A Novel Estimation Method in Generalized Single Index Models. Issue 2 (3rd April 2023) - Zhang, Dixin; Wang, Yulin; Liang, Hua
Abstract: The single index and generalized single index models have been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables in the low-dimensional case. In this article, we propose a new estimation approach for generalized single index models E ( Y | θ ⊤ X ) = ψ ( g ( θ ⊤ X ) ) with ψ ( · ) known but g ( · ) unknown. Specifically, we first obtain a consistent estimator of the regression function by using a local linear smoother, and then estimate the parametric components by treating ψ ( g ̂ ( θ ⊤ X i ) ) as our continuous response. The resulting estimators of θ are asymptotically normal. The proposed procedure can substantially overcome convergence problems encountered in generalized linear models with discrete response variables when sparseness occurs and misspecification. We conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze a financial dataset from a peer-to-peer lending platform of China as an illustration.