# Signed Sequential Rank CUSUMs

## TL;DR

This paper introduces distribution-free, self-starting CUSUM procedures based on signed sequential ranks for detecting shifts in symmetric distributions, effective for various distribution types including normal and heavy-tailed.

## Contribution

It develops new signed sequential rank CUSUMs that are distribution free and do not require parametric assumptions, suitable for detecting location and scale changes.

## Key findings

- Distribution-free and self-starting CUSUMs for symmetric distributions.
- Van der Waerden-based CUSUM performs well for normal distributions.
- Wilcoxon-based CUSUM is effective for heavy-tailed distributions.

## Abstract

CUSUMs based on the signed sequential ranks of observations are developed for detecting location and scale changes in symmetric distributions. The CUSUMs are distribution free and fully self-starting: given a specified in-control median and nominal in-control average run length, no parametric specification of the underlying distribution is required in order to find the correct control limits. If the underlying distribution is normal with unknown variance, a CUSUM based on the Van der Waerden signed rank score produces out-of-control average run lengths that are commensurate with those produced by the standard CUSUM for a normal distribution with known variance. For heavier tailed distributions, use of a CUSUM based on the Wilcoxon signed rank score is indicated. The methodology is illustrated by application to real data from an industrial environment.