Approximating the Distribution of the Median and other Robust Estimators on Uncertain Data

TL;DR

This paper develops methods to approximate the distribution of robust estimators like the median for uncertain data points with discrete distributions, enabling efficient analysis despite data uncertainty.

Contribution

It introduces algorithms to construct and estimate the distribution of robust estimators for uncertain data, including a general approximation technique for high-dimensional estimators.

Findings

Near-linear time for support construction

Quadratic time for probability assignment

Applicable to high-dimensional median and regression estimators

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.