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 P(Y≤y|X=x)=F0(θ0(x),y)$$ P\left(Y\le y|X=x\right)={F}_0\left({\theta}_0(x),y\right) $$ of a real‐valued outcome variable Y$$ Y $$ depend on d$$ d $$‐dimensional covariates X$$ X $$ through a univariate, parametric index function θ0(x)$$ {\theta}_0(x) $$, and increase stochastically as θ0(x)$$ {\theta}_0(x) $$ increases. We propose least squares approaches for the joint estimation of θ0$$ {\theta}_0 $$ and F0$$ {F}_0 $$ in the important case where θ0(x)=α0⊤x$$ {\theta}_0(x)={\alpha}_0^{\top }x $$ and obtain convergence rates of n−1/3$$ {n}^{-1/3} $$, thereby improving an existing result that gives a rate of n−1/6$$ {n}^{-1/6} $$. A simulation study indicates that the convergence rate for the estimation of α0$$ {\alpha}_0 $$ 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. [ABSTRACT FROM AUTHOR]