Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of [xi] in semiparametric models of the form Y=G([xi]TX,[epsilon]) where G([dot operator]) is an unknown link function, [epsilon] is an independent error, and [xi] is a pnx1 vector. To recover the direction of [xi], 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 [xi], we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G([dot operator]) or its derivative. When pn 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 pn follows the rate of pn=o(n1/4) 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.