The argmax theorem is a useful result for deriving the limiting distribution of estimators in many applications. The conclusion of the argmax theorem states that the argmax of a sequence of stochastic processes converges in distribution to the argmax of a limiting stochastic process. This paper generalizes the argmax theorem to allow the maximization to take place over a sequence of subsets of the domain. If the sequence of subsets converges to a limiting subset, then the conclusion of the argmax theorem continues to hold. We demonstrate the usefulness of this generalization in three applications: estimating a structural break, estimating a parameter on the boundary of the parameter space, and estimating a weakly identified parameter. The generalized argmax theorem simplifies the proofs for existing results and can be used to prove new results in these literatures.
翻译:armax 定理器是一个有用的结果,可以推断出许多应用中测算符的有限分布。 rgmax 定理器的结论指出, 随机过程序列的共性在分布到限制随机过程的趋同器中聚集在一起。 本文概括了 rgmax 定理器, 以便在域的一组子序列中实现最大化。 如果子集的序列聚集到一个限制子集中, 则Argmax 定理器的结论会继续维持下去。 我们用三种应用来证明这种一般化的有用性: 估计一个结构断裂, 估计参数空间边界上的参数, 估计一个薄弱的参数。 通用的argmax 定理器简化了现有结果的证据, 并可用于证明这些文献中的新结果。