Quick and Simple Kernel Differential Equation Regression Estimators for Data with Sparse Design

TL;DR

This paper introduces a nonparametric kernel regression method leveraging differential equations to improve estimation in sparse data regions, demonstrated on noisy biological growth data and supported by simulation studies.

Contribution

It proposes a novel kernel differential equation regression estimator that enhances sparse data modeling by integrating differential equation information, reducing bias and variance.

Findings

The new estimator outperforms local constant regression in sparse regions.

Simulation results show improved accuracy over traditional methods.

Asymptotic bias and variance analyses support the estimator's effectiveness.

Abstract

Local polynomial regression of order at least one often performs poorly in regions of sparse data. Local constant regression is exceptional in this regard, though it is the least accurate method in general, especially at the boundaries of the data. Incorporating information from differential equations which may approximately or exactly hold is one way of extending the sparse design capacity of local constant regression while reducing bias and variance. A nonparametric regression method that exploits first order differential equations is introduced in this paper and applied to noisy mouse tumour growth data. Asymptotic biases and variances of kernel estimators using Taylor polynomials with different degrees are discussed. Model comparison is performed for different estimators through simulation studies under various scenarios which simulate exponential-type growth.