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2020 Monotone least squares and isotonic quantiles
Alexandre Mösching, Lutz Dümbgen
Electron. J. Statist. 14(1): 24-49 (2020). DOI: 10.1214/19-EJS1659

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 $x$.

Citation

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Alexandre Mösching. Lutz Dümbgen. "Monotone least squares and isotonic quantiles." Electron. J. Statist. 14 (1) 24 - 49, 2020. https://doi.org/10.1214/19-EJS1659

Information

Received: 1 August 2019; Published: 2020
First available in Project Euclid: 3 January 2020

zbMATH: 1434.62056
MathSciNet: MR4047593
Digital Object Identifier: 10.1214/19-EJS1659

Subjects:
Primary: 62G08 , 62G20 , 62G30

Keywords: regression quantiles , stochastic order , uniform consistency

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