# Monotone Least Squares and Isotonic Quantiles

arXiv: https://arxiv.org/abs/1901.02398 · HTML: https://mlsanity.com/paper/1901.02398/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jan 2019.

## Built on by (most cited first, 7 of 7)

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- [Distributional regression and its evaluation with the CRPS: Bounds and convergence of the minimax risk](https://mlsanity.com/paper/2205.04360/) (2022)
- [Probabilistic measures afford fair comparisons of AIWP and NWP model output](https://mlsanity.com/paper/2506.03744/) (2025)
- [Revenue Maximization Under Sequential Price Competition Via The Estimation Of s-Concave Demand Functions](https://mlsanity.com/paper/2503.16737/) (2025)
- [Risk Bounds For Distributional Regression](https://mlsanity.com/paper/2505.09075/) (2025)
- [Isotonic Survival Regression: Calibrated Survival Distributions from Deep Cox Models](https://mlsanity.com/paper/2605.16571/) (2026)

## Abstract (arXiv)

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.