Claim 2 — union bound tighter than variance upper bound under asymmetric arrival
"Combining two arm-level anytime-valid confidence sequences via a union bound produces tighter intervals than the standard design-based variance upper bound when treatment induces asymmetric outcome arrival rates between arms (Section 4)."
This is Proposition 4.12: the union-bound width b(V₀;α/2)+b(V₁;α/2) is compared against the classical cumulative variance-upper-bound benchmark b(V₀/π+V₁/(1−π);α) (paper's Eq. 28), both using the exact stitching boundary b(V;α)=√[(Vη²+1)/η² · log((Vη²+1)/α²)] (paper's Eq. 21, α=0.05, η=0.5). We SIMULATE two DGP scenarios (N=500, R=250 replications, evaluated at t=25) and MEASURE V₀=Vₜ(0), V₁=Vₜ(1) from the real simulated potential-outcome trajectories (Eq. 18: the predictable quadratic variation, summed over units whose event has occurred) — the boundary formulas are then evaluated on these measured numbers, not fit to them.
Symmetric scenario (θ=(0.15,0.15), matched hazards and reward magnitudes): measured V̄₀=607.1, V̄₁=605.7 (ρ=V₀/(V₀+V₁)=0.501). Union width 183.51, variance-UB width 183.03, ratio 1.0026 — essentially 1.0, matching Proposition 4.12's ρ=0.5 theoretical value of exactly 1.0.
Asymmetric scenario (θ=(0.05,0.60), fast/large-magnitude control vs. rare/small-magnitude treatment): measured V̄₀=6110.7, V̄₁=23.6 (ρ=0.996 — one arm dominates the variance clock). Union width 330.91, variance-UB width 434.83, ratio 0.7610 — the union-bound CS is 23.9% narrower. The theoretical Proposition-4.12 asymptote at this measured ρ is 0.7496 — our finite-sample measured ratio (0.7610) is close to but not identical to the asymptote, exactly as expected since V₀+V₁ is finite (not literally →∞); the extreme limit √(1−π)=0.7071 is approached but not yet reached.
Verdict: VERIFIED — the union bound is measurably tighter under simulated asymmetric arrival, and essentially tied under simulated symmetric arrival, matching the direction and rough magnitude of Proposition 4.12 in both regimes.
{
"paper_id": "2603.25971",
"alpha": 0.05,
"eta": 0.5,
"pi": 0.5,
"scenarios": [
{
"scenario": "symmetric",
"N": 500,
"R": 250,
"t": 25.0,
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