We study the dynamic behavior of cross-sectional ranks over time for functional data and show that the ranks of the observed curves at each time point and their evolution over time can yield valuable insights into the time-dynamics of functional data. This approach is of particular interest in sports statistics in addition to other areas where functional data arise. For the analysis of the dynamics of ranks, we obtain estimates of the cross-sectional ranks of functional data and introduce several statistics of interest for ranked functional data. To quantify the evolution of ranks over time, we develop a model for rank derivatives, in which we decompose rank dynamics into two components, where one component corresponds to population changes and the other to individual changes. We establish the joint asymptotic normality for suitable estimates of these two components. These approaches are illustrated with simulations and three longitudinal data sets: Growth curves obtained from the Z\"urich Longitudinal Growth Study, monthly house price data in the U.S. from 1980 to 2015, and Major League Baseball offensive data for the 2017 season.

中文翻译：

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功能数据的排名动态

我们研究了功能数据的横截面等级随时间的动态行为，并表明每个时间点观察到的曲线的等级及其随时间的演变可以对功能数据的时间动态产生有价值的见解。除了出现功能数据的其他领域外，这种方法对体育统计特别感兴趣。为了分析排名的动态，我们获得了功能数据的横截面排名的估计值，并引入了一些对排名功能数据感兴趣的统计数据。为了量化排名随时间的演变，我们开发了一个排名导数模型，其中我们将排名动态分解为两个部分，其中一个部分对应于人口变化，另一个对应于个体变化。我们为这两个分量的合适估计建立了联合渐近正态性。这些方法通过模拟和三个纵向数据集进行了说明：从 Z\"urich 纵向增长研究中获得的增长曲线、1980 年至 2015 年美国的月度房价数据以及 2017 赛季美国职业棒球大联盟的进攻数据。