We consider the problem of constructing asymptotically valid confidence intervals for the change point in a high-dimensional covariance shift setting. A novel estimator for the change point parameter is developed, and its asymptotic distribution under high dimensional scaling obtained. We establish that the proposed estimator exhibits a sharp $O_p(\psi^{-2})$ rate of convergence, wherein $\psi$ represents the jump size between model parameters before and after the change point. Further, the form of the asymptotic distributions under both a vanishing and a non-vanishing regime of the jump size are characterized. In the former case, it corresponds to the argmax of an asymmetric Brownian motion, while in the latter case to the argmax of an asymmetric random walk. We then obtain the relationship between these distributions, which allows construction of regime (vanishing vs non-vanishing) adaptive confidence intervals. Easy to implement algorithms for the proposed methodology are developed and their performance illustrated on synthetic and real data sets.

We propose a two step algorithm based on $\ell_1/\ell_0$ regularization for the detection and estimation of parameters of a high dimensional change point regression model and provide the corresponding rates of convergence for the change point as well as the regression parameter estimates. Importantly, the computational cost of our estimator is only $2\cdotp$Lasso$(n,p)$, where Lasso$(n,p)$ represents the computational burden of one Lasso optimization in a model of size $(n,p)$. In comparison, existing grid search based approaches to this problem require a computational cost of at least $n\cdot Lasso(n,p)$ optimizations. Additionally, the proposed method is shown to be able to consistently detect the case of 'no change', i.e., where no finite change point exists in the model. We allow the true change point parameter $\tau_0$ to possibly move to the boundaries of its parametric space, and the jump size $\|b_0-g_0\|_2$ to possibly diverge as $n$ increases. We then characterize the corresponding effects on the rates of convergence of the change point and regression estimates. In particular, we show that, while an increasing jump size may have a beneficial effect on the change point estimate, however the optimal rate of regression parameter estimates are preserved only upto a certain rate of the increasing jump size. This behavior in the rate of regression parameter estimates is unique to high dimensional change point regression models only. Simulations are performed to empirically evaluate performance of the proposed estimators. The methodology is applied to community level socio-economic data of the U.S., collected from the 1990 U.S. census and other sources.