Asymptotics for estimating a diverging number of parameters -- with and without sparsity

TL;DR

This paper develops general asymptotic theory for high-dimensional parameter estimation where the number of parameters grows with the sample size, covering both penalized and unpenalized methods.

Contribution

It establishes weak, broad conditions for consistency, uniqueness, and asymptotic normality applicable to various estimation problems, including non-convex and structured penalties.

Findings

Conditions ensure consistency, uniqueness, and normality in high-dimensional settings.

Results apply to generalized linear models and multi-sample inference.

Framework accommodates non-convex and group-structured penalties.

Abstract

We consider high-dimensional estimation problems where the number of parameters diverges with the sample size. General conditions are established for consistency, uniqueness, and asymptotic normality in both unpenalized and penalized estimation settings. The conditions are weak and accommodate a broad class of estimation problems, including ones with non-convex and group structured penalties. The wide applicability of the results is illustrated through diverse examples, including generalized linear models, multi-sample inference, and stepwise estimation procedures.