Sequential probability ratio test

Remarks

Example: for coin tosses $X_i \in \set{0, 1}$, let $f_0$ be the Bernoulli law with parameter $1/2$ and $f_1$ the one with parameter $2/3$. Then each head multiplies $\Lambda_n$ by $4/3$ and each tail by $2/3$. With $A = 1/4$ and $B = 4$, the test stops the first time the running product leaves the interval $(1/4, 4)$.

Non-example: comparing $\Lambda_{100}$ with a single threshold after exactly $100$ tosses is a fixed-sample likelihood ratio test, not an SPRT, because the stopping rule of clause 1 is not used.

Since $f_0 \ne f_1$, the time $N$ is finite with probability one under both hypotheses and has finite expectation. In logarithms, $\log \Lambda_n$ is a random walk with the two absorbing barriers $\log A < 0 < \log B$. Let $\alpha$ and $\beta$ be the error probabilities of type I and type II. Wald's inequalities are $B \le (1 - \beta)/\alpha$ and $A \ge \beta/(1 - \alpha)$, and in practice the boundaries $B = (1 - \beta)/\alpha$ and $A = \beta/(1 - \alpha)$ are used as approximations.