Detection and inference of changes in high-dimensional linear regression with non-sparse structures

For the data segmentation problem in high-dimensional linear regression settings, a commonly made assumption is that the regression parameters are segment-wise sparse, which enables many existing methods to estimate the parameters locally via ₁-regularised maximum likelihood-type estimation and contrast them for change point detection. Contrary to the common belief, we show that the sparsity of neither regression parameters nor their differences, a. k. a. \ differential parameters, is necessary for achieving the consistency in multiple change point detection. In fact, both statistically and computationally, better efficiency is attained by a simple strategy that scans for large discrepancies in local covariance between the regressors and the response. We go a step further and propose a suite of tools for directly inferring about the differential parameters post-segmentation, which are applicable even when the regression parameters themselves are non-sparse. Theoretical investigations are conducted under general conditions permitting non-Gaussianity, temporal dependence and ultra-high dimensionality. Numerical experiments demonstrate the competitiveness of the proposed methodologies.