Data dependent wavelet thresholding in nonparametric regression with change-point applications

In one version of the change-point problem one has independent observations Y-1,..., Y-n which have the same mean under the null hypothesis. This problem is transformed into a nonparametric regression problem by considering each Y-i to have mean f(i/n), and then estimating the function f on [0, 1] from the data. Wavelets provide a useful tool for estimating such a function, which may have multiple abrupt jumps. A data dependent technique for selecting a threshold with which to shrink empirical wavelet coefficients is introduced. The technique, based on standard statistical tests of hypotheses, is shown to give good results both when the underlying function is constant, and when it undergoes multiple abrupt changes. By adjusting the level a of the tests of significance, it is possible to control the smoothness of the resulting estimator, allowing one to give preference to good expected MSE performance or to favor good visual appearance of the estimator in exploratory data analysis settings.