The geometric approximation to the cusum run length distribution

Abstract

It has long been known that at larger run lengths, the distribution of cusum run length, T , converges to the geometric form c λ T . Currently, the only approximation to c is 1 and one approximation is available for λ which goes back to Page (1954). The usefulness of these approximations has never been tested. Two new approximations for c and three for λ are derived. All are compared to numerically-derived values c and λ for a small but diverse set of cusums whose increments are normally-distributed. The resulting approximations to c λ T are finally compared to actual numerically-derived run length percentiles.