Empirical likelihood for heteroscedastic partially linear single-index models with growing dimensional data

In this paper, we propose a new approach to the empirical likelihood inference for the parameters in heteroscedastic partially linear single-index models. In the growing dimensional setting, it is proved that estimators based on semiparametric efficient score have the asymptotic consistency, and the limit distribution of the empirical log-likelihood ratio statistic for parameters $$(\beta ^{\top },\theta ^{\top })^{\top }$$ ( β ⊤ , θ ⊤ ) ⊤ is a normal distribution. Furthermore, we show that the empirical log-likelihood ratio based on the subvector of $$\beta $$ β is an asymptotic chi-square random variable, which can be used to construct the confidence interval or region for the subvector of $$\beta $$ β . The proposed method can naturally be applied to deal with pure single-index models and partially linear models with high-dimensional data. The performance of the proposed method is illustrated via a real data application and numerical simulations.