Monotone Least Squares and Isotonic Quantiles

We consider bivariate observations $(X_1,Y_1),\ldots,(X_n,Y_n)$ such that, conditional on the $X_i$, the $Y_i$ are independent random variables with distribution functions $F_{X_i}$, where $(F_x)_x$ is an unknown family of distribution functions. Under the sole assumption that $F_x$ is isotonic in $x$ with respect to stochastic order, one can estimate $(F_x)_x$ in two ways: (i) For any fixed $y$ one estimates the antitonic function $x \mapsto F_x(y)$ via nonparametric monotone least squares, replacing the responses $Y_i$ with the indicators $1_{[Y_i \le y]}$. (ii) For any fixed $\beta \in (0,1)$ one estimates the isotonic quantile function $x \mapsto F_x^{-1}(\beta)$ via a nonparametric version of regression quantiles. We show that these two approaches are closely related, with (i) being a bit more flexible than (ii). Then, under mild regularity conditions, we establish rates of convergence for the resulting estimators $\hat{F}_x(y)$ and $\hat{F}_x^{-1}(\beta)$, uniformly over $(x,y)$ and $(x,\beta)$ in certain rectangles.