Algorithm 1034: An Accelerated Algorithm to Compute the Q n Robust Statistic, with Corrections to Constants

Abstract

The robust scale estimator Q n developed by Croux and Rousseeuw [ 3 ], for the computation of which they provided a deterministic algorithm, has proven to be very useful in several domains including in quality management and time series analysis. It has interesting mathematical (50% breakdown, 82% Asymptotic Relative Efficiency) and computing ( O(nlogn) time, O ( n ) space) properties. While working on a faster algorithm to compute Q n , we have discovered an error in the computation of the d constant, and as a consequence in the d n constants that are used to scale the statistic for consistency with the variance of a normal sample. These errors have been reproduced in several articles including in the International Standard Organisation 13,528 [ 12 ] document. In this article, we fix the errors and present a new approach, which includes a new algorithm, allowing computations to run 1.3 to 4.5 times faster when n grows from 10 to 100,000.