Cusum Procedures for Detecting Changes in the Tail Probability of a Normal Distribution

Let x ₁, x ₂, … be independent random variables having normal distribution with unknown parameters, and consider the problem of detecting changes in the tail probability p = P ( x 1 > x 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203493205/efe52689-d7cd-469d-aa36-8f9c2548773d/content/eq1809.tif"/> , where x ₀ is a known constant. To monitor changes in p we consider some cusum procedures including a Shewhart control chart and its generalization based on a suitable integer-valued random walk. Both of these procedures are based on noncentral t-statistics under group sampling although a special case of the generalized version is also applicable without group sampling. Some cusum procedures based on noncentral t-statistics under group sampling are also considered, and the average run lengths of these procedures are evaluated and/or estimated by simulations to compare their relative merits.