the conversation was often on technical matters, and several joint papers owed their origin to such walking discussions.

Wolfowitz had a profound influence on professionals all over the world through his teaching, published research, and lectures. For the foreseeable future, the development of several fields will reflect this influence.