The investigators propose a comprehensive theory and methodology for adaptive estimation of linear functionals. A general framework is provided for the construction of both adaptive point estimators and adaptive confidence intervals. The theory makes precise exactly when adaptation is possible and also the minimum price that must be paid when adaptation is not possible. A new geometric quantity is shown to be instrumental in building this theory. This geometric approach also leads to effective adaptive algorithms in optimal recovery problems. The theory is extended to include simultaneous estimation of many linear functionals.
As data collection and computing power have grown exponentially there has been a trend to fit more complicated models using more flexible statistical tools. The investigators develop a general theory for the analysis of such models. A geometric framework is provided and shown to be useful in many engineering and computer science applications.
