同时考虑数值误差和测量误差的非线性常微分方程模型中常系数和时变系数的筛分估计
Sieve estimation of constant and time-varying coefficients in nonlinear ordinary differential equation models by considering both numerical error and measurement error
摘要：
本文研究了在没有解析解的情况下，非线性常微分方程（ODE）模型中常系数和时变系数的估计问题。本文研究了基于数值解的非线性最小二乘估计.数值算法如龙格-库塔法被用来近似常微分方程的解决方案。同时考虑数值误差和测量误差，建立了估计量的渐近性质。采用B样条函数逼近时变系数，并在筛法框架下研究了相应的渐近理论。我们的研究结果表明，如果$p$阶数值算法的最大步长以比$n^{-1/(p\wedge4)}$更快的速率变为零，则与测量误差相比，数值误差可以忽略不计。这一结果为常微分方程数值计算中步长的选取提供了理论指导。此外，我们还证明了基于数值解的NLS估计和基于筛分的NLS估计是强相合的。常数参数的筛法估计是渐近正态的，具有与已知常微分方程真解时相同的渐近协方差，而时变参数的筛法估计在一定的正则性条件下具有最优的收敛速度.当ODE数值解的步长不能足够快地归零或数值误差与测量误差相当时，理论结果也得到了发展。我们说明了我们的方法与模拟研究和HIV病毒动力学的临床数据。
英文摘要：
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 $n^{-1/(p\wedge4)}$, 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.
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