### BIOGRAPHY 23.2* Abraham Wald* (1702 -1761)

Abraham Wald was born in Klausenburg, Hungary (which later became Cluj, Rumania). He was privately educated and self-taught (a consequence of complications arising from his family's Jewish orthodoxy). In the 1930s, he moved to Vienna, worked at the Austrian Institute for Business Cycle Research, and, at the time of the Nazi annexation, accepted an invitation by the Cowles Commission to work in the United States. That move surely saved him from death in a gas chamber - a fate that befell all but one member of his numerous family. Eventually, Wald taught statistics at Columbia University (his students remember him as a superb teacher of unrivalled clarity and precision), but his career was cut short by an airplane crash during a lecture tour of India.

Above all else, Wald's name will forever be linked with statistical decision theory and sequential analysis. In one of his earliest and finest papers, he introduced many fundamental concepts of decision theory, along with a mathematical structure for single-sample decision making that was sufficiently general to embrace estimation, hypothesis testing, and even the design of experiments. As he states in the Preface to his monumental *Statistical Decision Functions *(1950): "A major advance beyond previous results is the treatment of the design of experimentation as a part of the general decision problem." This tendency of Wald's and of later decision theorists to view all of statistics as a science of decision making under uncertainty (and to view the user of statistical data as someone in a decision-making situation who seeks to make the one of several possible decisions that has the best expected payoff) invoked the ire of Ronald Fisher (Biography 13.1). Fisher thought that decision theory was appropriate, perhaps, for industry but certainly not for scientific work in which inferences about the truth were to be made rather than optimal payoffs were to be achieved. Yet Wald's approach fit in nicely with the Neyman-Pearson theory of hypothesis testing (see Biographies 13.2 and 13.3).

Although he had forerunners, Wald was the first to formulate mathematically and solve generally the problem of sequential tests of statistical hypotheses. If evidence in sequentially unfolding data is sharply one-sided, Wald asked, why not stop an inquiry early? Why insist on observing and analyzing all the data contained in a single sample of predetermined fixed size? This line of thought ultimately gave rise to a classic work, *Sequential Analysis *(1947).

Nor is this all. During his short life, Wald made major contributions in other fields as well. He published more than 90 books and papers! These include topology, measure and set theory, lattice theory, econometrics (the seasonal correction of time series, formulas for approximating economic index numbers), mathematical economics (indifference surfaces, proofs of the existence and uniqueness of solutions to the Walrasian system of equations), and more.

*Sources:* Adapted from Oskar Morgenstern, "Abraham Wald, 1902-1950," *Econometrica*, October 1951, pp. 361-367; *International Encyclopedia of Statistics*, vol. 2 (New York: Free Press, 1978), pp. 1235-1238.