Multivariate permutation tests for the \(k\)-sample problem with clustered data (Q1855632)

This article deals with the reduction of data sets to a partition \(B=(B_1,\ldots,B_{m})\). The quality of such a partition is then measured by the \(f\)-information. Let \(f R^{d}\to R\) be a convex function. The \(f\)-information of the partition \(B\) (with respect to the distribution \(P\)) is given by \(I_{f}(B)=\sum\limits_{j=1}^{m}P(B_{j})f(m(B_{j}))\), where \[ m(B_{j})=P(B_j)^{-1}\int\limits_{B_{j}}xP(dx),\quad \text{if} P(B_{j})>0. \] The goal of the clustering method is to maximize \(I_{f}(B)\) under \(|B|\leq m\). A fixpoint algorithm for searching of local optima of the considered problem for an arbitrary fixed convex function \(f\) is presented. To compare the properties and usefulness of the proposed data compression methods, the author uses the \(k\)-sample problem. First the data sets are compressed and then tests for the clustered data are evaluated. The \(F\)-test statistic that measures the inner variance of the clustered data set and a \(\chi^2\)-type statistic are used. To assure that the tests hold the desired significance level the author applies a multivariate permutation test. To treat the \(k\)-sample problem, the author analyzes data sets that consist of \(k\) identically distributed samples. For the number of samples, the sample size and the underling distribution of various choices are considered and simulation results are presented.