We study the distribution of the maximum likelihood estimate (MLE) in high-dimensional logistic models, where covariates are Gaussian with an arbitrary covariance structure. We prove that in the limit of large problems holding the ratio between the number p of covariates and the sample size n constant, every finite list of MLE coordinates follows a multivariate normal distribution. Concretely, the jth coordinate of the MLE is asymptotically normally distributed with mean and standard deviation ; here, is the value of the true regression coefficient, and the standard deviation of the jth predictor conditional on all the others. The numerical parameters and only depend upon the problem dimensionality and the overall signal strength, and can be accurately estimated. Our results imply that the MLE’s magnitude is biased upwards and that the MLE’s standard deviation is greater than that predicted by classical theory. We present a series of experiments on simulated and real data showing excellent agreement with the theory.
E. C. was supported by the National Science Foundation via DMS 1712800 and via the Stanford Data Science Collaboratory OAC 1934578, and by a generous gift from TwoSigma. P.S. was supported by the Center for Research on Computation and Society, Harvard John A. Paulson School of Engineering and Applied Sciences. Q. Z. would like to thank Stephen Bates for helpful comments about an early version of this paper.
"The asymptotic distribution of the MLE in high-dimensional logistic models: Arbitrary covariance." Bernoulli 28 (3) 1835 - 1861, August 2022. https://doi.org/10.3150/21-BEJ1401