Statistical inference for the extended non linear models

In this article, an orthogonality-projection-based estimation method is first employed to study an extended non linear model, which can separately estimate the linear and non linear parts without losing some efficiency and increasing computation burden. To overcome the non robustness of classical least squares methods, we further propose a robust and efficient estimation procedure based on the exponential squares loss function, which is robust against outliers or heavy-tailed errors while asymptotically efficient as the non linear least squares estimation under the normal error case. Under some regularity conditions, the asymptotic properties of the proposed estimators are established. In addition, simulation studies are conducted to examine the finite sample performance of the proposed estimation methods.