Estimation and convergence rates in the distributional single index model

Abstract

The distributional single index model is a semiparametric regression model in which the conditional distribution functions P(Y ≤ y | X = x) = F₀(θ₀(x), y) of a real-valued outcome variable Y depend on d-dimensional covariates X through a univariate, parametric index function θ₀(x), and increase stochastically as θ₀(x) increases. We propose least squares approaches for the joint estimation of θ₀ and F₀ in the important case where θ₀(x) = αᵀ₀x and obtain convergence rates of n⁻¹/³, thereby improving an existing result that gives a rate of n⁻¹/⁶. A simulation study indicates that the convergence rate for the estimation of α₀ 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.