This paper is concerned with deriving a new test on a covariance matrix which is based on its nonlinear shrinkage estimator. The distribution of the test statistic is deduced under the null hypothesis in the large-dimensional setting, that is, when p/n tend to some positive constant c with p variables and n samples both tending to infinity. The theoretical results are illustrated by means of an extensive simulation study where the new nonlinear shrinkage-based test is compared with existing approaches, in particular with the commonly used corrected likelihood ratio test, the corrected John test, and the test based on the linear shrinkage approach. It is demonstrated that the new nonlinear shrinkage test possesses better power properties under heteroscedastic alternative.

The chapter is concerned with finding the asymptotic distribution of the estimated shrinkage intensity used in the definition of the linear shrinkage estimator of the covariance matrix, derived by Bodnar et al. (J Multivar Anal 132:215–228, 2014). As a result, a new test statistic is proposed which is deduced from the linear shrinkage estimator. This result is a ready-to-use multivariate hypothesis test in the large-dimensional asymptotic framework and constitutes the main result of the chapter. The theoretical findings are compared by means of a simulation study with existing tests, in particular with the commonly used corrected likelihood ratio test and the corrected John test, both derived by Wang and Yao (Electron J Stat 7:2164–2192, 2013).