THE PHASE TRANSITION FOR THE EXISTENCE OF THE MAXIMUM LIKELIHOOD ESTIMATE IN HIGH-DIMENSIONAL LOGISTIC REGRESSION
成果类型：
Article
署名作者：
Candes, Emmanuel J.; Sur, Pragya
署名单位：
Stanford University; Harvard University
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/18-AOS1789
发表日期：
2020
页码：
27-42
关键词：
摘要：
This paper rigorously establishes that the existence of the maximum likelihood estimate (MLE) in high-dimensional logistic regression models with Gaussian covariates undergoes a sharp phase transition. We introduce an explicit boundary curve h(MLE), parameterized by two scalars measuring the overall magnitude of the unknown sequence of regression coefficients, with the following property: in the limit of large sample sizes n and number of features p proportioned in such a way that p/n -> kappa, we show that if the problem is sufficiently high dimensional in the sense that kappa > h(MLE), then the MLE does not exist with probability one. Conversely, if kappa < h(MLE), the MLE asymptotically exists with probability one.