A continuous mapping theorem for the smallest argmax functional

This paper introduces a version of the argmax continuous mapping theorem that applies to M-estimation problems in which the objective functions converge to a limiting process with multiple maximizers. The concept of the smallest maximizer of a function in the d-dimensional Skorohod space is introduced and its main properties are studied. The resulting continuous mapping theorem is applied to three problems arising in change point regression analysis. Some of the results proved in connection to the d-dimensional Skorohod space are also of independent interest.