Introduction to Wald (1945) Sequential Tests of Statistical Hypotheses

Probability theory came of age with the advent of Kolmogorov’s axiomatics in 1933 and the subsequent developments in limit theorems and stochastic processes. Statistical inference came of age with the advent of the Neyman-Pearson theory in 1933 and the subsequent formalization of hypothesis testing, estimation, and decision theory. In the present paper, Wald unified the two seemingly dissimilar areas most elegantly. To probabilists, he offered gems of new results in random walks, martingales, stochastic processes, and limit theorems. He also pointed out indirectly how they can seek probability questions from statistical inference and use the latter area as a testing ground for their abstract theorems. To statisticians, he showed that statistical inference is not just the analysis or “significance” of an existing body of data; it also entails their entry into the very process of experimentation and a continual analysis of the data as they become available. The sequential probability ratio test embodies this aspect. Wald also showed them how results from “pure” areas of mathematics, particularly probability theory, can be adapted to put statistical inference, in general, on stronger footing without losing sight of its practical nature.