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Multiscale methods; Dynamic programming; Honest confidence sets; Change point regression; Exponential families
Abstract:
We introduce a new estimator, the simultaneous multiscale change point estimator SMUCE, for the change point problem in exponential family regression. An unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level alpha. The probability of overestimating the true number of change points K is controlled by the asymptotic null distribution of the multiscale test statistic. Further, we derive exponential bounds for the probability of underestimating K. By balancing these quantities, alpha will be chosen such that the probability of correctly estimating K is maximized. All results are even non-asymptotic for the normal case. On the basis of these bounds, we construct (asymptotically) honest confidence sets for the unknown step function and its change points. At the same time, we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate O(n-1) up to a log-term. Finally, the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n ->infinity, even for an unbounded number of change points. We illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions. The performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy.
