We develop a constant-tracking likelihood theory for two nonregular models: the folded normal and finite Gaussian mixtures. For the folded normal, we prove boundary coercivity for the profiled likelihood, show that the profile path of the location parameter exists and is strictly decreasing by an implicit-function argument, and establish a unique profile maximizer in the scale parameter. Deterministic envelopes for the log-likelihood, the score, and the Hessian yield elementary uniform laws of large numbers with finite-sample bounds, avoiding covering numbers. Identification and Kullback-Leibler separation deliver consistency. A sixth-order expansion of the log hyperbolic cosine creates a quadratic-minus-quartic contrast around zero, leading to a nonstandard one-fourth-power rate for the location estimator at the kink and a standard square-root rate for the scale estimator, with a uniform remainder bound. For finite Gaussian mixtures with distinct components and positive weights, we give a short identifiability proof up to label permutations via Fourier and Vandermonde ideas, derive two-sided Gaussian envelopes and responsibility-based gradient bounds on compact sieves, and obtain almost-sure and high-probability uniform laws with explicit constants. Using a minimum-matching distance on permutation orbits, we prove Hausdorff consistency on fixed and growing sieves. We quantify variance-collapse spikes via an explicit spike-bonus bound and show that a quadratic penalty in location and log-scale dominates this bonus, making penalized likelihood coercive; when penalties shrink but sample size times penalty diverges, penalized estimators remain consistent. All proofs are constructive, track constants, verify measurability of maximizers, and provide practical guidance for tuning sieves, penalties, and EM-style optimization.

翻译：本文针对两种非正则模型——折叠正态分布与有限高斯混合模型——发展了常数追踪似然理论。对于折叠正态分布，我们证明了轮廓似然的边界强制性，通过隐函数论证表明位置参数的轮廓路径存在且严格递减，并在尺度参数上建立了唯一的轮廓极大化子。通过对数似然、得分函数及Hessian矩阵的确定性包络，我们获得了具有有限样本界的初等一致大数定律，避免了覆盖数的使用。识别性与Kullback-Leibler分离性保证了一致性。双曲余弦函数的六阶展开在零点附近产生了二次减四次的对比形式，导致位置估计量在拐点处呈现非标准的四分之一次幂收敛速率，而尺度估计量保持标准的平方根收敛速率，且具有一致余项界。对于具有互异分量和正权重的有限高斯混合模型，我们通过傅里叶和范德蒙德思想给出了关于标签置换可识别性的简洁证明，在紧筛上推导了双侧高斯包络和基于责任函数的梯度界，并获得了带有显式常数的几乎必然一致和高概率一致定律。利用置换轨道上的最小匹配距离，我们证明了在固定筛和增长筛上的Hausdorff一致性。我们通过显式的尖峰-奖励界量化了方差塌缩尖峰现象，并证明位置与对数尺度上的二次惩罚项主导该奖励，使得惩罚似然具有强制性；当惩罚项收缩但样本量乘以惩罚项发散时，惩罚估计量仍保持一致性。所有证明均具有构造性，追踪常数项，验证极大化子的可测性，并为筛的选取、惩罚项设置及EM类优化提供了实用指导。