Central Limit Theorems for Linear Spectral Statistics on Large Regularized Covariance Matrices

In this thesis we shall consider sample covariance matrices Sn in the case when the dimension of the data increases with the sample size to infinity ,while the ratio approaches a fixed constant. We will derive a new statistic based on the general linear shrinkage estimator by Bodnar et al. (2014)[1] We will show that the new statistic is normally distributed under the null hypothesis that the true covariance matrix is the identity, where we assume the existence of the fourth moment of our data.Furthermore, we will do simulation study that compares our new statistic to tests from finite dimensional statistics that have been altered to work in high dimensional statistics by Wang and Yao [3]. We will look at three different hypothesis, the equicorrelation case, the auto-regressive case and a fixed ratio case. After that, we will look at the non-linear shrinkage estimator based on the work by Ledoit and Peche (2011) [11], and show that, under the null hypothesis, constructing a test is not directly possible like it is in the linear case.Applied Mathematic