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<p>The Fisher information matrix is pivotal for determining the asymptotic behavior of maximum likelihood estimators, yet verifying its non-singularity in complex change-point and multipath change-point (MCP) models remains challenging. This paper provides a rigorous characterization of the singularity set, defined as the locus where the Fisher information matrix loses rank. Utilizing differential geometric techniques, we demonstrate that under standard smoothness and identifiability assumptions, the singularity set is a closed, nowhere-dense subset of the parameter space. Furthermore, when the likelihood function is real-analytic, this set is shown to have Lebesgue measure zero, residing within proper analytic subvarieties. These results generalize established findings from finite mixture models to a broader class of change-point problems. By proving that pathological parameters are theoretically rare, this study justifies the generic applicability of standard asymptotic theory and outlines pathways for extending this analysis to dependent and quasi-identifiable processes.</p>

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The Rarity of Singularities: A Topological and Measure-Theoretic Analysis of Fisher Information in Change-Point Models

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • Masoud Majidizadeh

摘要

The Fisher information matrix is pivotal for determining the asymptotic behavior of maximum likelihood estimators, yet verifying its non-singularity in complex change-point and multipath change-point (MCP) models remains challenging. This paper provides a rigorous characterization of the singularity set, defined as the locus where the Fisher information matrix loses rank. Utilizing differential geometric techniques, we demonstrate that under standard smoothness and identifiability assumptions, the singularity set is a closed, nowhere-dense subset of the parameter space. Furthermore, when the likelihood function is real-analytic, this set is shown to have Lebesgue measure zero, residing within proper analytic subvarieties. These results generalize established findings from finite mixture models to a broader class of change-point problems. By proving that pathological parameters are theoretically rare, this study justifies the generic applicability of standard asymptotic theory and outlines pathways for extending this analysis to dependent and quasi-identifiable processes.