Monotone Least Squares and Isotonic Quantiles

Abstract

We consider bivariate observations $(X_1,Y_1),\dots ,(X_n,Y_n)$ 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 $x$.

References in corpus (2)

• Extensions of smoothing via taut strings
• Optimal solutions to the isotonic regression problem

Cited by in corpus (7)

• Regression Diagnostics meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination
• Isotonic Distributional Regression
• Forecast Evaluation of Quantiles, Prediction Intervals, and other Set-Valued Functionals
• Honest calibration assessment for binary outcome predictions
• Accelerating the pool-adjacent-violators algorithm for isotonic distributional regression
• Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk
• Estimation of a Likelihood Ratio Ordered Family of Distributions