Abstract: The aim of this study is to develop a set of functional finite mixture regression models with functional predictors in the framework of the reproducing kernel Hilbert space. First, we show the consistency of a penalized likelihood model order estimator for the true model order, denoted as q*. We further show that the penalty of order q ^{2r / (2r + 1)} n ^{1 / (2r + 1)} log(n) yields a strong consistent estimator of q*, where n and q are the sample size and the model order, respectively, and r is the eigenvalue decay rate of an operator determined jointly by the reproducing and covariance kernels. Second, we establish the minimax rate of convergence for the estimation risk. We show that the optimal rate is determined by the alignment of the reproducing kernel and the covariance kernel and the true model order q*. An efficient algorithm is also developed to estimate all unknown components of the functional finite mixture model. Simulation studies and a real-data analysis illustrate the merits of the proposed method.

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