Monotone least squares and isotonic quantiles

Abstract

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. Precisely, the conditional distribution function of $Y_{i}$ equals $F_{X_{i}}$, 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 $x$.