Description: Testing homogeneity of k multivariate distributions is a classical and challenging problem in
statistics, and this becomes even more challenging when the dimension of the data exceeds the sample size.
We construct some tests for this purpose which are exact level (size) alpha tests based on clustering.
These tests are easy to implement and distribution-free in finite sample situations. Under appropriate
regularity conditions, these tests have the consistency property in HDLSS asymptotic regime, where the
dimension of data grows to infinity while the sample size remains fixed. We also consider a multiscale
approach, where the results for different number of partitions are aggregated judiciously. Details are in
Yoav Benjamini and Yosef Hochberg (1995) <doi: 10.2307/2346101>.

Authors@R: c(person("Biplab", "Paul", role = c("aut", "cre"), email = "paul.biplab497@gmail.com"),
Biplab Paul, Shyamal K De and Anil K Ghosh (2021) <doi:10.1016/j.jmva.2021.104897>; Soham Sarkar and Anil K Ghosh (2019)

Title: A Multiple Testing Procedure for High-Dimensional Mediation
Description: A multiple-testing procedure for high-dimensional mediation hypotheses. Mediation analysis is of rising interest in epidemiology and clinical trials. Among existing methods for mediation analyses, the popular joint significance (JS) test yields an overly conservative type I error rate and therefore low power. In the R package 'HDMT' we implement a multiple-testing procedure that accurately controls the family-wise error rate (FWER) and the false discovery rate (FDR) when using JS for testing high-dimensional mediation hypotheses. The core of our procedure is based on estimating the proportions of three component null hypotheses and deriving the corresponding mixture distribution of null p-values. Results of the data examples include better-behaved quantile-quantile plots and improved detection of novel mediation relationships on the role of DNA methylation in genetic regulation of gene expression. With increasing interest in mediation by molecular intermediaries such as gene expression and epigenetic markers, the proposed method addresses an unmet methodological challenge.