Wald's approximations to the average run length in cusum procedures

Wald's approximation to the ARL(average run length in cusum) (cumulative sum) procedures are given for an exponential family of densities. From these approximations it is shown that Page's (1954) cusum procedure is (in a sense) identical with a cusum procedure defined in terms of likelihood ratios. Moreover, these approximations are improved by estimating the excess over the boundary and their closeness is examined by numerical comparisons with some exact results. Some examples are also given.