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.

中文翻译：

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函数单指数模型中曲率的局部二次估计

环境变异对物种丰度的非线性影响在维持生态多样性方面发挥着重要作用。尽管如此，许多常见模型使用参数非线性项来预先确定生态结论。出于这种考虑，我们研究了函数单指数模型中链接函数 g 的二阶导数（曲率）的估计。由于系数函数和链接函数都是未知的，因此估计被表示为嵌套优化。对于固定和未知的系数函数，通过局部二次近似估计链接函数及其二阶导数，然后通过最小化模型的MSE来估计系数函数。在本文中，我们推导出估计的收敛速度。此外，我们证明 g 的参数，可以一致地估计 root-n。然而，该方法的实际实现需要解决非线性优化问题，我们的结果表明链接函数和系数函数的估计对起始值的选择非常敏感。