Estimation and convergence rates in the distributional single index model

The distributional single index model is a semiparametric regression model in which the conditional distribution functions of a real-valued outcome variable depend on -dimensional covariates through a univariate, parametric index function , and increase stochastically as increases. We propose least squares approaches for the joint estimation of and in the important case where and obtain convergence rates of , thereby improving an existing result that gives a rate of . A simulation study indicates that the convergence rate for the estimation of might be faster. Furthermore, we illustrate our methods in an application on house price data that demonstrates the advantages of shape restrictions in single index models.