  The nonlinear effects of environmental variability on species abundance plays
an important role in the maintenance of ecological diversity. Nonetheless, many
common models use parametric nonlinear terms pre-determining ecological
conclusions. Motivated by this concern, we study the estimate of the second
derivative (curvature) of the link function g in a functional single index
model. Since the coefficient function and the link function are both unknown,
the estimate is expressed as a nested optimization. For a fixed and unknown
coefficient function, the link function and its second derivative are estimated
by local quadratic approximation, then the coefficient function is estimated by
minimizing the MSE of the model. In this paper, we derive the rate of
convergence of the estimation. In addition, we prove that the argument of g,
can be estimated root-n consistently. However, practical implementation of the
method requires solving a nonlinear optimization problem, and our results show
that the estimates of the link function and the coefficient function are quite
sensitive to the choices of starting values.
