In Vienna, Birnbaum entered a gymnasium which had been set up for Polish speaking refugees so had lessons in Polish. However, he was soon bored at this school, so persuaded his parents to have him educated at home with a private tutor. The tutor spent one hour per day with Birnbaum and he was able to take examinations each year which the Viennese State set specifically for home educated pupils. He passed these examinations in 1915, 1916, 1917 and 1918 and after World War I ended in 1918 the family returned to Lwów. This was now a Polish city since Poland had declared itself an independent state in November 1918. He enrolled in the H Sienkiewicz State Gymnasium No. X in Lwów but, quickly becoming bored, spent a while being home educated again. He decided to spend the year 1920-21 at the Gymnasium preparing for the matura examination. Here his interests were turned towards mathematics by an inspiring young teacher :-
... mathematics was taught there by a young doctoral student, whose enthusiasm for set theory and topology exceeded his fear of the principal's reprimand for violating the official syllabus. He did not pay too much attention to the logarithmic tables and trigonometry, and talked in class about mathematics in a way that left a permanent impression on Wilhelm.Due to the war, the Birnbaum family's financial position was by now much less favourable so it was decided that Birnbaum needed to get a degree in a practical subject - certainly mathematics was not considered to be in this category. He therefore applied to Lwów University to enter the medical faculty and, in case that was unsuccessful, he also applied to enter the engineering faculty. However, Birnbaum was Jewish and there was a strict quota which severely limited the number of Jewish students who could enrol. Both his applications failed so he made a third application to study law and this time he was admitted. He took the necessary courses on law but also took mathematics courses since these were what he really loved. He did well in the law courses and was awarded a government fellowship in law which supported him after he completed the law course. He graduated with a law degree in 1925 and joined a law office as a legal clerk. He continued to officially hold this position for several years but he hardly practiced law at all for he now enrolled full time in the Faculty of Philosophy, attending lectures and seminars in mathematics by Kazimierz Kuratowski, Hugo Steinhaus and other leading Polish mathematicians. He explained in an interview :-
It was the student generation of Juliusz Schauder, Mark Kac, Stanisław Ulam, Władysław Orlicz, Marceli Stark, Henryk Auerbach, Ludwik Sternbach, Stanisław Mazur and Julian Schreier. ... I was a part of that generation. Mathematics in that group of infatuated young people was kind of a fever. We would get together at all times of day and night, talking incessantly mathematics.He qualified as a mathematics teacher in 1926 and taught for the year 1926-27 at the Dr Adela Karp-Fuchsowa private Gymnasium. It was not a successful year for Birnbaum who failed to keep order in his classes. He transferred to the private Lwów Evangelical coeducational gymnasium where he taught for the two years 1927-29. Given good advice by an experienced teacher, his teaching improved but he had already published his first paper Quelques remarques sur l'intégrale de Cauchy Ⓣ (1927) and was participating fully in the mathematical research taking place in Café Roma and the Scottish Café. These nightly sessions :-
... turned into addictive intellectual feasts. Problems were raised, entered in the problem book which was kept by the head waiter, scribbled on marble-top tables (in pencil - ink was forbidden), and often solved.During these years Birnbaum was working towards his doctorate advised by Hugo Steinhaus. His dissertation Zur Theorie der schlichten Funktionen Ⓣ on univalent functions of a complex variable was reviewed by Banach and Steinhaus and, after taking the rigorosum, a rigorous oral examination in which the examiners could ask any mathematical questions they chose, he was awarded a doctorate in 1929. Birnbaum, aware of the high reputation that Göttingen had in mathematics, had been saving up his teacher's salary so that he could spend two years there and, on the day after he was awarded his doctorate, he boarded a train in Lwów travelling to Göttingen.
After arriving in Göttingen, Edmund Landau became his advisor, and he attended several lecture courses: differential equations given by Courant; calculus of variations given by Courant; power series given by Landau; higher geometry given by Herglotz; probability calculus given by Bernays; analysis of infinitely many variables given by Wegner; and attended the mathematical seminar directed by Courant and Herglotz. Soon Birnbaum was undertaking research on topics different from those he worked on for his doctorate. He published results he obtained on Sturm-Liouville theorems in Abschätzung der Eigenwerte eines Sturm-Liouvilleschen Eigenwert-Problems mit Koeffizienten von beschraenkter Schwankung Ⓣ (1930) and work on approximation theorems in function space in two papers Über Approximation im Mittel Ⓣ (1930) (the first written with Władysław Orlicz, the second being single-authored). These papers, for the first time, considered approximation with regard to arbitrary means. Also with Władysław Orlicz he published the influential work Über die Verallgemeinerung des begriffes der zueinander konjugierten Funktionen Ⓣ (1931) where spaces now called Birnbaum-Orlicz spaces were studied. After being in Göttingen for a year, Edmund Landau offered him an assistantship. This was in 1930 but the signs that the Nazis were becoming increasing powerful was clear to Landau for, after offering the assistantship to Birnbaum, he suggested he might be unwise to accept since :-
... there was no future in Germany since he was not sure how long even he would be allowed to keep his chair.He took Landau's advice seriously and decided to protect his future employment prospects by taking a course given by Felix Bernstein on insurance mathematics and attended his mathematical statistics seminar. In June 1931 he was offered a job working as an actuary for the Phoenix Life assurance Company. The Company's head office was in Vienna and :-
I started working there on the same day as Eugene Lukacs. We were given desks facing each other and were both assigned to Eduard Helly who was to be our preceptor. This brilliant mathematician ... was a gentle, warm and cheerful human being, who made my stay at the Phoenix Life Insurance Company enjoyable and instructive in many ways. He showed me over and over again how mathematics could be used to look at a specific actuarial transaction, give it a general formulation, obtain a general solution, and make it part of an inventory of techniques to be used in similar situations in the future.He returned to Lwów in 1932 when appointed as chief actuary of a Polish subsidiary of the Phoenix Life Insurance Company. He rejoined the exciting mathematical life of Lwów but now had no time to undertake research of his own. He continued with this life until 1936 when, with the economy of Europe rapidly deteriorating, first the Phoenix Life Insurance Company in Vienna was declared bankrupt and then its Polish subsidiary was put into receivership. Birnbaum continued to work for the firm while the receiver attempted to find a buyer, but he decided that, given the political situation in Germany, he should emigrate :-
I left Lwów on May 3, 1937, and was not to see my parents nor my sister, Franciszka seven years my junior, again. They were seen for the last time in the Bergen-Belsen concentration camp. ... An American visa was not easy to obtain. People would wait for them for ten years after filing an application. However, my cousin Ludwik Rubel, who was the editor-in-chief of the Cracow paper 'Ilustrowany Kurjer Codzienny', ... took me to the American consulate and introduced me as his newspaper's reporter. My journey took me to Vienna where I bade farewell to my relatives and acquaintances, and to Paris, where with my reporter's credentials I managed to see the World Exhibition a few days before its official opening. ... I left for America from Le Havre on the liner M S Georgia of the Cunard White Line.Arriving in New York in June 1937, Birnbaum wrote news reports for the Cracow paper for a couple of months, but he actively looked for work as a mathematician or as an actuary. He tried to contact people he got to known while in Göttingen who were now in the United States. He made contact with Otto Neugebauer at Brown University and Richard Courant in New York. Courant was helpful but had nothing to offer Birnbaum while Neugebauer was keen that he wrote reviews for Mathematical Reviews which he was on the point of founding. Indeed Birnbaum began to write reviews and waited for Felix Bernstein, who was out of town, to return to New York. Eventually, he met up with Felix Bernstein whose course on insurance mathematics he had studied while at Göttingen. The anti-Semitic policies of the Nazis had forced Felix Bernstein to emigrate and he was, by this time, professor of biometry at New York University. He offered Birnbaum a research assistant position in his department which he gladly accepted. He was required to attend lectures and work on a research project concerning :-
... a study of the connection between old age farsighted- ness, presbyopia and life length. Bernstein had, for a long time, a very strong curiosity about measurable physiological functions in humans that could be used as a yardstick for physiological age. He had arrived at the conclusion that presbyopia, the old age changing in the flexibility of the eye lens, was very strongly correlated with the all-around aging of the organism. It therefore extended some promise of being a predictor for life lengths. Whether that was realistic, I don't know. I suspected at that time that it was not so.Later in 1937, Birnbaum began attending Harold Hotelling's seminar at Columbia University. However, after a year working for Felix Bernstein, the research funding came to an end and Birnbaum was left without an income. He started up his own statistics consulting firm, renting a room for an office and placing advertisements. He had a few customers but they only presented him with some rather strange projects. He now got to know Samuel Wilks, who was working at Princeton, tried to learn as much about statistics as he could, and began applying for every conceivable job. Hotelling advised him to apply for a job advertised at the University of Washington but said not to use him as a referee since he was not on good terms with the department there. Birnbaum got Richard Courant, Edmund Landau and Albert Einstein to act as his referees and, after an interview in New York by the president of the New School for Social research and by the chief executive of the Sun Oil Company, he was offered an assistant professor position at the University of Washington. The only criticism he received from the interviewers was about his English accent and so, before taking up the appointment, he took a summer course to improve it.
Birnbaum's appointment in 1939 at the University of Washington involved him in teaching mathematics but at the same time was given the task of setting up a mathematical statistics group. The group was set up in the Mathematics Department and only in 1948 got a more formal status as the Laboratory for Statistical Research. Shortly after taking up the appointment in Seattle, he met Hilde Merzbach who was a German who had studied law but, being Jewish, had been forced to leave Germany and had joined other members of her family in Seattle. They met since both were working to assist Jewish refugees coming from Europe. She married Birnbaum in 1940; they had two children Ann, born in 1941, and Richard, born in 1945. Hilde Birnbaum took a Master's Degree in economics at the University of Washington and looked for an academic position. The University had rules which prevented husband and wife both working at the University, so she was forced to take jobs in other colleges around the Seattle area.
Birnbaum remained at the University of Washington until his retirement in 1974, working for these 35 years in the statistics group within the Department of Mathematics. The group only became a separate Department of Statistics in 1978. To summarise his statistical contributions we quote from his own description in :-
Our work dealt with probabilistic inequalities, with effects of pre-selection or truncation and of non-response in sampling surveys, with properties of Kolmogorov-Smirnov type and related statistics and numerical tabulations of these statistics for small sample sizes, and with the use of computers in unconventional situations.In 1962 he published the book Introduction to probability and mathematical statistics. E Parzen writes:-
This is an introductory text on probability and statistics. The first 163 pages provide a careful mathematical development of probability theory .... The next 140 pages discuss some procedures of statistical inference ... . While the emphasis throughout the book is on the statement of theorems rather than the handling of data, a number of numerical examples and exercises are given. However, the author states that the book does not aim to teach the elements of statistical technique. The book satisfactorily achieves its aim as a text for students with good mathematical background being introduced to mathematical statistics.He published the report On the mathematics of competing risks in 1979. Bruce Turnbull writes:-
This report provides a concise account of the basic mathematical concepts and probabilistic theory of competing risks. It also describes some of the statistical methodology much of which has been developed only quite recently. ... [it] will serve excellently someone seeking a brief introduction or a brief review of the theory of competing risks.Birnbaum received a number of honours including being elected a fellow of the Institute of Mathematical Statistics in 1949, the American Statistical Association, and the International Statistical Institute. He was president of the Institute of Mathematical Statistics in 1964. His greatest honour was the award of the S S Wilks Medal of the American Statistical Association in 1984 for:-
... his theoretical research, wide applications, leadership, inspiration and teaching.Let us end by quoting from  the range of Birnbaum's activities:-
In the course of years I have served as consultant for airplane manufacturers and for companies engaged in managing radioactive waste, for government agencies, as expert witness in law suits, as editor of professional journals, as consulting editor to publishers, and have found this variety of activities stimulating and exciting, with a strong appeal to my sense of adventure.
Article by: J J O'Connor and E F Robertson
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