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In quantum optics, the quantum state of a light beam is represented through the Wigner function, a density on $mathbb R^2$ which may take negative values but must respect intrinsic positivity constraints imposed by quantum physics. In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < eta leq 1$, we study the theoretical performance of a kernel estimator of the Wigner function. We prove that it is minimax efficient, up to a logarithmic factor in the sample size, for the $mathbb L_infty$-risk over a class of infinitely differentiable. We compute also the lower bound for the $mathbb L_2$-risk. We construct adaptive estimator, i.e. which does not depend on the smoothness parameters, and prove that it attains the minimax rates for the corresponding smoothness class functions. Finite sample behaviour of our adaptive procedure are explored through numerical experiments.
Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the va
We study minimax density estimation on the product space $mathbb{R}^{d_1}timesmathbb{R}^{d_2}$. We consider $L^p$-risk for probability density functions defined over regularity spaces that allow for different level of smoothness in each of the variab
This paper is concerned with density estimation of directional data on the sphere. We introduce a procedure based on thresholding on a new type of spherical wavelets called {it needlets}. We establish a minimax result and prove its optimality. We are
We address the problem of adaptive minimax density estimation on $bR^d$ with $bL_p$--loss on the anisotropic Nikolskii classes. We fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm inde
We consider nonparametric inference of finite dimensional, potentially non-pathwise differentiable target parameters. In a nonparametric model, some examples of such parameters that are always non pathwise differentiable target parameters include pro