Least squares estimation in the monotone single index model

We study the monotone single index model where a real response variable Y is linked to a d-dimensional covariate X through the relationship E[Y | X] = Ψ₀(αT₀ X) almost surely. Both the ridge function, Ψ₀, and the index parameter, α₀, are unknown and the ridge function is assumed to be monotone on its interval of support. Under some regularity conditions, without imposing a particular distribution on the regression error, we show the n-1/3 rate of convergence in the ₂-norm for the least squares estimator of the bundled function ψ₀(αT₀ ·), and also that of the ridge function and the index separately. Furthermore, we show that the least squares estimator is nearly parametrically rate-adaptive to piecewise constant ridge functions.