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Testing the equality of two means is a fundamental inference problem. For high-dimensional data, the Hotelling's T2-test either performs poorly or becomes inapplicable. Several modifications have been proposed to address this issue. However, most of them are based on asymptotic normality of the null distributions of their test statistics which inevitably requires strong assumptions on the covariance. We study this problem thoroughly and propose an L2-norm based test that works under mild conditions and even when there are fewer observations than the dimension. Specially, to cope with general nonnormality of the null distribution we employ the Welch–Satterthwaite χ2-approximation. We derive a sharp upper bound on the approximation error and use it to justify that χ2-approximation is preferred to normal approximation. Simple ratio-consistent estimators for the parameters in the χ2-approximation are given. Importantly, our test can cope with singularity or near singularity of the covariance which is commonly seen in high dimensions and is the main cause of nonnormality. The power of the proposed test is also investigated. Extensive simulation studies and an application show that our test is at least comparable to and often outperforms several competitors in terms of size control, and the powers are comparable when their sizes are. for this article are available online. [ABSTRACT FROM AUTHOR]

In this article, we propose two new tests for the equality of the covariance functions of several functional populations, namely, a quasi-GPF test and a quasi-Fmax test whose test statistics are obtained via globalizing a pointwise quasi-F-test statistic with integration and taking its supremum over some time interval of interest, respectively. Unlike several existing tests, they are scale-invariant in the sense that their test statistics will not change if we multiply each of the observed functions by any nonzero function of time. We derive the asymptotic random expressions of the two tests under the null hypothesis and show that under some mild conditions, the asymptotic null distribution of the quasi-GPF test is a chi-squared-type mixture whose distribution can be well approximated by a simple-scaled chi-squared distribution. We also propose a random permutation method for approximating the null distributions of the quasi-GPF and Fmax tests. The asymptotic distributions of the two tests under a local alternative are also investigated and the two tests are shown to be root-n consistent. A theoretical power comparison between the quasi-GPF test and the L2-norm-based test proposed in the literature is also given. Simulation studies are presented to demonstrate the finite-sample performance of the new tests against five existing tests. An illustrative example is also presented. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]