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.
The Random Projection was first proposed by @cite_40 and was further discussed in later studies @cite_39 @cite_34 @cite_18 @cite_41 . For Gaussian data, the procedure projects high-dimensional data onto random subspaces of relatively low-dimensional spaces to allow the traditional Hotelling @math statistic to work well. This method can be viewed as a two-step procedure. First, a single random projection is drawn, and it is then used to map the samples from the high-dimensional space to a low-dimensional space. Second, the Hotelling @math test is applied to a new hypothesis-testing problem in the projected space. A decision is then returned to the original problem by simply rejecting @math whenever the Hotelling test rejects it in the projected spaces.
@cite_18 @cite_41 @cite_39 @cite_40 @cite_34
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A common problem in modern genetic research is that of comparing the mean vectors of two populations-typically in settings in which the data dimension is larger than the sample size-where Hotelling's test cannot be applied.Recently, a test using random subspaces was proposed, in which the data are randomly projected into several lower-dimensional subspaces, and Hotelling's test is well defined. Superior performance with competing tests was demonstrated when the variables were correlated.Following the research of random subspaces, a modified test was proposed that might make more efficient use of covariance structure at high dimension. Hierarchical clustering is performed first such that highly correlated variables are clustered together. Next, Hotelling's statistics are computed for every cluster-subspace and summed as the new test statistic. High performance was demonstrated via simulations and real data analysis. A two-sample test using hierarchical clustering was proposed.Hotelling's statistics are computed in cluster-subspaces and summed as the statistic.Highly correlated variables take priority for being processed.A cutoff distance is used to restrain the effect of statistical fluctuations.High performance was demonstrated in simulations and real data analysis. A common problem in genetics is that of testing whether a set of highly dependent gene expressions differ between two populations, typically in a high-dimensional setting where the data dimension is larger than the sample size. Most high-dimensional tests for the equality of two mean vectors rely on naive diagonal or trace estimators of the covariance matrix, ignoring dependences between variables. A test using random subspaces is proposed, which offers higher power when the variables are dependent and is invariant under linear transformations of the marginal distributions. The p-values for the test are obtained using permutations. The test does not rely on assumptions about normality or the structure of the covariance matrix. It is shown by simulation that the new test has higher power than competing tests in realistic settings motivated by microarray gene expression data. Computational aspects of high-dimensional permutation tests are also discussed and an efficient R implementation of the proposed test is provided. We consider the hypothesis testing problem of detecting a shift between the means of two multivariate normal distributions in the high-dimensional setting, allowing for the data dimension p to exceed the sample size n. Specifically, we propose a new test statistic for the two-sample test of means that integrates a random projection with the classical Hotelling T^2 statistic. Working under a high-dimensional framework with (p,n) tending to infinity, we first derive an asymptotic power function for our test, and then provide sufficient conditions for it to achieve greater power than other state-of-the-art tests. Using ROC curves generated from synthetic data, we demonstrate superior performance against competing tests in the parameter regimes anticipated by our theoretical results. Lastly, we illustrate an advantage of our procedure's false positive rate with comparisons on high-dimensional gene expression data involving the discrimination of different types of cancer. Motivated by the prevalence of high dimensional low sample size datasets in modern statistical applications, we propose a general nonparametric framework, Direction-Projection-Permutation (DiProPerm), for testing high dimensional hypotheses. The method is aimed at rigorous testing of whether lower dimensional visual differences are statistically significant. Theoretical analysis under the non-classical asymptotic regime of dimension going to infinity for fixed sample size reveals that certain natural variations of DiProPerm can have very different behaviors. An empirical power study both confirms the theoretical results and suggests DiProPerm is a powerful test in many settings. Finally DiProPerm is applied to a high dimensional gene expression dataset.