Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξ in semiparametric models of the form Y = G (ξ^T X, ε) where G ({dot operator}) is an unknown link function, ε is an independent error, and ξ is a p_n × 1 vector. To recover the direction of ξ, we first propose an inverse regression approach regardless of the link function G ({dot operator}); to construct a data-driven confidence region for the direction of ξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G ({dot operator}) or its derivative. When p_n remains fixed, the empirical likelihood ratio without bias correlation can be asymptotically standard chi-square. Moreover, the asymptotic normality of the empirical likelihood ratio holds true even when the dimension p_n follows the rate of p_n = o (n^{1 / 4}) where n is the sample size. Simulation studies are carried out to assess the performance of our proposal, and a real data set is analyzed for further illustration.