Documentation

A predictable bettor.

A bettor describes how, given the history of centred observations z_t (each test module specifies its own centring; see the per-module documentation), the next predictable bet lambda_t is chosen. Predictability -- that is, lambda_t depends only on data observed strictly before step t -- is what makes the resulting wealth process a nonnegative supermartingale under H_0.

For Adaptive and Newton, a safe-bet ceiling lambda_max derived from the test's admissible-observation range is enforced by clipping lambda to [0, lambda_max], so the wealth factor stays nonnegative.

• Fixed always bets the supplied constant lambda. The wager does not respond to observed data; this strategy is useful only as a baseline.
• Adaptive is the aGRAPA (approximate growth-rate adaptive predictable plug-in) bettor of Waudby-Smith & Ramdas (2024). It tracks the empirical mean mu and variance sigma^2 of centred observations and bets the Kelly-optimal plug-in lambda* = mu / (sigma^2 + mu^2) clipped to [0, lambda_max]. Fast to compute and competitive in practice.
• Newton is the online Newton step (ONS) bettor. The per-step log-wealth loss -log(1 + lambda * z) is convex in lambda; ONS performs one Newton step per observation, accumulating squared gradients to scale the update. Achieves logarithmic regret against the best constant bet in hindsight and is in practice the strongest of the three bettors under most signal regimes.

Test outcome at the current sample count.

Reject means the wealth process has crossed the rejection threshold, so H_0 is rejected at level alpha. Continue means there is not yet enough evidence; collect more samples (or stop and report no rejection -- the type-I error guarantee holds for any stopping rule).