Uniform-over-dimension convergence with application to location tests for high-dimensional data
arxiv(2024)
摘要
Asymptotic methods for hypothesis testing in high-dimensional data usually
require the dimension of the observations to increase to infinity, often with
an additional condition on its rate of increase compared to the sample size. On
the other hand, multivariate asymptotic methods are valid for fixed dimension
only, and their practical implementations in hypothesis testing methodology
typically require the sample size to be large compared to the dimension for
yielding desirable results. However, in practical scenarios, it is usually not
possible to determine whether the dimension of the data at hand conform to the
conditions required for the validity of the high-dimensional asymptotic
methods, or whether the sample size is large enough compared to the dimension
of the data. In this work, a theory of asymptotic convergence is proposed,
which holds uniformly over the dimension of the random vectors. This theory
attempts to unify the asymptotic results for fixed-dimensional multivariate
data and high-dimensional data, and accounts for the effect of the dimension of
the data on the performance of the hypothesis testing procedures. The
methodology developed based on this asymptotic theory can be applied to data of
any dimension. An application of this theory is demonstrated in the two-sample
test for the equality of locations. The test statistic proposed is unscaled by
the sample covariance, similar to usual tests for high-dimensional data. Using
simulated examples, it is demonstrated that the proposed test exhibits better
performance compared to several popular tests in the literature for
high-dimensional data. Further, it is demonstrated in simulated models that the
proposed unscaled test performs better than the usual scaled two-sample tests
for multivariate data, including the Hotelling's T^2 test for multivariate
Gaussian data.
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