Inference on Multiple Change Points in High Dimensional Linear Regression Models

Confidence intervals are constructed for multiple change points in high-dimensional linear regression models. Locally refitted estimators are developed, and their rate of convergence is evaluated. The componentwise rate of estimation obtained is optimal, and the simultaneous rate is the sharpest available in the literature. Limiting distributions of the considered estimates are provided in both vanishing and non-vanishing jump size regimes, along with the joint limiting distributions. The relationship between the distributions in the two regimes is further examined, and an adaptation property is illustrated to allow for inference without knowledge of the underlying regime. Theoretical results are supported by Monte Carlo simulations and further demonstrated by a real data example.