  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 $x \mapsto F_x$ is
isotonic 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 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 as well as uniformly in $y$ or $\beta$ for a fixed $x$.
