An Empirical Approximation to the Null Unbounded Steady-State Distribution of the Cumulative Sum Statistic

We develop an empirical approximation to the null steady-state distribution of the cumulative sum (cusum) statistic, defined as the distribution of values obtained by running a cusum with no threshold under the null state for an indefinite period. The derivation is part theoretical and part empirical, and the approximation is valid for cusum's applied to normal data with known variance (although the theoretical result is true in general for exponential family data). The result leads to a formula for steady-state p values corresponding to cusum values, where the steady-state p value is obtained as the tail area of the null steady-state distribution and represents the expected proportion of time, under repeated application of the cusum to null data, that the cusum statistic is greater than some particular value. When designing individual charts with fixed boundaries, this measure could be used alongside the average run length (ARL) value, which, as we show by way of examples may be approximately related to the steady-state p value. For multiple cusum schemes, using a p value enables application of a signaling procedure that adopts a false discovery rate approach to multiplicity control. Under this signaling procedure, boundaries on each individual chart change at every observation according to the ordering of cusum values across the group of charts. We demonstrate an application of the measure to a single chart in which the example data are number of earthquakes per year registering >7 on the Richter scale recorded between 1900 and 1998. Simulation results relating to the empirical approximation of the null steady-state distribution are summarized, and those relating to the statistical properties of the proposed signaling procedure for multiple cusum schemes are presented.