Detecting Spectral Breaks in Spiked Covariance Models

In this paper, the key objects of interest are the sequential covariance matrices $\mathbf{S}_{n,t}$ and their largest eigenvalues. Here, the matrix $\mathbf{S}_{n,t}$ is computed as the empirical covariance associated with observations $\{\mathbf{x}_1,\ldots,\mathbf{x}_{ \lfloor nt \rfloor } \}$, for $t\in [0,1]$. The observations $\mathbf{x}_1,\ldots,\mathbf{x}_n$ are assumed to be i.i.d. $p$-dimensional vectors with zero mean, and a covariance matrix that is a fixed-rank perturbation of the identity matrix. Treating $\{ \mathbf{S}_{n,t}\}_{t \in [0,1]}$ as a matrix-valued stochastic process indexed by $t$, we study the behavior of the largest eigenvalues of $\mathbf{S}_{n,t}$, as $t$ varies, with $n$ and $p$ increasing simultaneously, so that $p/n \to y \in (0,1)$. As a key contribution of this work, we establish the weak convergence of the stochastic process corresponding to the sample spiked eigenvalues, if their population counterparts exceed the critical phase-transition threshold. Our analysis of the limiting process is fully comprehensive revealing, in general, non-Gaussian limiting processes. As an application, we consider a class of change-point problems, where the interest is in detecting structural breaks in the covariance caused by a change in magnitude of the spiked eigenvalues. For this purpose, we propose two different maximal statistics corresponding to centered spiked eigenvalues of the sequential covariances. We show the existence of limiting null distributions for these statistics, and prove consistency of the test under fixed alternatives. Moreover, we compare the behavior of the proposed tests through a simulation study.