Estimation and empirical likelihood for single-index multiplicative models

In this paper, we study a class of multiplicative models with a single-index structure called single-index multiplicative models, which greatly extend the range of application of positive response data since it allows the presence of nonparametric link and also mitigates the curse of dimensionality. Firstly, based on the least product relative error (LPRE) method proposed recently by Chen et al. (2016), we propose an iterative LPRE approach for estimation of parameters and nonparametric function, where a local kernel-weighted LPRE approach is developed. Under some regularity conditions, we prove that proposed estimates have nice asymptotic normalities for both nonparametric function and parameter vector. Secondly, to make inference on the parameter vector, an empirical likelihood procedure is presented, which facilitates confidence region construction without estimating covariance matrix. Finally, extensively numerical studies are carried out to evaluate the finite sample performance of proposed approaches.