### Abstract

Robust estimators, like the median of a point set, are important for data analysis in the presence of outliers. We study robust estimators for locationally uncertain points with discrete distributions. That is, each point in a data set has a discrete probability distribution describing its location. The probabilistic nature of uncertain data makes it challenging to compute such estimators, since the true value of the estimator is now described by a distribution rather than a single point. We show how to construct and estimate the distribution of the median of a point set. Building the approximate support of the distribution takes near-linear time, and assigning probability to that support takes quadratic time. We also develop a general approximation technique for distributions of robust estimators with respect to ranges with bounded VC dimension. This includes the geometric median for high dimensions and the Siegel estimator for linear regression.

Original language | English |
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Title of host publication | 34th International Symposium on Computational Geometry, SoCG 2018 |

Editors | Csaba D. Toth, Bettina Speckmann |

Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Number of pages | 14 |

ISBN (Electronic) | 9783959770668 |

DOIs | |

Publication status | Published - 1 Jun 2018 |

Event | 34th International Symposium on Computational Geometry, SoCG 2018 - Budapest, Hungary Duration: 11 Jun 2018 → 14 Jun 2018 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 99 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 34th International Symposium on Computational Geometry, SoCG 2018 |
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Country | Hungary |

City | Budapest |

Period | 11/06/18 → 14/06/18 |

### Keywords

- Uncertain Data
- Robust Estimators
- Geometric Median
- Tukey Median
- Robust estimators
- Geometric median
- Tukey median
- Uncertain data

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## Cite this

Buchin, K. A., Phillips, J. M., & Tang, P. (2018). Approximating the distribution of the median and other robust estimators on uncertain data. In C. D. Toth, & B. Speckmann (Eds.),

*34th International Symposium on Computational Geometry, SoCG 2018*[16] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 99). Schloss Dagstuhl - Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SoCG.2018.16