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 ( ⋅ ) 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 ( ⋅ ) ; 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 ( ⋅ ) 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.