Source
Wald (1945), “Sequential Tests of Statistical Hypotheses”, The Annals of Mathematical Statistics 16(2): 117–186, LIT-1523, read in full in NOTE-1165 from the JSTOR scan the owner supplied. The test is (3.8)–(3.10), its Bayesian motivation (3.1)–(3.7), and termination (3.11). The inequalities are (3.12)–(3.19) and the practical boundaries (3.24)–(3.30). The derivations below follow Wald’s and are written out by the record.
What was actually shown
The test. Let p_im be the joint density of the first m observations under Hᵢ. Accept H₁ the first time p₁ₘ/p₀ₘ ≥ A, accept H₀ the first time p₁ₘ/p₀ₘ ≤ B, and otherwise observe again, with 0 < B < A. For i.i.d. observations, log p₁ₘ/p₀ₘ = Z_m = z₁ + … + z_m with zᵢ = log f₁(xᵢ)/f₀(xᵢ), and the test stops when Z_m leaves (log B, log A).
Where it comes from (§3.1). With prior probabilities g₀, g₁, Bayes gives posterior odds g₁ₘ/g₀ₘ = (g₁/g₀)·p₁ₘ/p₀ₘ. Accepting H₁ when g₁ₘ ≥ d₁ and H₀ when g₀ₘ ≥ d₀, with d₀, d₁ > ½, is the pair of thresholds (3.6)–(3.7) on p₁ₘ/p₀ₘ, constants that do not depend on m. So the posterior log odds are the walk shifted by the prior log odds, log(g₁/g₀) + Z_m. The SPRT is that rule with the prior absorbed into A and B.
The inequalities. Let Q₁ be the set of sample paths that stop accepting H₁. On each such path, stopped at n, p₁ₙ ≥ A p₀ₙ. Summing (integrating) over Q₁ gives P₁(Q₁) ≥ A P₀(Q₁), that is 1 − β ≥ Aα (3.16). Likewise P₁(Q₀) ≤ B P₀(Q₀), that is β ≤ B(1 − α) (3.17). The only other fact used is P(Q₀ ∪ Q₁) = 1 under each hypothesis, so that P₀(Q₀) = 1 − α and P₁(Q₁) = 1 − β. Independence enters only through termination, and Wald notes that (3.11) “can be shown to hold also for dependent observations under very general conditions” (p. 130).
The boundaries. Given target errors (α, β), put A = (1 − β)/α and B = β/(1 − α). The realized errors α′, β′ then satisfy α′/(1 − β′) ≤ α/(1 − β) and β′/(1 − α′) ≤ β/(1 − α) (3.26)–(3.27). The first gives α′ ≤ α′/(1 − β′) ≤ α/(1 − β), and the second β′ ≤ β/(1 − α) (3.28)–(3.29). Multiplying the first by (1 − β)(1 − β′) and the second by (1 − α)(1 − α′), then adding, gives α′ + β′ ≤ α + β (3.30). So the approximate boundaries never weaken the test in sum.
Why they are inequalities. The ratio jumps past the boundary when it stops. If it stopped exactly on A or B, every inequality above would be an equality. Wald: “If fractional observations were possible … A(α, β) and B(α, β) would be exactly equal to (1 − β)/α and β/(1 − α) respectively” (p. 132).
Checked by simulation (numpy, when filed; N(μ, 1), H₀ μ = 0 against H₁ μ = δ, the boundaries above):
| α, β, δ | runs | α′, β′ | α′/(1 − β′) vs 1/A | β′/(1 − α′) vs B | α′ + β′ |
|---|---|---|---|---|---|
| .05, .05, .5 | 20,000 | .035, .039 | .037 ≤ .053 | .040 ≤ .053 | .074 ≤ .10 |
| .01, .10, .5 | 20,000 | .007, .073 | .008 ≤ .011 | .073 ≤ .101 | .080 ≤ .11 |
| .05, .05, .2 | 20,000 | .046, .043 | .048 ≤ .053 | .045 ≤ .053 | .089 ≤ .10 |
| .05, .05, .05 | 4,000 | .047, .046 | .049 ≤ .053 | .048 ≤ .053 | .093 ≤ .10 |
The slack shrinks as δ falls and the overshoot with it, as Wald says it should. What could have failed: the realized errors could have exceeded the bounds when the overshoot is large (δ = .5 is a large step). They stayed below in every case.
What this does not say
- Not how long the test runs, or that it is optimal. Those need i.i.d. observations and the overshoot to be neglected (THEORY-951).
- Not that both errors fall. Only their sum is guaranteed not to rise. One error can exceed its target, by at most a factor 1/(1 − β) or 1/(1 − α).
- Not anything for composite hypotheses directly. For those, Wald tests against a chosen simple alternative, or against a mixture over a weight function, which bounds a weighted average of the second-kind errors (§6).
- Not that a walk without boundaries is a test. The SPRT is the log-likelihood-ratio walk together with its stopping rule.
Connections
- CASE-155 (note of 2026-10-25) and CLAIM-453. The replicator’s log share odds, moving by each observation’s log likelihood ratio from the prior log odds, are §3.1’s posterior log odds log(g₁/g₀) + Z_m. They are Wald’s walk, and the SPRT’s statistic. They become the SPRT when commitment is a fixed bound on them, which the claim lists as a condition.
- THEORY-068 takes the SPRT through Bogacz et al. (LIT-569). This account supplies the primary statement of the test and its boundaries.