Competitive prediction under additive noise

In this correspondence, we consider sequential prediction of a real-valued individual signal from its past noisy samples, under square error loss. We refrain from making any stochastic assumptions on the generation of the underlying desired signal and try to achieve uniformly good performance for any deterministic and arbitrary individual signal. We investigate this problem in a competitive framework, where we construct algorithms that perform as well as the best algorithm in a competing class of algorithms for each desired signal. Here, the best algorithm in the competition class can be tuned to the underlying desired clean signal even before processing any of the data. Three different frameworks under additive noise are considered: the class of a finite number of algorithms; the class of all pth order linear predictors (for some fixed order p); and finally the class of all switching pth order linear predictors.