Almost sure hypothesis testing

by W. B. Meitei, PhD

Almost sure hypothesis testing is a statistical framework that guarantees, with probability one, the correct decision as the sample size becomes very large. Specifically, an almost sure hypothesis test is designed so that:

• If the null hypothesis is true, the probability that the test incorrectly rejects it goes to zero as the sample size grows, eventually failing to reject the null with probability one for all large samples.
• Conversely, if the alternative hypothesis is true, the test will reject the null hypothesis with probability one for sufficiently large samples.

This concept leverages the idea of almost sure convergence, meaning the test's decision converges to the correct outcome with probability one in the long run.

Almost sure hypothesis testing is notable because it differs from the classical fixed-level tests that maintain a constant significance level (e.g., 0.05) regardless of sample size. Such fixed-level tests can be problematic in large samples, leading to paradoxes like the Jeffreys-Lindley paradox, where frequentist and Bayesian conclusions diverge. In contrast, almost sure hypothesis testing allows the significance level to decrease as the sample size increases, ensuring the probabilities of both Type I (false positive) and Type II (false negative) errors tend to zero.

Recent research highlights that these tests make only a finite number of erroneous decisions with probability one, providing a robust alternative to traditional significance testing. This methodology is applicable to a wide range of statistical settings, including independent and identically distributed data as well as dependent data with strong mixing properties.

Suggested Readings:

1. Naaman, M. (2016). Almost sure hypothesis testing and a resolution of the Jeffreys-Lindley paradox. Electronic Journal of Statistics. 10 (2016) 1526–1550
2. Dembo, A., & Peres, Y. (1994). A topological criterion for hypothesis testing. The Annals of Statistics, 106-117.

Suggested Citation: Meitei, W. B. (2025). Almost sure hypothesis testing. WBM STATS.