Conference Agenda

We derive the joint asymptotic behavior of the diagonal and off-diagonal elements of projection matrices, whose underlying dimension $m$ is asymptotically proportional to sample size $n$, so that $m/n\to\mu$ as $n\to\infty$, where the aspect ratio $\mu\in (0,1)$. The rate of convergence turns out to be $\sqrt{n}$, and the limiting distribution turns out to be multivariate centered Gaussian with an intriguing pattern in the asymptotic variance-covariance matrix.

Technically, using the celebrated Sherman–Morrison-Woodbury identity for an inverse of rank-k-perturbated matrices, we work out the formulas for quadratic and bilinear forms with respect to such an inverse. On the basis of these results, we figure out the connections between the elements of the projection matrix and their leave-k-analogs. This helps break the dependence between the projection matrix denominator, $Z’Z$, and the individual columns $z_i$ and $z_j$. The asymptotic Gaussianity of the elements of the leave-k projection matrix is then translated to that of the elements of the original projection matrix, using the Delta-method. In turn, the asymptotic Gaussianity of the elements of the leave-k projection matrix is derived from the central limit theorems for quadratic and bilinear forms of IID random elements.

In a multivariate setting, classical test statistics such as the Hotelling's-t-test or the multivariate CUSUM test are usually weighted with the inverse covariance operator. For functional data, i.e., random elements in infinite dimensional Hilbert spaces, one approach for taking the inverse covariance operator into account is based on dimension reduction techniques to a finite dimensional subspace, possibly with principle components, followed by classical multivariate procedures. Such an approach has often been criticized as not being fully functional and losing too much information. As an alternative, tests have been proposed directly based on the functional CUSUM but they fail to include the covariance structure.

In this talk, we propose an alternative that includes the covariance structure with an offset parameter as a middle ground to produce a scale-invariant test procedure and to increase power when the change is not aligned with the first components. Some asymptotic properties are provided under mild assumptions on the dependence structure. A simulation study investigates the behavior of the proposed methods, including detecting abrupt and gradual mean changes.

Drees and Sabourin (2021) examined the PCA projection of the angular part of a multivariate regular varying random vector. In particular, they derived uniform bounds for the risk of the PCA approximation, that is the expected squared norm of the difference between the angular part and its PCA approximation. In this talk we show that under mild conditions the true rate of convergence is much faster than it is suggested by these results. In addition, we establish limit distributions for the PCA projection matrix and the resulting risk in a setting with fixed dimensions. The asymptotic results are used to motivate a data-driven method to select the dimension of the PCA subspace.

We provide necessary and sufficient conditions for the uniqueness of the $k$-means set of a probability distribution. This uniqueness problem is related to the choice of $k$: depending on the underlying distribution, some values of this parameter could lead to multiple sets of $k$-means, which hampers the interpretation of the results and/or the stability of the algorithms. We give a general assessment on consistency of the empirical $k$-means adapted to the setting of non-uniqueness and determine the asymptotic distribution of the within cluster sum of squares. We also provide a statistical characterization of $k$-means uniqueness in terms of the asymptotic Gaussianity of the empirical WCSS. As a consequence, we derive a bootstrap test for uniqueness of the set of $k$-means. The results are illustrated with examples of different types of non-uniqueness that might arise. Finally, we check by simulations the performance of the proposed methodology.