Change-Point estimation using shape-restricted regression splines

摘要

Change-Point estimation is in need in fields like climate change, signal processing, economics, dose-response analysis etc, but it has not yet been fully discussed. We consider estimating a regression function and its change-point m, where m is a mode, an inflection point, or a jump point. Linear inequality constraints are used with spline regression functions to estimate m and the regression function simultaneously using profile methods. For a given m, the maximum-likelihood estimate of the regression function is found using constrained regression methods, then the set of possible change-points is searched to find the m that maximizes the likelihood. Convergence rates are obtained for each type of change-point estimator, and we show an oracle property, that the convergence rate of the regression function estimator is as if m were known. Parametrically modeled covariates are easily incorporated in the model. Simulations show that for small and moderate sample sizes, these methods compare well to existing methods. The scenario when the random error is from a stationary autoregressive process is also presented. Under such a scenario, the change-point and parameters of the stationary autoregressive process, such as autoregressive coefficients and the model variance, are estimated together via Cochran-Orcutt-type iterations. Simulations are conducted and it is shown that the change-point estimator performs well in terms of choosing the right order of the autoregressive process. Penalized spline-based regression is also discussed as an extension. Given a large number of knots and a penalty parameter which controls the effective degrees of freedom of a shape-restricted model, penalized methods give smoother fits while balance between under- and over-fitting. A bootstrap confidence interval for a change-point is established. By generating random change-points from a curve on the unit interval, we compute the coverage rate of the bootstrap confidence interval using penalized estimators, which shows advantages such as robustness over competitors.