Rank Dynamics for Functional Data
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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 Zürich 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.
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