SPRTs

Sequential Probability Ratio Test (SPRT) is a statistical method for testing two competing hypotheses as data arrives, introduced by Abraham Wald in 1948. It aims to decide between two hypotheses with a minimum expected sample size subject to specified error rates, making decisions sequentially rather than after a fixed sample.

In the canonical form, H0 and H1 are simple hypotheses with known distributions f0 and f1 for the observed data. After observing the first n samples, compute the likelihood ratio Λn = ∏i=1..n f1(xi) / f0(xi). Two stopping boundaries are set by the desired Type I error α and Type II error β: A = (1−β)/α and B = β/(1−α). If Λn ≥ A, accept H1; if Λn ≤ B, accept H0; otherwise continue collecting observations and repeating the test.

The SPRT has the property of minimizing the expected number of observations required to reach a decision among all tests with the same α and β for simple hypotheses. In practice, log-likelihood ratios are accumulated to improve numerical stability: compare ln Λn to ln A and ln B.

Extensions include SPRTs for composite hypotheses or unknown parameters, using generalized or adaptive likelihood ratio approaches, and variants for discrete or continuous data. The SPRT framework is widely used in quality control, clinical trials, sequential A/B testing, and other settings where data accrue over time and rapid decision making is valued.

Limitations include reliance on correctly specified distributions under H0 and H1, independence assumptions, and potential biases under model misspecification or dependent data. Despite these caveats, SPRTs remain a foundational tool for efficient sequential decision making in statistics.