标题： 纵向数据的经验动态 显示英文标题

标题： Empirical dynamics for longitudinal data

摘要： 我们证明了在线拍卖价格出价以及许多其他纵向数据背后的进程可以由一个具有时变系数和光滑漂移过程的经验一阶随机常微分方程来表示。 该方程可以从样本中受试者的纵向观测中经验获得，并不假设对底层进程有特定的知识。 对于微分方程各部分的非参数估计，只需提供稀疏观测的纵向测量值，这些测量值可能是噪声的，并且是由样本中每个受试者或实验单元的底层光滑随机轨迹生成的。 驱动方程的漂移过程决定了个体过程轨迹如何紧密地遵循微分方程的确定性近似。 我们提供了轨迹的估计值，特别是漂移过程的方差函数的估计值。 在每个固定的时刻点，所提出的经验动态模型意味着对底层纵向数据的过程导数进行分解，其中一部分由由变化系数函数动态方程确定的线性部分解释，另一部分是正交补，对应于漂移过程。 增强的扰动结果使我们能够获得对特征函数导数估计的改进渐近收敛速度，以及变化系数函数和漂移过程各部分的一致性。 我们通过在线拍卖数据的动力学应用来说明该微分方程。 显示英文摘要

摘要： We demonstrate that the processes underlying on-line auction price bids and many other longitudinal data can be represented by an empirical first order stochastic ordinary differential equation with time-varying coefficients and a smooth drift process. This equation may be empirically obtained from longitudinal observations for a sample of subjects and does not presuppose specific knowledge of the underlying processes. For the nonparametric estimation of the components of the differential equation, it suffices to have available sparsely observed longitudinal measurements which may be noisy and are generated by underlying smooth random trajectories for each subject or experimental unit in the sample. The drift process that drives the equation determines how closely individual process trajectories follow a deterministic approximation of the differential equation. We provide estimates for trajectories and especially the variance function of the drift process. At each fixed time point, the proposed empirical dynamic model implies a decomposition of the derivative of the process underlying the longitudinal data into a component explained by a linear component determined by a varying coefficient function dynamic equation and an orthogonal complement that corresponds to the drift process. An enhanced perturbation result enables us to obtain improved asymptotic convergence rates for eigenfunction derivative estimation and consistency for the varying coefficient function and the components of the drift process. We illustrate the differential equation with an application to the dynamics of on-line auction data.