Monotone Least Squares and Isotonic Quantiles

We consider bivariate observations (X1,Y1), ldots, (Xn,Yn) such that, conditional on the Xi, the Yi are independent random variables with distribution functions FXi, where (Fx)x is an unknown family of distribution functions. Under the sole assumption that x mapsto Fx is isotonic with respect to stochastic order, one can estimate (Fx)x in two ways: (i) For any fixed y one estimates the antitonic function x mapsto Fx(y) via nonparametric monotone least squares, replacing the responses Yi with the indicators 1[Yi ≤ y]. (ii) For any fixed β ∈ (0,1) one estimates the isotonic quantile function x mapsto Fx-1(β) 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 hatFx(y) and hatFx-1(β), uniformly over (x,y) and (x,β) in certain rectangles as well as uniformly in y or β for a fixed x.