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Phase transitions for the existence of unregularized M-estimators in single index models

Authors :
Bellec, Pierre C.
Koriyama, Takuya
Publication Year :
2025

Abstract

This paper studies phase transitions for the existence of unregularized M-estimators under proportional asymptotics where the sample size $n$ and feature dimension $p$ grow proportionally with $n/p \to \delta \in (1, \infty)$. We study the existence of M-estimators in single-index models where the response $y_i$ depends on covariates $x_i \sim N(0, I_p)$ through an unknown index ${w} \in \mathbb{R}^p$ and an unknown link function. An explicit expression is derived for the critical threshold $\delta_\infty$ that determines the phase transition for the existence of the M-estimator, generalizing the results of Cand\'es & Sur (2020) for binary logistic regression to other single-index models. Furthermore, we investigate the existence of a solution to the nonlinear system of equations governing the asymptotic behavior of the M-estimator when it exists. The existence of solution to this system for $\delta > \delta_\infty$ remains largely unproven outside the global null in binary logistic regression. We address this gap with a proof that the system admits a solution if and only if $\delta > \delta_\infty$, providing a comprehensive theoretical foundation for proportional asymptotic results that require as a prerequisite the existence of a solution to the system.<br />Comment: 31 pages, 3 figures

Subjects

Subjects :
Mathematics - Statistics Theory

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2501.03163
Document Type :
Working Paper