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 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 $x$.