标题： 局部拟指数增长模型：核微分方程回归与稀疏数据 显示英文标题

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

摘要： 局部多项式回归在处理稀疏数据时面临多个挑战。捕捉潜在函数的局部特征的难度可能导致对真实关系的潜在误表征。此外，在局部邻域内数据点有限的情况下，估计量的方差可能会显著增加。局部多项式回归还需要大量的数据来生成良好的模型，这使得它对于稀疏数据集效率较低。本文采用了一种由 \citet{ding2014estimation}提出的受微分方程约束的回归方法，用于局部拟指数增长模型。 通过引入一阶微分方程，该方法扩展了局部多项式回归的稀疏设计能力，同时减少了偏差和方差。我们讨论了使用一次泰勒多项式的一阶核估计器的渐近偏差和方差。通过小鼠肿瘤生长数据以及在不同场景下的模拟研究，这些研究模拟了具有不同噪声水平和增长率的拟指数增长，进行了模型比较。 显示英文摘要

摘要： 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.