Adaptive confidence region for the direction in semiparametric regressions

In this paper we aim to construct adaptive confidence region for the direction of ξξ in semiparametric models of the form Y=G(ξTX,ε)Y=G(ξTX,ε) where G(⋅)G(⋅) is an unknown link function, εε is an independent error, and ξξ is a pn×1pn×1 vector. To recover the direction of ξξ, we first propose an inverse regression approach regardless of the link function G(⋅)G(⋅); to construct a data-driven confidence region for the direction of ξξ, we implement the empirical likelihood method. Unlike many existing literature, we need not estimate the link function G(⋅)G(⋅) or its derivative. When pnpn 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 pnpn follows the rate of pn=o(n1/4)pn=o(n1/4) where nn 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.