  This paper studies a \textit{partial functional partially linear single-index
model} that consists of a functional linear component as well as a linear
single-index component. This model generalizes many well-known existing models
and is suitable for more complicated data structures. However, its estimation
inherits the difficulties and complexities from both components and makes it a
challenging problem, which calls for new methodology. We propose a novel
profile B-spline method to estimate the parameters by approximating the unknown
nonparametric link function in the single-index component part with B-spline,
while the linear slope function in the functional component part is estimated
by the functional principal component basis. The consistency and asymptotic
normality of the parametric estimators are derived, and the global convergence
of the proposed estimator of the linear slope function is also established.
More excitingly, the latter convergence is optimal in the minimax sense. A
two-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 a
predictor is also obtained. Empirical properties of the proposed procedures are
studied through Monte Carlo simulations. A real data example is also analyzed
to illustrate the power and flexibility of the proposed methodology.
