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OPTIMAL ADAPTIVE ESTIMATION OF A QUADRATIC FUNCTIONAL
, 2006
"... Adaptive estimation of a quadratic functional over both Besov and Lp balls is considered. A collection of nonquadratic estimators are developed which have useful bias and variance properties over individual Besov and Lp balls. An adaptive procedure is then constructed based on penalized maximization ..."
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Cited by 11 (3 self)
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Adaptive estimation of a quadratic functional over both Besov and Lp balls is considered. A collection of nonquadratic estimators are developed which have useful bias and variance properties over individual Besov and Lp balls. An adaptive procedure is then constructed based on penalized maximization over this collection of nonquadratic estimators. This procedure is shown to be optimally rate adaptive over the entire range of Besov and Lp balls in the sense that it attains certain constrained risk bounds.
ASYMPTOTIC EQUIVALENCE AND ADAPTIVE ESTIMATION FOR ROBUST NONPARAMETRIC REGRESSION
, 2009
"... Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for rob ..."
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Cited by 6 (4 self)
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Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the regression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.
Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data
 Mathematical Methods of Statistics
"... The aim of this paper is to answer an important issue in quantum mechanics, namely to determine if a quantum state of a light beam is pure or mixed. The estimation of the purity is done from measurements by Quantum Homodyne Tomography performed on identically prepared quantum systems. The quantum s ..."
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Cited by 3 (2 self)
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The aim of this paper is to answer an important issue in quantum mechanics, namely to determine if a quantum state of a light beam is pure or mixed. The estimation of the purity is done from measurements by Quantum Homodyne Tomography performed on identically prepared quantum systems. The quantum state of the light is entirely characterized by the Wigner function, a density of generalized joint probability which can take negative values and which must respect certain constraints of positivity imposed by quantum physics. We propose to estimate a quadratic functional of the Wigner function by a kernel method as the physical measure of the purity of the state. We give also an adaptive estimator that does not depend on the smoothness parameters and we establish upper bound on the minimax risk over a class of infinitely differentiable functions. 1 ha l0
Submitted to the Annals of Statistics ESTIMATION OF FUNCTIONALS OF SPARSE COVARIANCE MATRICES
"... Highdimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other `r norms. Motivated by the computation of critical values of such tests, we investigat ..."
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Highdimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other `r norms. Motivated by the computation of critical values of such tests, we investigate the difficulty of estimation the functionals of sparse correlation matrices. Specifically, we show that simple plugin procedures based on thresholded estimators of correlation matrices are sparsityadaptive and minimax optimal over a large class of correlation matrices. Akin to previous results on functional estimation, the minimax rates exhibit an elbow phenomenon. Our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics. 1. Introduction. Covariance
Quadratic functional estimation in inverse problems
, 2009
"... We consider in this paper a Gaussian sequence model of observations Yi, i ≥ 1 having mean (or signal) θi and variance σi which is growing polynomially like i γ, γ> 0. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal ∑ i≥1 θ2 i when ..."
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We consider in this paper a Gaussian sequence model of observations Yi, i ≥ 1 having mean (or signal) θi and variance σi which is growing polynomially like i γ, γ> 0. This model describes a large panel of inverse problems. We estimate the quadratic functional of the unknown signal ∑ i≥1 θ2 i when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights iα, α> 0) and infinite smoothness (exponential weights eβir, β> 0, 0 < r ≤ 2). We propose a Pinsker type projection estimator in each case and study its quadratic risk. When the signal is sufficiently smoother than the difficulty of the inverse problem (α> γ + 1/4 or in the case of exponential weights), we obtain the parametric rate and the efficiency constant associated to it. Moreover, we give upper bounds of the second order term in the risk and conjecture that they are asymptotically sharp minimax. When the signal is finitely smooth with α ≤ γ + 1/4, we compute non parametric upper bounds of the risk of and we presume also that the constant is asymptotically sharp.
Optimal Recovery and Statistical Estimation in Lp Sobolev Classes
"... We present algorithms for the optimal recovery of a partial derivative of a function at a point when we have approximate measurements of the Riesz transform of the function, and this function belongs to a Lp Sobolev class. The algorithms are exactly optimal among linear algorithms for the cases p = ..."
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We present algorithms for the optimal recovery of a partial derivative of a function at a point when we have approximate measurements of the Riesz transform of the function, and this function belongs to a Lp Sobolev class. The algorithms are exactly optimal among linear algorithms for the cases p = 1 and p = 2 and we give tight bounds for the performance of the algorithms when p> 1, p 6 = 2. Previously only the case p = 2 has been studied. Algorithms for the optimal recovery problem provide optimal estimators for several statistical problems, when we calibrate algorithms suitably. As examples we construct a nearly minimax estimator in the Gaussian white noise model and an asymptotically nearly minimax estimator for the problem of regression function estimation with i.i.d. data. We give also bounds for the asymptotic adaptive risk in the Gaussian white noise model.
NONQUADRATIC ESTIMATORS OF A QUADRATIC FUNCTIONAL1
"... Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often rate suboptimal. In such cases minimax rate optimal procedures are con ..."
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Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often rate suboptimal. In such cases minimax rate optimal procedures are constructed based on local thresholding. These nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence. Moreover, it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures. 1. Introduction. The Gaussian
Estimation of Functionals of Sparse Covariance Matrices
"... Abstract. Highdimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other `r norms. Motivated by the computation of critical values of such tests, we ..."
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Abstract. Highdimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other `r norms. Motivated by the computation of critical values of such tests, we investigate the difficulty of estimation the functionals of sparse correlation matrices. Specifically, we show that simple plugin procedures based on thresholded estimators of correlation matrices are sparsityadaptive and minimax optimal over a large class of correlation matrices. Akin to previous results on functional estimation, the minimax rates exhibit an elbow phenomenon. Our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics.