Scan Statistic Tail Probability Assessment Based on Process Covariance and Window Size

A scan statistic is examined for the purpose of testing the existence of a global peak in a random process with dependent variables of any distribution. The scan statistic tail probability is obtained based on the covariance of the moving sums process, thereby accounting for the spatial nature of the data as well as the size of the searching window. Exact formulas linking this covariance to the window size and the correlation coefficient are developed under general, common and auto covariance structures of the variables in the original process. The implementation and applicability of the formulas are demonstrated on multiple processes of t-statistics, treating also the case of unknown covariance. A sensitivity analysis provides further insight into the variant interaction of the tail probability with the influence parameters. An R code for the tail probability computation and the data analysis is offered within the supplementary material.