Local Quasi-Exponential Growth Models: Kernel Differential Equation Regression and Sparse Data

Local polynomial regression struggles with several challenges when dealing with sparse data. The difficulty in capturing local features of the underlying function can lead to a potential misrepresentation of the true relationship. Additionally, with limited data points in local neighborhoods, the variance of estimators can increase significantly. Local polynomial regression also requires a substantial amount of data to produce good models, making it less efficient for sparse datasets. This paper employs a differential equation-constrained regression approach, introduced by \citet{ding2014estimation}, for local quasi-exponential growth models. By incorporating first-order differential equations, this method extends the sparse design capacity of local polynomial regression while reducing bias and variance. We discuss the asymptotic biases and variances of kernel estimators using first-degree Taylor polynomials. Model comparisons are conducted using mouse tumor growth data, along with simulation studies under various scenarios that simulate quasi-exponential growth with different noise levels and growth rates.