Modality, runs, strings and wavelets

The paper considers the problem of non-parametric regression with emphasis on controlling the number of local extrema. Two methods, the run method and the taut string-wavelet method, are introduced and analysed on standard test beds. It is shown that the number and location of local extreme values are consistently estimated. Rates of convergence are proved for both methods. The run method has a slow rate but can withstand blocks as well as a high proportion of isolated outliers. The rate of convergence of the taut string-wavelet method is almost optimal and the method is extremely sensitive being able to detect very low power peaks. Section 1 contains a short introduction with special reference to modality. The run method is described in Section 2 and the taut string-wavelet method in Section 3. Low power peaks are considered in Section 4. Section 5 contains a short conclusion and the proofs are given in Section 6.