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LIT-1523: Sequential Tests of Statistical Hypotheses

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StatusActive — read 2026-10-25 (NOTE-1165) in full, from the copy the owner supplied on 2026-10-25, a JSTOR scan of the Annals article (this client had been stopped by Project Euclid's bot check the same day). Worth reading as the source of the sequential probability ratio test: its boundaries follow from the error rates alone, A = (1 − β)/α and B = β/(1 − α) (THEORY-952), and its operating characteristic, expected sample size and optimality against every test of the same strength are derived, with savings of about half against the fixed-sample test (THEORY-951). Read it knowing that optimality is proved only when the overshoot of the boundaries is neglected, the exact case being a conjecture (footnote 12), that the operating characteristic and sample-size formulas are approximations that need i.i.d. observations, and that the paper never mentions diffusion: the drift-diffusion limit is later work's, and is the case in which these approximations are exact.
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Tagsmathematical-statistics
Cited as “paper” byNOTE-1165 — Wald — Sequential tests of statistical hypotheses
Cited as “source” byTHEORY-951 — For independent identically distributed observations, and neglecting how far the log-likelihood-ratio walk overshoots its boundaries, Wald's sequential probability ratio test accepts the null hypothesis with probability (A^h − 1)/(A^h − B^h), where h ≠ 0 solves E e^{hz} = 1, and takes on average [(1 − γ) log B + γ log A]/E z observations; no sequential test with the same error probabilities needs fewer on average under either hypothesis than Wald's lower bounds, which the test attains when the overshoot is zero, and for a normal mean it saves 47–63% of the fixed-sample test's observations at error rates from .01 to .05; the overshoot-free formulas are exact for Brownian motion with drift, which the paper never mentions, THEORY-952 — In Wald's sequential probability ratio test, which samples until the likelihood ratio of two simple hypotheses leaves an interval (B, A), the realized error probabilities satisfy α/(1 − β) ≤ 1/A and β/(1 − α) ≤ B whenever the test terminates with probability one, with or without independent observations; so the boundaries A = (1 − β)/α and B = β/(1 − α), set with no distribution theory, give errors α′ ≤ α/(1 − β), β′ ≤ β/(1 − α) and α′ + β′ ≤ α + β, and they would be exact only if the ratio stopped on the boundary

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Wald, A. (1945). Sequential tests of statistical hypotheses. The Annals of Mathematical Statistics 16(2): 117–186. doi:10.1214/aoms/1177731118.

Read in full on 2026-10-25 from the copy the owner supplied (NOTE-1165). Part I gives the test, its fundamental inequalities α/(1 − β) ≤ 1/A and β/(1 − α) ≤ B, the operating characteristic, the expected sample size and its distribution, truncation, and the optimality theorem. Part II applies the test to binomial proportions, double dichotomies and a normal mean, and outlines composite hypotheses through weight functions. THEORY-952 and THEORY-951 re-derive the main results.

Why this is here

At U34 the owner hypothesized that when the perspectives advocate one of n = 2 interpretations, a properly set-up simulation should show drift–diffusion dynamics, and at U35 the reply cited the sequential probability ratio test as the classical link between accumulated likelihood evidence and binary drift–diffusion. The record already held the link only as restated: Bogacz et al. (LIT-569) state that the SPRT “of Barnard and Wald” is the fastest test for a given error rate and that its continuum limit is the drift–diffusion model, and THEORY-068 reads it. CASE-155 part 4 shows that the replicator in log-odds form with no feedback term is the SPRT walk. The claim CLAIM-453 and the construction CASE-160 rest on that limit; CLAIM-262 is the owner’s earlier, continuous statement of the hypothesis. This is the primary source, held since the owner supplied it on 2026-10-25.

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