Scan Statistics for Detecting a Local Change in Mean for Normal Data

In this article, we review the approximations and inequalities that have been derived in the scientific literature for fixed-, multiple-, and variable-window-length scan statistics, for detecting a local change in the population mean, for one-dimensional normal data. We assume that the variance of the underlying distribution is known and remains unchanged. Monitoring processes based on a fixed-window scan statistic via fixed and sequential sampling schemes are discussed as well. In the context of sequential sampling schemes for the monitoring process, we discuss a repeated significance test and evaluate its properties. The implementation of two multiple-window-length scan statistics are based on the minimum p-value statistic and the generalized likelihood ratio test statistic, respectively. The implementation of the variable-window scan statistic is based on the generalized likelihood ratio test statistic. Simulation algorithms and numerical results are presented to evaluate the performance of the multiple and variable-window-type scan statistics and compare them with fixed-window scan statistics.