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A theorem on the principal components inference

A method of multivariate data compression and dimension reduction is established, which is based on principal components and avoids all overfitting effects. This method allows the use of ‘compressed’ data for exact level-alpha tests of hypotheses on the mean vectors. It is a particularity of the method that the coefficients of the constructed linear scores depend solely on the residual sums of products matrix; the empirical means are not necessary to determine the compression. Thus, novel and very simple confidence regions of the unknown multivariate mean vectors are also obtained. The method can be combined with strategies of selecting variables. Furthermore, multiple testing procedures are derived, which serve for finding all sets of variables with deviations from the null hypothesis. The methods are evaluated by computer simulations.