Estimation for SLS models: finite sample guarantees
arxiv(2024)
摘要
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 𝕡 which allows us to identify the so-called
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 𝕡^3≪ n from Spokoiny
(2023a) to 𝕡^3/2≪ n. Examples for classical models like
logistic regression, log-density and precision matrix estimation illustrate the
applicability of general results.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要