Work

Nonregular and minimax estimation of individualized thresholds in high dimension with binary responses

Year: 2022

Type: article

Abstract: Given a large number of covariates Z, we consider the estimation of a high-dimensional parameter θ in an individualized linear threshold θTZ for a continuous variable X, which minimizes the disagreement between sign(X−θTZ) and a binary response Y. While the problem can be formulated into the M-estimation framework, minimizing the corresponding empirical risk function is computationally intractable due to discontinuity of the sign function. Moreover, estimating θ even in the fixed-dimensional setting is known as a nonregular problem leading to nonstandard asymptotic theory. To tackle the computational and theoretical challenges in the estimation of the high-dimensional parameter θ, we propose an empirical risk minimization approach based on a regularized smoothed non-convex loss function. The Fisher consistency of the proposed method is guaranteed as the bandwidth of the smoothed loss is shrunk to 0. Statistically, we show that the finite sample error bound for estimating θ in ℓ2 norm is (slogd/n)β/(2β+1), where d is the dimension of θ, s is the sparsity level, n is the sample size and β is the smoothness of the conditional density of X given the response Y and the covariates Z. The convergence rate is nonstandard and slower than that in the classical Lasso problems. Furthermore, we prove that the resulting estimator is minimax rate optimal up to a logarithmic factor. The Lepski’s method is developed to achieve the adaption to the unknown sparsity s or smoothness β. Computationally, an efficient path-following algorithm is proposed to compute the solution path. We show that this algorithm achieves geometric rate of convergence for computing the whole path. Finally, we evaluate the finite sample performance of the proposed estimator in simulation studies and a real data analysis from the ChAMP (Chondral Lesions And Meniscus Procedures) Trial.

Source: The Annals of Statistics

DOI: 10.1214/22-aos2188

Authors: Huijie Feng, Yang Ning, Jiwei Zhao

Institutions: Cornell University, University of Wisconsin–Madison

Language: English

Topic: Statistical Methods and Inference

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: bronze

Funders: National Science Foundation, University of Wisconsin-Madison

Awards: A Robust and Efficient Statistical Framework for Handling Missing-Not-At-Random Data in Patient Reported Outcomes and Beyond, CDS&E: Graph-Based Learning and Uncertainty Quantification for Large-Scale Complex Data, G6671297155, CAREER: High-Dimensional M-Estimation Under Nonstandard Conditions