WAVELET METHODS FOR CURVE ESTIMATION

The theory of wavelets is a developing branch of mathematics with a wi de range of potential applications. Compactly supported wavelets are p articularly interesting because of their natural ability to represent data with intrinsically local properties. They are useful for the dete ction of edges and singularities in image and sound analysis and for d ata compression. But most of the wavelet-based procedures currently av ailable do not explicitly account for the presence of noise in the dat a. A discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given. Wavelet analogies of some familiar kernel and orthogonal series estimators are introduced, and their finite sample and asymptotic properties are studied. We dis cover that there is a fundamental instability in the asymptotic varian ce of wavelet estimators caused by the lack of translation invariance of the wavelet transform. This is related to the properties of certain lacunary sequences. The practical consequences of this instability ar e assessed.