Abraham Wald

Who they are

Wald was born on 31 October 1902 in Kolozsvár, then in Hungary and now Cluj in Romania, into a Jewish family. He was educated at home, because his family would not let him attend school on the Sabbath. He entered the University of Vienna in 1927 to study under Karl Menger and took his doctorate there in 1931, with work in geometry.

From 1931 to 1937 he worked as a mathematics tutor to the banker and economist Karl Schlesinger, which led him into econometrics. After the Nazi invasion of Austria in 1938 he left for the United States on an invitation from the Cowles Commission, became a Carnegie Corporation Fellow, and by September was studying statistics at Columbia University. He began lecturing there in 1939 and 1940, joined the faculty in 1941, and stayed until his death.

On 13 December 1950, after accepting an invitation from the Indian government to lecture on statistics, he and his wife were killed in a plane crash in Travancore.

Key papers

During the war Wald worked with Columbia's Statistical Research Group on military problems. His memoranda on aircraft survivability were never published externally at the time. Copies circulated, and the Center for Naval Analyses reprinted them on 7 July 1980 as 'A Method of Estimating Plane Vulnerability Based on Damage of Survivors', on the grounds of their historical interest and because the method remained relevant. The reprint notes that the methodology was later used to analyse data from the Korean and Vietnam Wars.

A historical review by Bill Casselman for the American Mathematical Society's Feature Column in 2016 describes the set as eight memoranda. Five address the probability that a plane survives a given number of hits, using data only from returning planes. One addresses the vulnerability of different sections of a plane.

Separately, Wald's work in sequential analysis, published as a book in 1947, grew out of wartime demand for efficient industrial quality control.

Key ideas

The central device in the aircraft memoranda is what Casselman calls Wald's basic equation. It lets an analyst estimate the distribution of hits on planes that were shot down from the distribution on planes that survived. The data are biased by construction, since the downed planes cannot be examined, and the method corrects for that instead of ignoring it. This is the technical core of the idea Survivorship Bias in Success Stories applies to funds, firms and pilots.

Casselman's review also corrects the legend. The popular story has Wald advising the military to armour the engines, because returning planes showed no holes there, and it circulates with diagrams. Casselman describes that version as a plausible reconstruction, writes that Wald says nothing in the memoranda about what the military should do to improve things, and says there is no evidence that Wald drew the pictures attributed to him. A postscript to the column softens the criticism by noting that a 1980 article by W. Allen Wallis in the Journal of the American Statistical Association connects Wald's work to survivorship bias explicitly.

What the record supports is narrower than the legend and more useful. Wald recognised that survivors are a filtered sample and wrote down how to reason about the part that is missing.

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