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arXiv · 2611.04453

Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant

Yakir Berchenko
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Abstract

A system contains an unknown number $N$ of units, each failing at an unknown common rate, each unit is removed once it fails. The failure-rate declines as units are removed, but after only a few failures the decline is too slight to distinguish from a constant rate, and the maximum-likelihood estimate of $N$ is infinite with substantial probability. This is a well-known feature of the model, but its probability has not been quantified. We show that about $N^{2/3}$ failures are needed before a finite population can be distinguished from an inexhaustible one, and that at this scale the profile log-likelihood ratio converges to that of a Gaussian shift experiment, in which an infinite estimate occurs with a calculable probability. The exponent is not a property of the model alone. Profiling annihilates the Taylor terms that the nuisance-parameters can reproduce, and detectability is governed by the first surviving term. For example, unknown drift in testing-effort reproduces the linear term, leaving a quadratic survivor and raising the requirement to $N^{4/5}$; in general the critical scale is $N^{2q/(2q+1)}$ for a survivor of order $q$. We obtain the limiting-law of the estimator, the null likelihood ratio distribution, and confidence sets that may be unbounded.

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Yakir Berchenko. 2026-11-03. Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant. https://arxiv.org/abs/2611.04453

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