Inference In High-dimensional Single-Index Models Under Symmetric Designs

We consider the problem of statistical inference for a finite number of covariates in a generalized single-index model with p > n covariates and unknown (potentially random) link function under an elliptically symmetric design. Under elliptical symmetry, the problem can be reformulated as a proxy linear model in terms of an identifiable parameter, which characterization is then used to construct estimates of the regression coefficients of interest that are similar to the de-biased lasso estimates in the standard linear model and exhibit similar properties: square-root consistency and asymptotic normality. The procedure is agnostic in the sense that it completely bypasses the estimation of the link function, which can be extremely challenging depending on the underlying structure of the problem. Our method allows testing for the importance of pre-fixed covariates in the single-index model, as well as testing for the relative importance of coefficients via straightforward application of the delta method. Furthermore, under Gaussianity, we extend our approach to prescribe improved, i.e., more efficient estimates of the coefficients using a sieved strategy that involves an expansion of the true regression function in terms of Hermite polynomials. Finally, we illustrate our approach via carefully designed simulation experiments.