Wavelet Methods for Curve Estimation

The theory of wavelets is a developing branch of mathematics with a wide range of potential applications. Compactly supported wavelets are particularly interesting because of their natural ability to represent data with intrinsically local properties. They are useful for the detection of edges and singularities in image and sound analysis, and for data compression. However, most of the wavelet based procedures currently available do not explicitly account for the presence of noise in the data. A discussion of how this can be done in the setting of some simple nonparametric curve estimation problems is given. Wavelet analogues of some familiar kernel and orthogonal series estimators are introduced and their finite sample and asymptotic properties are studied. We discover that there is a fundamental instability in the asymptotic variance 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 art assessed.