A note on Left-Spherically Distributed Test with covariates

In this note we extend the Left-Spherically Distributed linear scores test (LSD) of Lauter, Glimm and Kropf test (1998, The Annals of Statistics). The LSD test is a method for multivariate testing also applicable to the p >> n (much more variables than observations). As a key feature, the score cofficients are chosen such that a left-spherical distribution of the scores is reached under the null hypothesis. Here the test is extended to account for nuisance parameters, particularly for covariates that are assumed to explain (part of) the response variables but are not under test. Moreover it is shown how to deal with the random matrix D which is a Borel function of the residuals under the tested null hypothesis instead of a Borel function of total residuals. This provides a D matrix - and a test as well - which is more focused on the effects of the predictors under test. A R code is available on the someMTP package in CRAN.