# Local Quadratic Estimation of the Curvature in a Functional Single Index Model

## TL;DR

This paper develops a method for estimating the curvature of the link function in a functional single index model, addressing the challenge of unknown coefficient and link functions, with theoretical convergence results and practical considerations.

## Contribution

It introduces a nested optimization approach for estimating the second derivative of the link function in a functional single index model, including convergence rates and root-n consistency.

## Key findings

- The second derivative of the link function can be estimated with a specific convergence rate.
- The argument of the link function can be estimated with root-n consistency.
- Estimates are sensitive to initial values in practical implementation.

## Abstract

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.