Quantum statistical models and inference
Freitag, 13.1.17, 12:00-13:00, Raum 404, Eckerstr. 1
Quantum statistics is concerned with the inference for systems\ndescribed by quantum mechanics. After an introduction to the\nmain mathematical notions of quantum statistics: quantum states,\nmeasurements, channels, we describe nonparametric quantum models.\nWe prove the local asymptotic equivalence (LAE) in the sense of\nLe Cam theory of i.i.d. quantum pure states and a quantum Gaussian\nstate. We show nonparametric rates for the estimation of the quantum\nstates, of some quadratic functionals and for the testing of pure\nstates. The LAE allows to transfer proofs to a different model.\nSurprisingly, a sharp testing rate of order n^{-1/2} is\nobtained in a nonparametric quantum setup.\nThis is joint work with M. Guta and M. Nussbaum.
Asymptotic equivalence between density estimation and Gaussian white noise revisited
Freitag, 20.1.17, 12:00-13:00, Raum 404, Eckerstr. 1
Asymptotic equivalence between two statistical models means that they\nhave the same asymptotic properties with respect to all decision\nproblems with bounded loss. A key result by Nussbaum states that\nnonparametric density estimation is asymptotically equivalent to a\nsuitable Gaussian shift model, provided that the densities are smooth\nenough and uniformly bounded away from zero.\n\nWe study the case when the latter assumption does not hold and the\ndensity is possibly small. We further derive the optimal Le Cam distance\nbetween these models, which quantifies how close they are. As an\napplication, we also consider Poisson intensity estimation with low\ncount data. This is joint work with Johannes Schmidt-Hieber.