  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.
