My early work in the mid-90s in nonparametric regression was concerned with wavelet shrinkage. Wavelet shrinkage is a technique for nonparametric function estimation that has a number of advantages over other techniques. Wavelet shrinkage is highly efficient: fast in computation and frugal with storage. It has the potential to achieve extremely good results both on classically smooth functions, but also functions that have discontinuities or other `abnormalities'. Moreover, this level of performance is typically guaranteed by theory.
A great advantage of wavelet shrinkage techniques is that, unlike other methods, one does not need to know in advance of the likely smoothness of the function you are trying to estimate. If the function is smooth then wavelet shrinkage typically performs as well, sometimes better, compared to other methods. However, if the function contains irregularities then the wavelets will still work, whereas competitor methods might fail disasterously.
I contributed to early work in wavelet shrinkage, particularly cross-validation, parameter selection, survival function estimation, density estimation and confidence intervals for functions via empirical Bayes techniques.
I produced the first available software package for the R (originally S-Plus) package to carry out wavelet analysis and wavelet shrinkage for the statistical community. The package is called WaveThresh and is available from the CRAN archive.
More recently, I have been working on statistical methodology based on the the lifting transform. Lifting is a second-generation version of wavelets that can work with non-regular designs and other situations.
A key development in this area is the `lifting one-coefficient-at-a-time' or LOCAAT, which is an extremely flexible paradiagm that permits lifting to be used not only on irregular multivariate data but also data defined on networks or graphs.
Another lifting development that I was involved in was that of `adaptive lifting'. This technique locally chooses the basic function that achieves the best signal compression on a coefficient by coefficient basis. This techique achieves extremely good results compared to classical methods.
Lifting techniques such as the ones above have a number of advantages: fast, efficient, simple to code and extremely good empirical performance. Unfortunately, developing mathematical theory to explain their performance is extremely challenging and this aspect does not always find favour with some people!