  The distributional single index model is a semiparametric regression model in
which the conditional distribution functions $P(Y \leq y | X = x) =
F_0(\theta_0(x), y)$ of a real-valued outcome variable $Y$ depend on
$d$-dimensional covariates $X$ through a univariate, parametric index function
$\theta_0(x)$, and increase stochastically as $\theta_0(x)$ increases. We
propose least squares approaches for the joint estimation of $\theta_0$ and
$F_0$ in the important case where $\theta_0(x) = \alpha_0^{\top}x$ 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 $\alpha_0$ might be faster. Furthermore, we
illustrate our methods in a real data application that demonstrates the
advantages of shape restrictions in single index models.
