Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error
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兼顾数值误差与测量误差的非线性常微分方程系数筛估计
- 研究问题
- 当非线性常微分方程没有解析解时，如何估计常数与时变系数，并判断数值求解误差是否影响统计估计性质？
- 方法与思路
- 作者以数值求解得到的轨迹构造非线性最小二乘估计，用 B 样条逼近时变系数，并在筛估计框架下同时分析两类误差。
- 材料陈述的贡献
- 摘要称建立了步长选择指导、强一致性和相关渐近理论，并讨论数值误差不可忽略的情形，辅以模拟及 HIV 临床数据。
- 阅读关注点
- 核对数值算法阶数与步长条件、样条维度选择，以及常数参数渐近正态性和时变参数最优收敛率分别需要的条件。
- 阅读边界
- 仅据稿件版本未注明的 OpenAlex 索引摘要；具体正则条件和步长阈值须核对原文，时变系数本身不构成宏观时间序列证据。
原文摘要
This article considers estimation of constant and time-varying coefficients in nonlinear ordinary differential equation (ODE) models where analytic closed-form solutions are not available. The numerical solution-based nonlinear least squares (NLS) estimator is investigated in this study. A numerical algorithm such as the Runge-Kutta method is used to approximate the ODE solution. The asymptotic properties are established for the proposed estimators considering both numerical error and measurement error. The B-spline is used to approximate the time-varying coefficients, and the corresponding asymptotic theories in this case are investigated under the framework of the sieve approach. Our results show that if the maximum step size of the p-order numerical algorithm goes to zero at a rate faster than n(-1/(p∧4)), the numerical error is negligible compared to the measurement error. This result provides a theoretical guidance in selection of the step size for numerical evaluations of ODEs. Moreover, we have shown that the numerical solution-based NLS estimator and the sieve NLS estimator are strongly consistent. The sieve estimator of constant parameters is asymptotically normal with the same asymptotic co-variance as that of the case where the true ODE solution is exactly known, while the estimator of the time-varying parameter has the optimal convergence rate under some regularity conditions. The theoretical results are also developed for the case when the step size of the ODE numerical solver does not go to zero fast enough or the numerical error is comparable to the measurement error. We illustrate our approach with both simulation studies and clinical data on HIV viral dynamics.