Abraham Wald Hans Schneewei¼ Department of Statistics Akademiestrasse 1 80799 MÄunchen,Germany [email protected] Abraham Wald is best known for having given birth to Statistical Decision Theory, the paradigm of modern Statistics, and also for having developed the theory of sequential sampling. But he also contributed to many other ¯elds of Statistics often giving decisive impulses or even originating new directions of research. In Statistics proper one might mention: a very general consistency proof of maximum likelihood, a nonparametric runs test, tolerance intervals, power considerations for the Â2- goodness-of-¯t test, optimal experimental designs, discriminance analysis, the problem of inci- dental parameters, and linear models with errors in the variables. Econometricians know him for his work on seasonal adjustment, on index number theory, on the problem of estimating econometric models, and on the famous Wald test as an alternative to the likelihood ratio test. But he also made major contributions to mathematical economic theory and to game theory. Interesting enough, he started his academic career as a pure mathematician working in the ¯eld of geometry. Abraham Wald was born in 1902 in Klausenburg, at that time belonging to Hungary. So he was a citizen of the Austrian-Hungarian Empire. He studied Mathematics with Karl Menger in Vienna, where he got his Ph. D. in 1931 for his work in geometry. He would have liked to stay at the university, but could not get a position due to the adverse political and economic situation of the time. Through the intervention of Oskar Mor- genstern he was given a position as a statistical consultant to the Austrian Institute for Business Cycle Research in Vienna. It was here that he wrote his book on seasonal adjustment methodology "Berechnung und Ausschaltung von Saisonschwankungen" (1936), where he invented a new method of eliminating seasonal movements in an economic time series, which greatly improved on earlier methods by allowing for a slowly varying seasonal pattern. A similarly path-breaking contribution to economic statistics was his work on the cost of living index (1937, 1939). By assuming a (locally) quadratic utility function he derived a formula for the cost of living index which can be implemented from surveys of family budget data. In the mean time the political situation in Vienna became unbearable for Abraham Wald and he decided to leave Austria. In 1938 just after the Nazis took over Austria he went to the United States and accepted an o®er of a position with the Cowles Commission, where he continued his work in econometrics. But in the same year he became research associate and later professor at Columbia University, New York, following an invitation by Harold Hotelling. It was then and there that his statistical career really took o®. In 1948 he was president of the Institute of Mathematical Statistics and vice-president of the American Statistical Associa- tion. During this time he laid the foundations to Statistical Decision Theory, ¯rst in his seminal 1939 paper, which however went largely unnoticed at the time, and later (1950) in his famous book on Statistical Decision Functions. All the important concepts of decision theory, like risk function, minimax solution, Bayes solutions, and admissibility were present in his earlier paper, whereas in his book he incorporated concepts of game theory and of sequential decisions. In game theory it is often found that an optimal (minimax) strategy can only be realized as a randomized mixture of pure strategies. An important problem that Wald tackled was to ¯nd conditions under which optimal decision functions exist without the need to randomize. In expounding decision theory Wald took advantage of topological concepts, which he had ¯rst come across in his Viennese years. During the war, Wald was a member of the Statistical Research Group that was founded at Columbia University. In the course of his work with this group he developed the theory of sequential analysis. Sequential sampling was put into practice during the war, but its theoretical background was classi¯ed and could only be published after the war (1947). Instead of sampling with a ¯xed sample size, it is suggested to select and inspect items one by one. Each time a decision is made of whether to accept or to reject the lot or to continue with sampling. Eventually sampling will end. The actual sample size is random. It turns out that the expected sample size is smaller, sometimes very much smaller, than the sample size for a ¯xed sample of equal e±ciency. When inspection is very expensive, in particular when it is destructive, a sequential plan may reduce sampling costs considerably. Among the many other contributions to Statistics I should like to mention his paper (1940) on estimating a straight line by simply joining the centers of gravity of two subgroups of the sample points (xi; yi). Under some general conditions, which however were often misunderstood, one can prove that this estimate is consistent even if measurement errors in both variables x and y are present. Abraham Wald died in a plane accident in India in 1950 while being on a lecturing tour at the invitation of the Indian Government. A number of papers by renowned authors have appeared shortly after his death dedicated to his life and his scienti¯c work. REFERENCES Publications on Abraham Wald 1. Hotelling, H. (1951), Abraham Wald, American Statistician 5, 18-19. 2. Menger, K. (1952), The formative years of Abraham Wald and his work in geometry, Annals of Mathematical Statistics 23, 14-20. 3. Morgenstern, O. (1951), Abraham Wald, 1902-1950, Econometrica 19, 361-367. 4. Tintner, G. (1952), Abraham Wald's contributions to econometrics, Ann. Math. Statis- tics 23, 21-28. 5. Wolfowitz, J. (1952), Abraham Wald, 1902-1950, Ann. Math. Statistics 23, 1-13. Publications of Abraham Wald 1. The publications of Abraham Wald, Ann. Math. Statistics 23 (1952), 29-33. 2. Berechnung und Ausschaltung von Saisonschwankungen, (1936) Springer, Wien. 3. Zur Theorie der Preisindexzi®ern, Zeitschrift fÄurNationalÄokonomie 8 (1937), 179-219. 4. A new formula for the index of cost of living, Econometrica 7 (1939), 319-331. 5. Contributions to the theory of statistical estimation and testing hypotheses, Annals of Math. Stat. 10 (1939), 299-326. 6. The ¯tting of straight lines if both variables are subject to error, Annals of Math. Stat. 11 (1940), 284-300. 7. Sequential Analysis, (1947) John Wiley, New York. 8. Statistical Decision Functions, (1950) John Wiley, New York. RESUM¶ E¶ Abraham Wald, ne¶een 1902 avec la nationalit¶eaustro-hongroise, a ¶emigr¶ede Vienne vers les Etats-Unis en 1938 juste apr`esle d¶ebutde l'occupation nazie de l'Autriche. Aux Etats-Unis il est devenu c¶el`ebre pour avoir cr¶e¶ela th¶eoriede la d¶ecisionstatistique et pour le d¶eveloppement de l'analyse s¶equentielle. Il est d¶ec¶ed¶een 1950 dans un accident d'avion en Inde..