New Horizons in Statistical Decision Theory

Cornell University, Ithaca NY

Investigators

Abstract

The classical metric theory of statistical models (experiments) has been extended towards an asymptotic equivalence paradigm, allowing to classify and relate problems which are essentially infinite dimensional and ill-posed. Asymptotic equivalence theory is emerging as a recognizable research area in statistics. The theoretical possibility to carry over optimal procedures from one model to another allows a better conceptual understanding of asymptotic inference. This area is still under vigorous development, and new problems arise in the context of asymptotic inference based on high-dimensional data. Modern statistical concepts like these are also being integrated into the emerging field of quantum statistics, which is developing on the background of technological breakthroughs in quantum engineering. The analysis of quantum statistical models points towards an underlying non-commutative statistical decision theory with connections to operator algebra, quantum information and quantum probability. By focusing on this growing research area at the interface of Statistics and Mathematical Physics, the project will have the side effect of fostering interaction between the different scientific communities. One of the unexplored topics in asymptotic equivalence theory is the impact of additional observations in nonparametric models. Le Cam showed that in the regular parametric case, additional observations are negligible in an asymptotic information sense if their number is one order of magnitude below the original sample size, but for larger parameter spaces this critical threshold seems to be lower. Reasoning via additional observations has been applied in the past to prove equivalence of spectral density to white noise models, and may prove useful again for problems related to estimation of high-dimensional matrices. In a related topic, it is of interest to further refine sharp nonparametric risk bounds like the Pinsker bound, which have motivated the development of equivalence theory. Here the research program aims at confirming a conjecture about a bound of this type for sharp adaptive nonparametric testing, a possible complement to results in adaptive estimation. In the problem of symmetric quantum hypothesis testing, or discrimination between two quantum states, the program focuses on applications of the quantum Chernoff bound pertaining to the exponential rate of decay of the error probability. In that connection, several new problems appear, such as attainability of the bound by realizable receivers in quantum optics, and variants for discriminating states given by quantum Markov chains. An area of particular interest is local asymptotic normality for quantum statistical models. Some elements of such a theory have already been put forward in the literature; these notions will be further developed in the spirit of Le Cam's classical theory, focusing at first on the quantum analog of Gaussian stationary sequences.