The Asymptotics of Wide Remedians

TL;DR

This paper analyzes the asymptotic behavior of the remedian, a streaming median approximation method, deriving its distribution, efficiency, and extensions for multiple quantiles as parameters grow large.

Contribution

It provides a detailed asymptotic analysis of the remedian's distribution, efficiency, and robustness, including extensions to multivariate quantile estimation.

Findings

Remedian's distribution approaches normal as buffer size increases.

Asymptotic efficiency of the remedian relative to mean and median is derived.

Proposes distribution for multivariate quantile estimation using remedian in parallel.

Abstract

The remedian uses a $k\times b$ matrix to approximate the median of $n\leq b^{k}$ streaming input values by recursively replacing buffers of $b$ values with their medians, thereby ignoring its $200(\lceil b/2\rceil / b)^{k}%$ most extreme inputs. Rousseeuw & Bassett (1990) and Chao & Lin (1993); Chen & Chen (2005) study the remedian's distribution as $k\rightarrow\infty$ and as $k,b\rightarrow\infty$. The remedian's breakdown point vanishes as $k\rightarrow\infty$, but approaches $(1/2)^{k}$ as $b\rightarrow\infty$. We study the remedian's robust-regime distribution as $b\rightarrow\infty$, deriving a normal distribution for standardized (mean, median, remedian, remedian rank) as $b\rightarrow\infty$, thereby illuminating the remedian's accuracy in approximating the sample median. We derive the asymptotic efficiency of the remedian relative to the mean and the median. Finally, we discuss the estimation of more than one quantile at once, proposing an asymptotic distribution for the random vector that results when we apply remedian estimation in parallel to the components of i.i.d. random vectors.