The order in which nodes are explored in a (depth-first) iterative deepening search strategy is principally determined by the condition under which a path of the search tree is cut off in each search phase. A corresponding criterion, which has a strong influence on the performance of the overall (heuristic) search procedure, is generally realized in form of an upper cost bound. In this paper, we develop an effective and computationally efficient termination criterion based on statistical methods of change detection. The criterion is local in the sense that it depends on properties of a path itself, rather than on the comparison with other paths. Loosely spoken, the idea is to take a systematic change in the (heuristic) evaluation of nodes along a search path as an indication of suboptimality. An expected utility criterion which also takes the consequence of the suboptimal search decision on the solution quality into account is proposed as a generalization of this idea.