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. 2020 Jul 1;20:175. doi: 10.1186/s12874-020-01055-2

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© The Author(s) 2020

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.

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Fig. 1 — An illustrative example of the estimated CDFs, F^∗, under RQRs (top panels) and MQR (bottom panels). Panels in the left column present the residuals under the true model and panels in the right column present the residuals under the wrong model. The curved grey lines correspond to the theoretical CDF of F(k|x_i)versusx_i at each value of k. The points are the randomized CDFs $F^{*} (y_{i}; {\hat{μ}}_{i}, \hat{ϕ}, u_{i})$ with each colour corresponding to a unique value of y_i