This note 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 $n/p \to \delta \in (0, \infty)$. 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 $w \in R^p$ 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 and Sur (2020) for binary logistic regression to other single-index linear 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.

翻译：本文研究了在比例渐近条件下非正则化M估计量存在的相变，其中样本量$n$与特征维度$p$成比例增长，满足$n/p \to \delta \in (0, \infty)$。我们考察单指标模型中M估计量的存在性，该模型中响应变量$y_i$通过未知指标$w \in R^p$与未知链接函数依赖于协变量$x_i \sim N(0, I_p)$。我们推导出决定M估计量存在性相变的临界阈值$\delta_\infty$的显式表达式，将Cand`es与Sur（2020）针对二元逻辑回归的结果推广至其他单指标线性模型。此外，当M估计量存在时，我们研究了控制其渐近行为的非线性方程组解的存在性。在二元逻辑回归的全局零假设之外，对于$\delta > \delta_\infty$时该方程组解的存在性迄今尚未得到充分证明。我们通过证明该方程组当且仅当$\delta > \delta_\infty$时存在解，填补了这一理论空白，为需要以方程组解存在为前提的比例渐近结果提供了完整的理论基础。