Quantile and Probability Curves without Crossing

@article{Chernozhukov2007QuantileAP,
  title={Quantile and Probability Curves without Crossing},
  author={Victor Chernozhukov and Iv{\'a}n Fern{\'a}ndez-Val and Alfred Galichon},
  journal={MIT Economics Department Working Paper Series},
  year={2007}
}
This paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions, also known as the quantile crossing problem. The method consists in sorting or monotone rearranging the original estimated non-monotone curve into a monotone rearranged curve. We show that the rearranged curve is closer to the true quantile curve in finite samples than the original curve, establish a functional delta method for rearrangement… 
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