In the paper, we derive two types of the multivariate CUSUM control charts for stationary and variance nonstationary Gaussian processes based on maximizing the generalized likelihood ratio. As partial cases, the control schemes for independent observations and autoregressive processes are obtained. The second application leads to the two multivariate CUSUM control charts when the covariance matrix is separable. We show that the proposed schemes possess the invariance property. The performance of the control charts is studied within a simulation study.

In the chapter we consider the optimal portfolio choice problem under parameter uncertainty when the covariance matrix of asset returns is singular. Very useful stochastic representations are deduced for the characteristics of the expected utility optimal portfolio. Using these stochastic representations, we derive the moments of higher order of the estimated expected return and the estimated variance of the expected utility optimal portfolio. Another line of applications leads to their asymptotic distributions obtained in the high-dimensional setting. Via a simulation study, it is shown that the derived high-dimensional asymptotic distributions provide good approximations of the exact ones even for moderate sample sizes.

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).