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).