Local quadratic estimation of the curvature in a functional single index model
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.