## 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.