Mösching, Alexandre; Dümbgen, Lutz (2020). Monotone least squares and isotonic quantiles. Electronic journal of statistics, 14(1), pp. 2449. Institute of Mathematical Statistics 10.1214/19EJS1659

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We consider bivariate observations (X₁,Y₁),…,(Xn,Yn) such that, conditional on the Xi, the Yi are independent random variables. Precisely, the conditional distribution function of Yi equals FXi, where (Fx)x is an unknown family of distribution functions. Under the sole assumption that x↦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↦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↦F−1x(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 F^x(y) and F^−1x(β), uniformly over (x,y) and (x,β) in certain rectangles as well as uniformly in y or β for a fixed x.
Item Type: 
Journal Article (Original Article) 

Division/Institute: 
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science 
UniBE Contributor: 
Mösching, Alexandre and Dümbgen, Lutz 
Subjects: 
500 Science > 510 Mathematics 
ISSN: 
19357524 
Publisher: 
Institute of Mathematical Statistics 
Funders: 
[4] Swiss National Science Foundation 
Language: 
English 
Submitter: 
Lutz Dümbgen 
Date Deposited: 
14 Jan 2020 08:58 
Last Modified: 
14 Jan 2020 08:58 
Publisher DOI: 
10.1214/19EJS1659 
BORIS DOI: 
10.7892/boris.137857 
URI: 
https://boris.unibe.ch/id/eprint/137857 