Scalar-on-Function Mode Estimation Using Entropy and Ergodic Properties of Functional Time Series Data

In this paper, we investigate the recursive L1 estimator of the conditional mode when the input variable takes values in a pseudo-metric space. The new proposed estimator is constructed under an ergodicity assumption, which provides a robust alternative to the standard mixing processes in various practical settings. The particular interest of this contribution arises from the difficulty in incorporating the mathematical properties of a functional mixing process. In contrast, ergodicity is characterized by the Kolmogorov-Sinai entropy, which measures the dynamics, the sparsity, and the microscopic fluctuations of the functional process. Using an observation sampled from ergodic functional time series (fts), we establish the asymptotic properties of this estimator. In particular, we derive its convergence rate and show Borel-Cantelli (BC) consistency. The general expression for the convergence rate is then specialized to several notable scenarios, including the independence case, the classical kernel method, and the vector-valued case. Finally, numerical experiments on both simulated and real-world datasets demonstrate the superiority of the L1-recursive estimator compared to existing competitors.