Projection tests for high-dimensional spiked covariance matrices
Let \(X_1,X_2,\ldots, X_n\) be independent and identically distributed \(p\)-dimensional random vectors with covariance matrix \(\Sigma=\mathbb{I}_p+\mathbb{V}\Omega \mathbb{V}^{T}\), where \(\mathbb{I}_p\) is the \(p\)-dimensional identity matrix, \(\Omega=\text{diag}(\omega_1,\ldots,\omega_K)\) with \(\omega_1\geq\ldots\geq\omega_K>0\), and \(\mathbb{V}\) is a \(p\times K\) matrix with orthogonal and unit columns. The authors of the paper consider problems related to the testing of the following hypotheses: \[ \begin{aligned} & \Big\{\mathcal{H}_0:\Sigma=\mathbb{I}_p \;\text{vs.}\;\mathcal{H}_1 : \Sigma=\mathbb{I}_p+\mathbb{V}\Omega \mathbb{V}^{T}\Big\},\\ & \Big\{\mathcal{H}_0:\Sigma=\sigma^2\mathbb{I}_p \;\text{vs.}\;\mathcal{H}_1 : \Sigma=\sigma^2(\mathbb{I}_p+\mathbb{V}\Omega \mathbb{V}^{T})\Big\}, \end{aligned} \] where \(\sigma^2\) is an unknown but finite positive constant. New tests based on a projection approach are developed for the high-dimensional case. The asymptotic distribution of the proposed tests are obtained under suitable regularity conditions.
scientific article; zbMATH DE number 6997742
scientific article