# Asymptotic Properties of Neural Network Sieve Estimators

## TL;DR

This paper investigates the asymptotic behavior of neural network estimators within a nonparametric regression framework, establishing their consistency, convergence rates, and normality, supported by simulation evidence.

## Contribution

It extends sieve estimation theory to neural networks, providing the first rigorous asymptotic analysis of neural network estimators in a nonparametric setting.

## Key findings

- Neural network estimators are consistent.
- Convergence rates are established.
- Asymptotic normality is demonstrated.

## Abstract

Neural networks are one of the most popularly used methods in machine learning and artificial intelligence nowadays. Due to the universal approximation theorem (Hornik et al. (1989)), a neural network with one hidden layer can approximate any continuous function on a compact support as long as the number of hidden units is sufficiently large. Statistically, a neural network can be classified into a nonlinear regression framework. However, if we consider it parametrically, due to the unidentifiability of the parameters, it is difficult to derive its asymptotic properties. Instead, we considered the estimation problem in a nonparametric regression framework and use the results from sieve estimation to establish the consistency, the rates of convergence and the asymptotic normality of the neural network estimators. We also illustrate the validity of the theories via simulations.