In this paper, a method for multivariate testing based on
low-dimensional, data-dependent, linear scores is proposed. The new approach
reduces the dimensionality of observations and increases the stability of the
solutions. The method is reliable, even if there are many redundant variables.
As a key feature, the score coefficients are chosen such that a left-spherical
distribution of the scores is reached under the null hypothesis. Therefore,
well-known tests become applicable in high-dimensional situations, too. The
presented strategy is an alternative to least squares and maximum likelihood
approaches. In a natural way, standard problems of multivariate analysis thus
induce the occurrence of left-spherical, nonnormal distributions. Hence, new
fields of application are opened up to the generalized multivariate analysis.
The proposed methodology is not restricted to normally distributed data, but
can also be extended to any left-spherically distributed observations.