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Local quadratic estimation of the curvature in a functional single index model
Author(s) -
Ye Zi,
Hooker Giles
Publication year - 2020
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12481
Subject(s) - mathematics , estimator , function (biology) , curvature , nonlinear system , quadratic function , quadratic equation , parametric statistics , mean squared error , rate of convergence , statistics , mathematical optimization , computer science , computer network , channel (broadcasting) , physics , geometry , quantum mechanics , evolutionary biology , biology
Abstract The nonlinear responses of species to environmental variability can play an important role in the maintenance of ecological diversity. Nonetheless, many models use parametric nonlinear terms which pre‐determine the ecological conclusions. Motivated by this concern, we study the estimate of the second derivative (curvature) of the link function 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. We first estimate the coefficient function by minimizing squared error where the link function is estimated with a Nadaraya‐Watson estimator for each candidate coefficient function. The first and second derivatives of the link function are then estimated via local‐quadratic regression using the estimated coefficient function. In this paper, we derive a convergence rate for the curvature of the nonlinear response. In addition, we prove that the argument of the linear predictor 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.