OPTIMAL CHANGE-POINT DETECTION IN TREND MODELS WITH INTEGRATED MOVING AVERAGE ERRORS

ABSTRACT

Let {X_n, n ε N} be generated by a polynomial trend model with integrated moving average errors. It is assumed that the intercept of the model is μ up to time instant ν and it is μ + δ at ν and remains constant thereafter, where ν is an N-valued random variable (see Eq. (1.11) below). All parameters of the model are assumed to be known. Suppose {X_n } is observed sequentially and the objective is to declare the occurrence of the change quickly. It is shown in this paper that the sequential change-point detection rule N, which is optimal in the sense of minimising E(N −ν)⁺ in the class of rules, whose false-alarm probability is bounded by a preassigned value, is the Bayes rule of Shiryaev.

Keywords:

• Change-point problem
• Sequential detection rule
• Optimal stopping
• Shiryaev's rule
• Neutral shift in trend
• Integrated moving average process

ACKNOWLEDGMENT

The second author wishes to thank Bangalore University for facilitating a visit during his sabbatical leave from the University of Madras during which period this work was completed.