This note continues and extends the study from Spokoiny (2023a) about estimation for parametric models with possibly large or even infinite parameter dimension. We consider a special class of stochastically linear smooth (SLS) models satisfying three major conditions: the stochastic component of the log-likelihood is linear in the model parameter, while the expected log-likelihood is a smooth and concave function. For the penalized maximum likelihood estimators (pMLE), we establish several finite sample bounds about its concentration and large deviations as well as the Fisher and Wilks expansions and risk bounds. In all results, the remainder is given explicitly and can be evaluated in terms of the effective sample size $ n $ and effective parameter dimension $ \mathbb{p} $ which allows us to identify the so-called \emph{critical parameter dimension}. The results are also dimension and coordinate-free. Despite generality, all the presented bounds are nearly sharp and the classical asymptotic results can be obtained as simple corollaries. Our results indicate that the use of advanced fourth-order expansions allows to relax the critical dimension condition $ \mathbb{p}^{3} \ll n $ from Spokoiny (2023a) to $ \mathbb{p}^{3/2} \ll n $. Examples for classical models like logistic regression, log-density and precision matrix estimation illustrate the applicability of general results.

翻译：本文延续并拓展了Spokoiny（2023a）关于参数维度可能较大甚至无穷的参数模型估计研究。我们考虑一类特殊的随机线性光滑（SLS）模型，该类模型满足三个主要条件：对数似然函数的随机分量关于模型参数呈线性，而期望对数似然函数则为光滑凹函数。针对惩罚最大似然估计（pMLE），我们建立了关于其集中性、大偏差以及Fisher与Wilks展开和风险界的若干有限样本界。所有结果中，余项均显式给出，并可通过有效样本量 $ n $ 和有效参数维度 $ \mathbb{p} $ 进行评估，从而识别出所谓的“临界参数维度”。这些结果还具有维度无关性和坐标无关性。尽管具有一般性，但本文给出的所有界均接近最优，且经典渐近结果可作为简单推论获得。我们的结果表明，采用先进的四阶展开可将Spokoiny（2023a）中的临界维度条件 $ \mathbb{p}^{3} \ll n $ 放松至 $ \mathbb{p}^{3/2} \ll n $。逻辑回归、对数密度及精度矩阵估计等经典模型示例验证了通用结果的适用性。