We introduce the so-called naive tests and give a brief review of the new developments. Naive testing methods are easy to understand and perform robustly, especially when the dimension is large. We focus mainly on reviewing some naive testing methods for the mean vectors and covariance matrices of high-dimensional populations, and we believe that this naive testing approach can be used widely in many other testing problems.
Some other related work on tests of high-dimensional locations can be found in @cite_12 @cite_47 @cite_33 @cite_43 @cite_52 @cite_37 @cite_16 , which we do not discuss at length in this paper.
@cite_37 @cite_33 @cite_52 @cite_43 @cite_47 @cite_16 @cite_12
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This article is concerned with simultaneous tests on linear regression coefficients in high-dimensional settings. When the dimensionality is larger than the sample size, the classic @math -test is not applicable since the sample covariance matrix is not invertible. In order to overcome this issue, both Goeman, Finos and van Houwelingen (2011) and Zhong and Chen (2011) proposed their test procedures after excluding the @math term in @math -statistics. However, both these two test are not invariant under the group of scalar transformations. In order to treat those variables in a fair' way, we proposed a new test statistic and establish its asymptotically normal under certain mild conditions. Simulation studies showed that our test procedure performs very well in many cases. Summary The structural information in high-dimensional transposable data allows us to write the data recorded for each subject in a matrix such that both the rows and the columns correspond to variables of interest. One important problem is to test the null hypothesis that the mean matrix has a particular structure without ignoring the dependence structure among and or between the row and column variables. To address this, we develop a generic and computationally inexpensive nonparametric testing procedure to assess the hypothesis that, in each predefined subset of columns (rows), the column (row) mean vector remains constant. In simulation studies, the proposed testing procedure seems to have good performance and, unlike simple practical approaches, it preserves the nominal size and remains powerful even if the row and or column variables are not independent. Finally, we illustrate the use of the proposed methodology via two empirical examples from gene expression microarrays. In this article, we propose new multivariate two-sample tests based on nearest neighbor type coincidences. While several existing tests for the multivariate two-sample problem perform poorly for high dimensional data, and many of them are not applicable when the dimension exceeds the sample size, these proposed tests can be conveniently used in the high dimension low sample size (HDLSS) situations. Unlike Schilling (1986) [26] and Henze’s (1988) test based on nearest neighbors, under fairly general conditions, these new tests are found to be consistent in HDLSS asymptotic regime, where the sample size remains fixed and the dimension grows to infinity. Several high dimensional simulated and real data sets are analyzed to compare their empirical performance with some popular two-sample tests available in the literature. We further investigate the behavior of these proposed tests in classical asymptotic regime, where the dimension of the data remains fixed and the sample size tends to infinity. In such cases, they turn out to be asymptotically distribution-free and consistent under general alternatives. The Wilcoxon–Mann–Whitney test is a robust competitor of the @math test in the univariate setting. For finite-dimensional multivariate non-Gaussian data, several extensions of the Wilcoxon–Mann–Whitney test have been shown to outperform Hotelling's @math test. In this paper, we study a Wilcoxon–Mann–Whitney-type test based on spatial ranks in infinite-dimensional spaces, we investigate its asymptotic properties and compare it with several existing tests. The proposed test is shown to be robust with respect to outliers and to have better power than some competitors for certain distributions with heavy tails. We study its performance using real and simulated data. The multivariate two-sample testing problem has been well investigated in the literature, and several parametric and nonparametric methods are available for it. However, most of these two-sample tests perform poorly for high dimensional data, and many of them are not applicable when the dimension of the data exceeds the sample size. In this article, we propose a multivariate two-sample test that can be conveniently used in the high dimension low sample size setup. Asymptotic results on the power properties of our proposed test are derived when the sample size remains fixed, and the dimension of the data grows to infinity. We investigate the performance of this test on several high-dimensional simulated and real data sets, and demonstrate its superiority over several other existing two-sample tests. We also study some theoretical properties of the proposed test for situations when the dimension of the data remains fixed and the sample size tends to infinity. In such cases, it turns out to be asymptotically distribution-free and consistent under general alternatives. We propose a new scalar and shift transform invariant test statistic for the high-dimensional two-sample location problem. Theoretical results and simulation studies show the good performance of our test under certain circumstances. We discuss a one-sample location test that can be used in the case of highdimensional data. For high-dimensional data, the power of Hotelling’s test decreases when the dimension is close to the sample size. To address this loss of power, some non-exact approaches were proposed, e.g., Dempster (1958, 1960), Bai and Saranadasa (1996) and Srivastava and Du (2006). In this paper, we focus on Hotelling’s test and Dempster’s test. The comparative merits and demerits of these two tests vary according to the local parameters. In particular, we consider the situation where it is difficult to determine which test should be used, that is, where the two tests are asymptotically equivalent in terms of local power. We propose a new statistic based on the weighted averaging of Hotelling’s T 2 statistic and Dempster’s statistic that can be applied in such a situation. Our weight is determined on the basis of the maximum local asymptotic power on a restricted parameter space that induces local asymptotic equivalence between Hotelling’s test and Dempster’s test. In addition, some good asymptotic properties with respect to the local power are shown. Numerical results show that our test is more stable than Hotelling’s T 2 statistic and Dempster’s statistic in most parameter settings.