Abraham Wald

Sequential Tests of Statistical Hypotheses

By a sequential test of a statistical hypothesis is meant any statistical test procedure which gives a specific rule, at any stage of the experiment (at the n-th trial for each integral value of n), for making one of the following three decisions: (1) to accept the hypothesis being tested (null hypothesis), (2) to reject the null hypothesis, (3) to continue the experiment by making an additional observation. Thus, such a test procedure is carried out sequentially. On the basis of the first trial, one of the three decisions mentioned above is made. If the first or the second decision is made, the process is terminated. If the third decision is made, a second trial is performed. Again on the basis of the first two trials one of the three decisions is made and if the third decision is reached a third trial is performed, etc. This process is continued until either the first or the second decision is made.

Generalization of a Theorem By v. Neumann Concerning Zero Sum Two Person Games

In the theory of games developed by John v. Neumann [1], [2] the normalized form of a zero sum two person game is defined as follows (see section 14.1 in [2]): There are two players and there is a function K(ri, T2) of two variables mr and T2 given where Ti and T2 can take only a finite number of values. Player 1 chooses a value 1ri and player 2 chooses a value T2, each choice being made in complete ignorance of the other, and then players 1 and 2 get the amounts K(ri, T2) and -K(-ri, T2), respectively. Obviously, player 1 wishes to maximize K(ri, TO) and player 2 wishes to minimize K(Ti, T2). As v. Neumann has shown (see section 14.5 in [2]), the choice of ri by player 1 and the choice of T2 by player 2 can be rationalized if the game is strictly determined, i.e., if