The determination of the number of mixture components (the order) of a finite mixture model has been an enduring problem in statistical inference. We prove that the closed testing principle leads to a sequential testing procedure (STP) that allows for confidence statements to be made regarding the order of a finite mixture model. We construct finite sample tests, via data splitting and data swapping, for use in the STP, and we prove that such tests are consistent against fixed alternatives. Simulation studies are conducted to demonstrate the performance of the finite sample tests-based STP, yielding practical recommendations, and extensions to the STP are considered. In particular, we demonstrate that a modification of the STP yields a method that consistently selects the order of a finite mixture model, in the asymptotic sense. Our STP not only applicable for order selection of finite mixture models, but is also useful for making confidence statements regarding any sequence of nested models.

翻译：在统计推断中,确定一定混合物模型的混合物成分数量(顺序)是一个长期问题。我们证明,封闭测试原则导致一个连续测试程序(STP),允许就一定混合物模型的顺序做出信任声明。我们通过数据分离和数据交换,为在STP中使用而建立有限的样本测试,并且我们证明,这种测试与固定替代品是一致的。我们进行了模拟研究,以证明基于有限样品测试的STP的性能,提出了切实可行的建议,并考虑了对STP的扩展。特别是,我们证明,对STP的修改产生了一种方法,从简单意义上始终选择有限混合物模型的顺序。我们的STP不仅适用于限定混合物模型的顺序选择,而且还有助于就任何嵌套模型序列做出信任声明。