A robust and efficient estimation method for partially nonlinear models via a new MM algorithm

When the observed data set contains outliers, it is well known that the classical least squares method is not robust. To overcome this difficulty, Wang et al. (J Am Stat Assoc 108(502): 632–643, 2013) proposed a robust variable selection method by using the exponential squared loss (ESL) function with a tuning parameter. Although many important statistical models are investigated, to date, in the presence of outliers there is no paper to study the partially nonlinear model by using the ESL function. To fill in this gap, in this paper, we propose a robust and efficient estimation method for the partially nonlinear model based on the ESL function. Under certain conditions, we have shown that the proposed estimators can achieve the best convergence rates. Next, the asymptotic normality of the proposed estimators is established. In addition, we develop a new minorization–maximization algorithm to calculate the estimates for both non-parametric and parametric parts and present a procedure for deriving initial values. Finally, we provide a data-driven approach to select the tuning parameters. Numerical simulations and a real data analysis are used to illustrate that when there are outliers, the proposed ESL method is more robust and efficient for partially nonlinear models than the existing linear approximation method and the composite quantile regression method.