SPRT

SPRT stands for the Sequential Probability Ratio Test. It is a statistical method for testing two simple hypotheses by evaluating data as they arrive and stopping when evidence is strong enough to decide between the hypotheses. Introduced by Abraham Wald in 1947, the SPRT is designed to minimize the expected sample size for specified error rates under simple hypotheses.

Procedure: Let Λn denote the likelihood ratio based on the first n observations, Λn = f1(x1, ..., xn) / f0(x1, ..., xn), where f0 and f1 are the probability models under H0 and H1. After each observation, compare Λn to two predefined boundaries A and B. If Λn ≥ A, accept H1; if Λn ≤ B, accept H0; if B < Λn < A, continue sampling. The boundaries are chosen to achieve desired type I error α and type II error β, commonly A = (1 − β)/α and B = β/(1 − α). In log form, decisions are based on the cumulative log-likelihood ratio crossing thresholds log A and log B.

Properties: The stopping time is random, since sampling continues until the evidence crosses a boundary. Under the two simple hypotheses, the SPRT is optimal in the sense that it minimizes the expected number of observations among all tests with the same error probabilities α and β. The method requires known distributions under H0 and H1 and typically assumes independence of observations.

Extensions and applications: SPRT has variants for different data types, including Bernoulli, normal with known variance, and Poisson data, and can be extended to composite hypotheses or adapted to group-sequential or approximate implementations for practical monitoring. It is widely used in quality control, clinical trials, signal processing, and other settings where rapid, sequential decision-making is valuable.