Multivariate tests based on left-spherically distributed linear scores

TLDR

The presented strategy offers an alternative to least squares and maximum likelihood approaches. The paper proposes a method for multivariate testing using low‑dimensional, data‑dependent linear scores. The method selects score coefficients to achieve a left‑spherical distribution under the null and applies to any left‑spherically distributed data, not just normal. The approach reduces dimensionality, improves stability, remains reliable with redundant variables, and enables well‑known tests in high‑dimensional settings, opening new application areas.

Abstract

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.