P-hacking, Peeking & mSPRT Sequential

The Continuous Monitoring / Peeking Problem & Mixture Sequential Probability Ratio Test (mSPRT) establishes rigorous martingales-based defenses against false-positive rate inflation during continuous experimental monitoring; Peeking Pathology: repeatedly computing $p$-values daily and terminating early upon reaching $p < 0.05$ inflates the nominal $5\%$ false positive rate to $>30-40\%$ within 30 days under the null hypothesis $H_0$ due to random walk fluctuations crossing critical boundaries; Statistical Remedies: 1) Pre-registered Fixed-Horizon Testing (enforcing full-duration runs); 2) Group Sequential Boundaries (O'Brien-Fleming / Pocock alpha-spending functions); 3) Continuous Sequential Testing via mSPRT (constructing martingale-based Always-Valid $p$-values and time-uniform Confidence Sequences) enabling safe continuous monitoring and anytime early stopping.

Real-time experiment monitoring in enterprise A/B platforms (Netflix, Uber, ByteDance) and rapid negative-impact kill switches.

Eliminates rampant $p$-hacking caused by stakeholders prematurely halting experiments on transient random fluctuations, ensuring mathematically provable Type-I error control.

Derive why Brownian Motion first hitting times cause cumulative Type-I error rates to asymptotically approach 100% under continuous unadjusted peeking?

Derive the mathematical proof of mSPRT's error bounds using non-negative martingales and Doob's Maximal Inequality?

Explain how Lan-DeMets alpha-spending functions allocate Type-I error budgets conservatively across $K$ interim looks?

Why do Always-Valid Confidence Sequences guarantee $\ge 1-\alpha$ coverage probability uniformly across all stopping times?

When to deploy mSPRT for rapid disaster mitigation vs standard fixed-horizon testing for retention metrics with long carryover?