Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models
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| Format: | Preprint |
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2026
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| author | Chang, Yuan-chin Ivan |
| author_facet | Chang, Yuan-chin Ivan |
| contents | 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, $τ_{\mathrm{RM}}$, that requires three conditions simultaneously -- boundary closeness $B_n\leq\varepsilon$, uncertainty width $W_n\leq w$, and trajectory stability $r_n\leqη$ -- 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_02274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models Chang, Yuan-chin Ivan Methodology 60G42, 62L12, 62F03, 62J12 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, $τ_{\mathrm{RM}}$, that requires three conditions simultaneously -- boundary closeness $B_n\leq\varepsilon$, uncertainty width $W_n\leq w$, and trajectory stability $r_n\leqη$ -- 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. |
| title | Practical Boundary Degeneracy and Reverse-Martingale Limits in Sequential Binary Models |
| topic | Methodology 60G42, 62L12, 62F03, 62J12 |
| url | https://arxiv.org/abs/2605.02274 |