sequential probability ratio test

We are doing sequential hypothesis testing for a simple null $P$ vs simple alternative $Q$ with densities $p$ and $q$, respectively. We examine the likelihood ratio at time $t$,

$${\Lambda }_t=\frac{{\prod }_{i\le t}q(X_i)}{{\prod }_{i\le t}p(X_i)}.$$

We reject the null of ${\Lambda }_t\ge m_1$ and accepts the null if ${\Lambda }_t\le m_0$, otherwise we keep sampling. let $\varphi$ be the outcome of the procedure: $\varphi =1$ if the null is rejected, $\varphi =0$ if accepted. $m_1$ and $m_0$ are chosen such that the type-I error and type-II error satisfy

$$P(\varphi =1)=\alpha ,Q(\varphi =0)=\beta ,$$

for some pre-specified $\alpha ,\beta \in (0,1)$. Wald introduced the test in the 1930s and showed that its stopping time is almost surely finite. and Wald and Wolfowitz proved its optimality: If another test has lower type-I and type-II error then the expected stopping time is larger.

Unfortunately, $m_0$ and $m_1$ are often impossible to solve for, so approximations are necessary. One usually sets $m_1=(1-\beta )/\alpha$ and $m_0=\beta /(1-\alpha )$. But this loosens the guarantees. Fischer and Ramdas show how to improve the SPRT when approximating $m_1$ and $m_0$.