873 research outputs found
Minimax estimation of low-rank quantum states and their linear functionals
In classical statistics, a well known paradigm consists in establishing
asymptotic equivalence between an experiment of i.i.d. observations and a
Gaussian shift experiment, with the aim of obtaining optimal estimators in the
former complicated model from the latter simpler model. In particular, a
statistical experiment consisting of i.i.d observations from d-dimensional
multinomial distributions can be well approximated by an experiment consisting
of dimensional Gaussian distributions. In a quantum version of the
result, it has been shown that a collection of qudits (d-dimensional
quantum states) of full rank can be well approximated by a quantum system
containing a classical part, which is a dimensional Gaussian
distribution, and a quantum part containing an ensemble of shifted
thermal states. In this paper, we obtain a generalization of this result when
the qudits are not of full rank. We show that when the rank of the qudits is
, then the limiting experiment consists of an dimensional Gaussian
distribution and an ensemble of both shifted pure and shifted thermal states.
For estimation purposes, we establish an asymptotic minimax result in the
limiting Gaussian model. Analogous results are then obtained for estimation of
a low-rank qudit from an ensemble of identically prepared, independent quantum
systems, using the local asymptotic equivalence result. We also consider the
problem of estimation of a linear functional of the quantum state. We construct
an estimator for the functional, analyze the risk and use quantum local
asymptotic equivalence to show that our estimator is also optimal in the
minimax sense.Comment: arXiv admin note: text overlap with arXiv:0804.3876 by other author
Asymptotic equivalence for nonparametric generalized linear models
We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value ƒ(ti) of a regression function ƒ at a grid point ti (nonparametric GLM). When ƒ is in a Hölder ball with exponent β > 1⁄2, we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process
The degrees of ill-posedness in stochastic and deterministic noise models
The degrees of ill-posedness for inverse estimation in Hilbert scales in the presence of deterministic and random noise are compared. For Gaussian random noise with different "smoothness" the optimal order of the rate of convergence for above mentioned estimation is indicated
Maximum likelihood estimate for nonparametric signal in white noise by optimal control
The paper is devoted to questions of constructing the maximum likelihood estimate for a nonparametric signal in white noise by considering corresponding problems of optimal control. For signals with bounded derivatives, sensitivity theorems are proved. The theorems state a stability of the maximum likelihood estimate with respect to changing output data. They make possible to reduce the original problem to a standard problem of optimal control which is solved by iterative procedure. For signals of Sobolev type the maximum likelihood estimate is obtained to within a parameter which can be found from a transcendental equation
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