On the Use of the SPRT in Determining the Properties of Some CUSUM Procedures

Let Y 1, Y 1,…, Y ν−1 be iid random variables with a distribution function (df) F(y, θ0) = F 0(y), and let Y ν, Y ν+1,… be iid random variables with a df F(y, θ1) = F 1(y), where ν (ν ≥ 1) is an unknown time index of a change in distribution caused by the change in parametric value. For a suitable function ψ let X j = ψ(Y j ) (j ≥ 1) denote some convenient data reduction or it is defined by certain optimality consideration such as the SPRT. Let W 0 = 0, W n = max(0, W n−1 + X n ), S 0 = 0, , n ≥ 1. For detecting a change in distribution, Page [Page, E.S. Continuous inspection schemes. Biometrika 1954, 41, 100–115] defined his famous CUSUM (cumulative sum) procedure by N = inf{n ≥ 1 : W n ≥ h}, h > 0, and noted that N is equivalent to a series of SPRT of the form M = inf{n ≥ 1 : S n ≤ 0orS n ≥ h} until there is exit at the upper boundary h. Moreover, from the inherent renewal property he noted that EN = EM/P(S M ≥ h), which is quite useful in approximating and/or evaluating EN. However, a deeper connection between N and M was noted by Khan [Khan, R.A. On cumulative sum procedures and the SPRT with applications. Journal of the Royal Statistical Society B 1984, 46, 79–85]. The purpose of this exposition paper is to further exploit such a relationship to study the properties of some more CUSUMs with several applicable examples. Exact results are attempted whenever possible otherwise suitable approximations are outlined.