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This paper studies a \textit{partial functional partially linear single-indexmodel} that consists of a functional linear component as well as a linearsingle-index component. This model generalizes many well-known existing modelsand is suitable for more complicated data structures. However, its estimationinherits the difficulties and complexities from both components and makes it achallenging problem, which calls for new methodology. We propose a novelprofile B-spline method to estimate the parameters by approximating the unknownnonparametric link function in the single-index component part with B-spline,while the linear slope function in the functional component part is estimatedby the functional principal component basis. The consistency and asymptoticnormality of the parametric estimators are derived, and the global convergenceof the proposed estimator of the linear slope function is also established.More excitingly, the latter convergence is optimal in the minimax sense. Atwo-stage procedure is implemented to estimate the nonparametric link function,and the resulting estimator possesses the optimal global rate of convergence.Furthermore, the convergence rate of the mean squared prediction error for apredictor is also obtained. Empirical properties of the proposed procedures arestudied through Monte Carlo simulations. A real data example is also analyzedto illustrate the power and flexibility of the proposed methodology.

Profile Estimation for Partial Functional Partially Linear Single-Index Model