On SPRT and CUSUM Procedures and Some Open Problems

Abstract : Let $Y_1, Y_1, \cdots , Y_{\nu -1}$ be iid random variables with a distribution function (df) $F_0(y)$, and let $Y_{\nu}, Y_{\nu+1}, \cdots$ be iid random variables with a df $F_1(y)$, where $\nu$ is an unknown time index of a change in distribution with the change in parametric value. For a suitable function $\psi$ let $X_j = \psi(Y_j)$ denote some convenient data reduction or it may be defined by certain optimality consideration such as the Sequential Probability Ratio Test. For detecting a change-point in the distribution, Page (1954) defined his famous cusum (cumulative sum) procedure. From the inherent renewal property of the cusum, Page noted that $EN = EM/P(S_M \geq h)$, where $N$ is the cusum stopping rule and $M$ is the SPRT and $S_n$ represents the partial sum of the information $X_1, X_2,...$. The constant $h$ is the trigering constant for the cusum. This link between the SPRT and the cusum is quite useful in approximating and/or evaluating $EN$. However, a deeper connection between $N$ and $M$ is known that I will try to present. The purpose of this exposition is to further exploit such a relationship between $N$ and $M$ to study the properties of some one sided and two sided cusums with several applicable examples. We will see how exact results can be computed in discrete time settings. There are some open problems which I will outline.