Direct estimation of the index coefficient in a single-index model

Abstract

Single-index modeling is widely applied in,for example,econometric studies as a compromise between too restrictive parametric models and flexible but hardly estimable purely nonparametric models. By such modeling the statistical analysis usually focuses on estimating the index coefficients. The average derivative estimator (ADE) of the index vector is based on the fact that the average gradient of a single index function $f(x^{\top}\beta)$ is proportional to the index vector $\beta$. Unfortunately,a straightforward application of this idea meets the so-called “curse of dimensionality” problem if the dimensionality $d$ of the model is larger than 2. However, prior information about the vector $\beta$ can be used for improving the quality of gradient estimation by extending the weighting kernel in a direction of small directional derivative. The method proposed in this paper consists of such iterative improvements of the original ADE. The whole procedure requires at most 2 $\log n$ iterations and the resulting estimator is $\sqrt{n}$-consistent under relatively mild assumptions on the model independently of the dimensionality $d$.