Delta Theorem in the Age of High Dimensions

TL;DR

This paper extends the delta theorem to high-dimensional settings where the number of parameters exceeds the sample size, providing new insights into the convergence rates of functions of estimators in such contexts.

Contribution

It introduces a novel version of the delta theorem applicable to high-dimensional parameter estimation, accounting for different convergence rates based on function structure.

Findings

The new delta theorem applies when p > n.

Convergence rates vary depending on the function structure.

Illustrated with high-dimensional testing and portfolio risk estimation.

Abstract

We provide a new version of delta theorem, that takes into account of high dimensional parameter estimation. We show that depending on the structure of the function, the limits of functions of estimators have faster or slower rate of convergence than the limits of estimators. We illustrate this via two examples. First, we use it for testing in high dimensions, and second in estimating large portfolio risk. Our theorem works in the case of larger number of parameters, $p$, than the sample size, $n$: $p>n$.