Estimation for SLS models: finite sample guarantees

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.