### Abstract

Suppose that a target function is monotonic and an available original estimate of this target function is not monotonic. Rearrangements, univariate and multivariate, transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function. Furthermore, suppose an original confidence interval, which covers the target function with probability at least 1-α, is defined by an upper and lower endpoint functions that are not monotonic. Then the rearranged confidence interval, defined by the rearranged upper and lower endpoint functions, is monotonic, shorter in length in common norms than the original interval, and covers the target function with probability at least 1-α. We illustrate the results with a growth chart example.

Original language | English (US) |
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Pages (from-to) | 559-575 |

Number of pages | 17 |

Journal | Biometrika |

Volume | 96 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2009 |

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### Keywords

- Growth chart
- Improved estimation
- Improved inference
- Isotonization
- Lorentz inequality
- Monotone function
- Multivariate
- Quantile regression
- Rearrangement

### ASJC Scopus subject areas

- Statistics and Probability
- Mathematics(all)
- Agricultural and Biological Sciences (miscellaneous)
- Agricultural and Biological Sciences(all)
- Statistics, Probability and Uncertainty
- Applied Mathematics

### Cite this

*Biometrika*,

*96*(3), 559-575. https://doi.org/10.1093/biomet/asp030