An Upper Bound for Functions of Estimators in High Dimensions

TL;DR

This paper introduces an upper bound for functions of estimators in high-dimensional settings, aiding in understanding convergence rates when the number of parameters exceeds sample size.

Contribution

It provides a novel upper bound framework for functions of estimators in high dimensions, applicable even when parameters outnumber samples.

Findings

Upper bound can converge faster, slower, or at the same rate as estimators.

Applications include high-dimensional testing and portfolio variance estimation.

Results accommodate more parameters than samples.

Abstract

We provide an upper bound as a random variable for the functions of estimators in high dimensions. This upper bound may help establish the rate of convergence of functions in high dimensions. The upper bound random variable may converge faster, slower, or at the same rate as estimators depending on the behavior of the partial derivative of the function. We illustrate this via three examples. The first two examples use the upper bound for testing in high dimensions, and third example derives the estimated out-of-sample variance of large portfolios. All our results allow for a larger number of parameters, p, than the sample size, n.