Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models

TL;DR

This paper unifies boundary degeneracy phenomena in sequential binary models under a reverse-martingale framework, proposing a new stopping rule that considers boundary closeness, uncertainty, and stability to better identify genuine limiting degeneracy.

Contribution

It introduces a unified reverse-martingale approach and a stopping rule requiring multiple conditions, improving the detection of boundary degeneracy in sequential binary data analysis.

Findings

Boundary closeness alone is unreliable as a stopping signal.

Stability condition distinguishes transient from genuine degeneracy.

Numerical studies validate the effectiveness of the proposed stopping rule.

Abstract

A run of all failures, a run of all successes, or complete separation in a logistic regression each tempts the analyst to declare a probability of exactly zero or one. The central message of this paper is that all three phenomena share a common structure: finite sequential data justify practical boundary statements of the form $p\leq\varepsilon$ or $p\geq1-\varepsilon$, but not exact boundary probabilities. The paper's contribution is to unify these three settings under a single reverse-martingale framework and to derive a stopping rule, $\tau_{\mathrm{RM}}$, that requires three conditions simultaneously -- boundary closeness $B_n\leq\varepsilon$, uncertainty width $W_n\leq w$, and trajectory stability $r_n\leq\eta$ -- rather than boundary closeness alone. The reverse-martingale view recasts boundary degeneracy as a property of the limiting conditional law $M_\infty=\E(Y\given\G_\infty)$ rather than a finite-sample event, complementing classical one-sided binomial tests and Wald's sequential probability ratio test without replacing them. Numerical studies across Bernoulli rare-event trials, low- and high-dimensional logistic regression, controlled risk trajectories, and a real health-economics data set demonstrate that boundary closeness alone is an unreliable stopping signal, and that the stability condition separates transient apparent certainty from genuine limiting degeneracy.