# Near-Linear Time Local Polynomial Nonparametric Estimation with Box Kernels

## TL;DR

This paper introduces a nearly linear time algorithm for local polynomial nonparametric estimation using box kernels, significantly improving computational efficiency for large datasets.

## Contribution

The paper presents a novel algorithm leveraging multi-dimensional binary indexed trees to accelerate local polynomial estimation, reducing complexity from quadratic to nearly linear.

## Key findings

- Algorithm achieves near-linear time complexity.
- Simulation confirms improved efficiency and accuracy.
- Applicable to large-scale nonparametric density estimation.

## Abstract

Local polynomial regression (Fan and Gijbels 1996) is an important class of methods for nonparametric density estimation and regression problems. However, straightforward implementation of local polynomial regression has quadratic time complexity which hinders its applicability in large-scale data analysis. In this paper, we significantly accelerate the computation of local polynomial estimates by novel applications of multi-dimensional binary indexed trees (Fenwick 1994). Both time and space complexity of our proposed algorithm is nearly linear in the number of input data points. Simulation results confirm the efficiency and effectiveness of our proposed approach.