WALD, ABRAHAM

(b. Cluj, Rumania, 31 October 1902; d. India, 13 December 1950), mathematical statistics, mathematical economics, geometry.

Wald was born into a familywhich had consider able intellectual interests but had to earn it s livelihood in petty trade because of anti-jewish restrictions. These restrictions made his education at the University of Cluj, and later at the University of Vienna, very difficult. After Hitler occupied Austria in 1938, Wald moved to the United States, which saved him from death in a German concentration camp—the fate of all but one other member of his numerous family. He later married Lucille Lang, an American, who died with him in an airplane crash.

At Vienna, Wald was a student and a protégé, and later a friend, of Karl Menger. His work in pure mathematics was largely, although not wholly, in geometry. Menger later directed Wald toward mathematical statistics and mathematical economics, so that he was able to find employment with the distinguished economist Oskar Morgenstern.

It seems reasonable to say that Wald’s most important work was in statistics, both because of his relative importance in the field and because of the current assessment of the field’s importance. One of his great contributions to statistics was to bring to it mathematical precision in the formulation of problems and mathematical rigor in argument. These qualities, which were often lacking when he began his statistical career in 1938, have transformed the subject—although not necessarily to the satisfaction of everyone. It should be emphasized, however, that these accomplishments were a by-product and consequence of his extraordinary ability and the breadth of his statistical interests. Wald wrote lucidly and unambiguously on many statistical subjects, and there is scarcely a branch of modern statistics to which he did not contribute. His writings and lectures were so lucid and so unambiguous because of this precision, and he achieved so much in the way of results, that the superiority of mathematical precision became apparent to all. It is impossible to discuss Wald’s statistical results in detail; rather, we shall single out the two most important fields of his work, which he founded and in which his results still dominate: sequential analysis and the theory of decision functions.

In sequential analysis one takes observations seriatim until the evidence is sufficiently strong, bearing in mind certain previously imposed bounds on the probabilities of error. When there are only two possible hypotheses, the “Wald sequential probaitlity ratio test” has the property that it requires the smallest average number of observations under either hypothesis. This famous “optimum property of the sequential probability ration test” was brilliantly conjectured by Wald in 1943 and proved jointly by him and a colleague in 1948. Wald proved many theorems on the distribution of the required number of observations and obtained many approximations on probabilities of error and average required numbers of observations, that are still used in applications. Most, although not all, of his results were summed up in Sequential Analysis (1947). With minor exceptions, the entire contents of this book were obtained by him. Such a phenomenon is rare in mathematical books and indicates the extent to which he founded and dominated the field of sequential analysis.

When Wald began his work in statistics, a large part of the field was concerned with the theory of testing hypotheses. He regarded this theory as, at best (when properly interpreted), one of deciding between exactly two courses of action. Consequently very many statistical problems actually fall outside the scope of this theory. There was no consistent theory for deciding among more than two courses of action, and attempts to force such problems into the framework of the theory of testing hypotheses had yielded very unsatisfactory results. It is interesting that these objections were clearly realized by a theoretician like Wald and not by the practical statisticians in industrial and agricultural laboratories who applied the theory. (For a recent criticism of the theory see Wolfowitz, “Remarks on the Theory of Testing Hypotheses.”) Wald’s theory of statistical decision functions considers the problem of deciding among any number of (possibly infinitely many) courses of action, both sequentially and nonsequentially. The statistician introduces a loss function that measures the consequences of various actions under different situations. With each statistical procedure (decision function) there is associated a vector, or function, of average loss under the various possible situations (the risk function). The statistical procedures of which the risk functions are not inferior to those of any other form a “complete” class, and the statistician can properly ignore the procedures not in the complete class.

At Wald’s death the theory of statistical decision functions was far from the point of application to everyday, practical statistical problems; and little progress has been made in this direction since then. The theory is still of great conceptual and theoretical importance, and provides a logical basis for the formulation of many research problems. Recent research in the theory itself has, however, been chiefly in the direction of very technical mathematical refinements and has not achieved any essential breakthroughs.

Some of Wald’s work in statistics originated in economic problems and properly belongs to both subjects. One such example is his work on the identification of economic relations—roughly speaking, the problem whether the distributions, which result from a model of the observed chance variables, uniquely determine all or certain specified parameters of the model. Also included in this category is his work on stochastic difference equations—models involving sequences of change variables.connected by difference equations with “error” chance variables. Wald also proved theorems on the existence of unique solutions for systems of equations for several types of economic systems and studied cost-of-living index numbers, the empirical determination of indifference surfaces, and the elimination of seasonal variation in time series. In all these his methods were ingenious and his contributions very important.

In pure mathematics, Wald’s first three published papers and “Zur Axiomatik des Zwischenbegriffes” dealt with the characterization of “betweenness” in metric spaces. He also extended Steinitz’s theorem to vectors with infinitely many elements; the theorem states that a divergent series, the elements of which are finite vectors, can, by a permutation of its terms, be made to converge to any element of a linear manifold. Perhaps his best result was the development of a differential geometry that starts from the assumption of a convex, compact metric space that at every point admits what should be called a Wald curvature. From this he was able to derive properties of differential geometry that are postulated in other systems.

Relatively uninterested in mathematical elegance, Wald spent little time in polishing a paper after a problem was solved to his satisfaction. In his masterly hands simple methods sometimes yielded the most amazing results. Although he was readily accessible, he had very few students. With one of these, J. Wolfowitz, who became his friend and colleague, he wrote fifteen joint papers. His American, and largely statistical, period was relatively brief (1938-1950) and extraordinarily productive. During this time he learned mathematical statistics, contributed deeply to it, changed it essentially, and dominated the subject. It has borne his impress since, and the paths he opened are still being pursued.

BIBLIOGRAPHY

I. Original Works. A comprehensive bibliography of Wald’s writings follows Tintner’s memoir (see below). His works include “Zur Axiomatik des Zwischenbegriffes.” in Ergebnisse eines mathematischen Kolloquiums4 (1933), 23–24: Sequential Analysis (New York—London, 1947); and “Oprtimum Character of the Sequential Probability Ratio Test,” in Annals of Mathematical Statistics, 19 (1948), 326–339, written with J. Wolfowitz.

II. Secondary Literature. On Wald and his work, see J. Wolfowitz, “Abraham Wald, 1902-1950,” in Annals of Mathematical Statistics, 23 (1952), 1–13; Karl Menger, “The Formative Years of Abraham Wald and His Work in Geometry,” ibid., 14–20; G. Tintner, “Abraham Wald’s Contributions to Econometrics,” ibid., 21–28; and “The Publications of Abraham Wald,” ibid., 29–33, which lists 103 works (1931-1952).

See also J. Wolfowitz, “Remarks on the Theory of Testing Hypotheses,” in New York Statistician18 , no. 7 (Mar. 1967), 1–3.

J. Wolfowitz