Wavelets in statistics: beyond the standard assumptions

The original application of wavelets in statistics was to the estimation of a curve given observations of the curve plus white noise at 2(J) regularly spaced points. The rationale for the use of wavelet methods in this contex t is reviewed briefly. Various extensions of the standard statistical metho dology are discussed. These include curve estimation in the presence of cor related and non-stationary noise, the estimation of (0-1) functions, the ha ndling of irregularly spaced data and data with heavy-tailed noise, and def ormable templates in image and shape analysis. Important tools are a Bayesi an approach, where a suitable prior is placed on the wavelet expansion, enc apsulating the notion that most of the wavelet coefficients are zero; the u se of the non-decimated, or translation-invariant, wavelet transform; and a fast algorithm for finding all the within-level covariances within the tab le of wavelet coefficients of a sequence with arbitrary band-limited covari ance structure. Practical applications drawn from neurophysiology, meteorol ogy and palaeopathology are presented. Finally, some directions for possibl e future research are outlined.