An application of universal data compression to statistical analysis of time series

It is shown that universal data compressors (or universal codes) can be used for solving some of the most important statistical problems for time series. By definition, a universal lossless data compressor (or a universal code) can compress any sequence generated by a stationary and ergodic source asymptotically to the Shannon entropy, which, in turn, is the best achievable ratio for lossless data compressors. We consider finite-alphabet and real-valued time series and the following problems: estimation of the limiting probabilities for finite-alphabet time series and estimation of the density for real-valued time series and the on-line prediction for both types of the time series and the following problems of hypothesis testing: goodness-of-fit testing and testing of serial independence. On the one hand, all problems are considered in the framework of classical mathematical statistics and, on the other hand, everyday methods of data compression (or archivers) can be used as a tool for the estimation and testing. It turns out, that quite often the suggested methods and tests are more powerful than known ones when they are applied in practice.