  Solving semiparametric models can be computationally challenging because the
dimension of parameter space may grow large with increasing sample size.
Classical Newton's method becomes quite slow and unstable with intensive
calculation of the large Hessian matrix and its inverse. Iterative methods
separately update parameters for finite dimensional component and infinite
dimensional component have been developed to speed up single iteration, but
they often take more steps until convergence or even sometimes sacrifice
estimation precision due to sub-optimal update direction. We propose a
computationally efficient implicit profiling algorithm that achieves
simultaneously the fast iteration step in iterative methods and the optimal
update direction in the Newton's method by profiling out the infinite
dimensional component as the function of the finite dimensional component. We
devise a first order approximation when the profiling function has no explicit
analytical form. We show that our implicit profiling method always solve any
local quadratic programming problem in two steps. In two numerical experiments
under semiparametric transformation models and GARCH-M models, we demonstrated
the computational efficiency and statistical precision of our implicit
profiling method.
