Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant
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 N2/3 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 N4/5; in general the critical scale is N2q/(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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BibTeX
@misc{berchenko2026nuisance,
title = {{Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant}},
author = {Yakir Berchenko},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04453},
eprint = {2611.04453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nuisance structure and the detection boundary for population size: how profiling can change a rate, not only a constant
AU - Yakir Berchenko
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04453
UR - https://arxiv.org/abs/2611.04453
AB - 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.
ER -