# CUSUM ARL - Conditional or Unconditional?

## TL;DR

This paper discusses the importance of using unconditional average run length (ARL) over conditional ARL (CARL) for self-starting CUSUM charts, arguing that CARL's dependence on initial readings is less relevant in practice.

## Contribution

The paper challenges the relevance of CARL for practitioners and advocates for the use of unconditional ARL as the primary performance measure for self-starting CUSUM charts.

## Key findings

- Unconditional ARL is more relevant for practitioners than CARL.
- CARL depends heavily on initial process readings, which may not reflect typical performance.
- The paper questions the practical significance of CARL in quality control applications.

## Abstract

The behavior of CUSUM charts depends strongly on how they are initialized. Recent work has suggested that self-starting CUSUM methods retain some dependence on their very first readings, and introduced the concept of "conditional average run length" (CARL) -- the average run length conditioned on the first few process readings -- as a result of which is it claimed that different practitioners using the same methodology could experience different ARLs because of the random differences in their earliest readings. We cast doubt on whether CARL is relevant to practitioners who use self-starting methods and argue that the unconditional ARL is the relevant measure there.