On the detection of changes in the variance in high-dimensional $K$-sample situations

• Über die Erkennung von Strukturbrüchen in der Varianz in Hochdimensionalen $K$ - Stichproben Situationen

Mause, Nils; Steland, Ansgar (Thesis advisor); von Sachs, Rainer (Thesis advisor)

Aachen : RWTH Aachen University (2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

In this thesis we suppose that at time $T>0$ we observe $K \in \mathbb N$ independent samples of $d_T \in \mathbb N$ - dimensional vector time series $Y_{T,1},\dots,Y_{T,K}$ with (possibly different) lengths $N_j \in \mathbb N, j=1,\dots,K$. While having applications to a sensor monitoring problem in mind, we suppose that the coordinates of the high-dimensional random vectors $Y_{T,j,i},i=1,\dots,N_j,$ follow a causal linear process structure. Projections $w_T'Y_{T,j,i}$ based on such high-dimensional time series and some appropriate weighting vector $w_T \in \mathbb R^{d_T}$ appear naturally in many statistical procedures and applications, such as the Principal Component Analysis and Portfolio Optimization, and are a common method of dealing with high-dimensional data sets. Now, when one is interested in drawing inference on the second moment structure of these projections, quadratic forms $w_T' \Sigma^{(j)}_{T} w_T$, and, more generally, bilinear forms $v_T' \Sigma^{(j)}_{T} w_T$ as well as their nonparametric estimator $v_T' \hat{\Sigma}^{(j)}_{T} w_T$ for $v_T,w_T \in \mathbb R^{d_T}$ need to be studied, where $\Sigma^{(j)}_{T}$ is the $j$th variance-covariance matrix and $\hat{\Sigma}_T^{(j)}$ is the $j$th sample variance-covariance matrix. In order to detect structural breaks (change-points) in the variance of these projections, which may only affect a small subset of all of the $K$ samples, we developed new test statistics which are either based on a sum of squared errors approach or based on a bilinear form of the pooled sample variance-covariance matrix. Within the high-dimensional framework where not only the time horizon $T$ (or the sample sizes $N_j,j=1,\dots,K$) shall go to infinity but also the dimension $d_T$ of the data is allowed to grow with the time horizon we establish new weak approximation results for functions of these bilinear forms in terms of functions of Brownian motions. These approximation results do not depend on any constraints on the ratio of dimension and sample size and thus also hold for situations where the dimension $d_T$ grows much faster than the sample sizes $N_j$ or the time horizon $T$. While these approximations take place on a richer probability space, we also show how they can be returned onto the original probability space, as well as that these approximations still hold when replacing the asymptotic variance parameters of the approximating Brownian motions with weakly consistent Bartlett - type estimators. These statistics are then empirically evaluated and analyzed through simulations and applications on a real data set. In particular, the empirical evaluations show that both approaches can be advantageous, depending on which specific situation one is analyzing (early change, late change or a change in the middle of the time interval, change in the standard deviations of innovations or a change in the coefficients)Moreover, the weak approximation results also allow us to test for the homogeneity of covariance matrices in a $K$ - sample situations, as well as analyze the case when one is no longer dealing with a single bilinear form, but an increasing and possibly infinite amount of bilinear forms based on the pooled sample variance-covariance matrix. For each of these topics the asymptotic behaviour is studied and the finite sample performance is illustrated through a simulation study.