Asymptotic efficiency and finite-sample properties of the generalized profiling estimation of parameters in ordinary differential equations
常微分方程参数广义剖面估计的渐近效率与有限样本性质AI 解读 ↓
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常微分方程参数广义剖面估计的渐近效率与有限样本性质
- 研究问题
- 利用基函数和微分方程惩罚估计动态系统参数时，如何建立广义剖面估计的统计性质，并选择平滑参数？
- 方法与思路
- 作者先界定真实解与基函数近似解的统一误差上界，再证明估计性质，并分析固定样本与基函数下的惩罚极限。
- 材料陈述的贡献
- 摘要称，方法具有一致性和渐近正态性，渐近协方差与最大似然相同；另提出平滑参数选择和无需解方程的偏差计算方法。
- 阅读关注点
- 应核对观测模型、基函数逼近与惩罚参数的联合条件，以及效率结论、固定样本极限和偏离量计算的适用范围。
- 阅读边界
- 仅据OpenAlex索引摘要，稿件版本未注明；效率结论的具体条件和有限样本表现未给出，不能推广为任意动态系统上的优势。
原文摘要
Ordinary differential equations (ODEs) are commonly used to model dynamic behavior of a system. Because many parameters are unknown and have to be estimated from the observed data, there is growing interest in statistics to develop efficient estimation procedures for these parameters. Among the proposed methods in the literature, the generalized profiling estimation method developed by Ramsay and colleagues is particularly promising for its computational efficiency and good performance. In this approach, the ODE solution is approximated with a linear combination of basis functions. The coefficients of the basis functions are estimated by a penalized smoothing procedure with an ODE-defined penalty. However, the statistical properties of this procedure are not known. In this paper, we first give an upper bound on the uniform norm of the difference between the true solutions and their approximations. Then we use this bound to prove the consistency and asymptotic normality of this estimation procedure. We show that the asymptotic covariance matrix is the same as that of the maximum likelihood estimation. Therefore, this procedure is asymptotically efficient. For a fixed sample and fixed basis functions, we study the limiting behavior of the approximation when the smoothing parameter tends to infinity. We propose an algorithm to choose the smoothing parameters and a method to compute the deviation of the spline approximation from solution without solving the ODEs.