Optimal sequentially planned change-point detection procedures

Aiming at practical applications, the idea of sequential sampling has been popular since its first introduction by Abraham Wald in 1940s. However, the limitations of classical sequential procedures , which take only one observation at a time, have been criticized as expensive or impractical. The main reason is that standard linear cost functions include only the cost of each observation. Many applied problems take more factors into account, such as the time and risk, cost of equipment, cost of personnel, group insurance in Phase III clinical trials, and other fixed costs, in addition to variable costs required to conduct each sampling stage. These needs lead to a generalized technique, sequentially planned statistical procedures. It allows to sample in groups of variable sizes, determined sequentially, based on the data collected so far.;The main goal of this research is focused on the constructive theory and practice of optimal sequential planning for change-point detection .;The change-point detection problem, sequential in its nature, is described and formulated. We propose to apply sequentially planned probability ratio tests to change-point detection in order to achieve optimality under the suitable cost functions and risk functions. The goal is to detect the change as soon as possible while keeping an eye on the rate of false alarms. Reflecting this, a suitable cost function increases significantly after the change-point. That is, a practitioner faces high costs for each day after the occurrence of a change point, when the process runs out of control, until this change-point is detected and the process is terminated or re-calibrated.;Optimal sequential plans are designed accordingly. Two types of optimization problems are considered, the variational problem and the Bayesian problem. They reflect different types of the optimization criteria: minimization of the expected detection delay cost under the fixed rate of false alarms and minimization of the total unrestricted Bayes risk.;Constructive solutions to both problems are obtained. Numerical algorithms for risk evaluation and risk comparison are proposed. Solutions of the variational problem result in the cumulative sum process while Bayesian sequential plans are based on the posterior distribution of the change-point.