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Register a new userMinimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data
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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.
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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 variables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this paper, we focus on two commonly used types of parameter spaces, and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based thresholding rule. Lepskis method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model. Numerical studies are carried out to confirm our theoretical findings.
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 variables. Precisely, we study probabilities on Sobolev spaces with dominating mixed-smoothness. We provide the rate of convergence that is optimal even for the classical Sobolev spaces.
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 motivated by astrophysical applications, in particular in connection with the analysis of ultra high energy cosmic rays.
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 indexes in definitions of the functional class and of the risk. In particular, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikolskii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.
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 probability density functions at a point, or regression functions at a point. In causal inference, under appropriate causal assumptions, mean counterfactual outcomes can be pathwise differentiable or not, depending on the degree at which the positivity assumption holds. In this paper, given a potentially non-pathwise differentiable target parameter, we introduce a family of approximating parameters, that are pathwise differentiable. This family is indexed by a scalar. In kernel regression or density estimation for instance, a natural choice for such a family is obtained by kernel smoothing and is indexed by the smoothing level. For the counterfactual mean outcome, a possible approximating family is obtained through truncation of the propensity score, and the truncation level then plays the role of the index. We propose a method to data-adaptively select the index in the family, so as to optimize mean squared error. We prove an asymptotic normality result, which allows us to derive confidence intervals. Under some conditions, our estimator achieves an optimal mean squared error convergence rate. Confidence intervals are data-adaptive and have almost optimal width. A simulation study demonstrates the practical performance of our estimators for the inference of a causal dose-response curve at a given treatment dose.
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