Sensitivity of eigenvectors and eigenvalues of symmetric matrix estimates to the removal of a single observation have been well documented in the literature. However, a complicating factor can exist in that the rank of the eigenvalues may change due to the removal of an observation, and with that so too does the perceived importance of the corresponding eigenvector. We refer to this problem as "switching of eigenvalues". Since there is not enough information in the new eigenvalues post observation removal to indicate that this has happened, how do we know that this switching has occurred? In this paper we show that approximations to the eigenvalues can be used to help determine when switching may have occurred. We then discuss possible actions researchers can take based on this knowledge, for example making better choices when it comes to deciding how many principal components should be retained and adjustments to approximate influence diagnostics that perform poorly when switching has occurred. Our results are easily applied to any eigenvalue problem involving symmetric matrix estimators. We highlight our approach with application to a real data example.

It has previously been shown that ordinary least squares can be used to estimate the coefficients of the single-index model under only mild conditions. However, the estimator is non-robust leading to poor estimates for some models. In this paper we propose a new sliced least-squares estimator that utilizes ideas from Sliced Inverse Regression. Slices with problematic observations that contribute to high variability in the estimator can easily be down-weighted to robustify the procedure. The estimator is simple to implement and can result in vast improvements for some models when compared to the usual least-squares approach. While the estimator was initially conceived with the single-index model in mind, we also show that multiple directions can be obtained, therefore providing another notable advantage of using slicing with least squares. Several simulation studies and a real data example are included, as well as some comparisons with some other recent methods.