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26
Semiparametric mixtures of symmetric distributions
 Scand. J. Statist
, 2013
"... We consider in this paper the semiparametric mixture of two distributions equal up to a shift parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. In order to insure the identifiability of the model it is assumed ..."
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Cited by 6 (2 self)
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We consider in this paper the semiparametric mixture of two distributions equal up to a shift parameter. The model is said to be semiparametric in the sense that the mixed distribution is not supposed to belong to a parametric family. In order to insure the identifiability of the model it is assumed that the mixed distribution is symmetric, the model being then defined by the mixing proportion, two location parameters, and the probability density function of the mixed distribution. We propose a new class of Mestimators of these parameters based on a Fourier approach, and prove that they are nconsistent under mild regularity conditions. Their finitesample properties are illustrated by a Monte Carlo study and a benchmark real dataset is also studied with our method. AMS 2000 subject classifications. Primary 62G05, 62G20; secondary 62E10. Key words and phrases. Asymptotic normality, consistency, contrast estimators,
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
Noisy quantization Anisotropic oracle inequalities in noisy quantization
"... The effect of errors in variables in quantization is investigated. We prove general exact and nonexact oracle inequalities with fast rates for an empirical minimization based on a noisy sample Zi = Xi + i, i = 1,..., n, where Xi are i.i.d. with density f and i are i.i.d. with density η. These rates ..."
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The effect of errors in variables in quantization is investigated. We prove general exact and nonexact oracle inequalities with fast rates for an empirical minimization based on a noisy sample Zi = Xi + i, i = 1,..., n, where Xi are i.i.d. with density f and i are i.i.d. with density η. These rates depend on the geometry of the density f and the asymptotic behaviour of the characteristic function of η. This general study can be applied to the problem of kmeans clustering with noisy data. For this purpose, we introduce a deconvolution kmeans stochastic minimization which reaches fast rates of convergence under standard Pollard’s regularity assumptions.
Goodnessoffit test for noisy directional data
, 2013
"... We consider spherical data Xi noised by a random rotation εi ∈ SO(3) so that only the sample Zi = εiXi, i = 1,..., N is observed. We define a nonparametric test procedure to distinguish H0: ”the density f of Xi is the uniform density f0 on the sphere ” and H1: ”‖f − f0 ‖ 2 2 ≥ CψN and f is in a Sobo ..."
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We consider spherical data Xi noised by a random rotation εi ∈ SO(3) so that only the sample Zi = εiXi, i = 1,..., N is observed. We define a nonparametric test procedure to distinguish H0: ”the density f of Xi is the uniform density f0 on the sphere ” and H1: ”‖f − f0 ‖ 2 2 ≥ CψN and f is in a Sobolev space with smoothness s”. For a noise density fε with smoothness index ν, we show that an adaptive procedure (i.e. s is not assumed to be known) cannot have a faster rate of separation than ψ ad N (s) = (N/ √ log log(N)) −2s/(2s+2ν+1) and we provide a procedure which reaches this rate. We also deal with the case of super smooth noise. We illustrate the theory by implementing our test procedure for various kinds of noise on SO(3) and by comparing it to other procedures. Applications to real data in astrophysics and paleomagnetism are provided.
Statistical learning with indirect observations
"... Abstract: Let (X, Y) ∈ X × Y be a random couple with unknown distribution P. Let G be a class of measurable functions and ` a loss function. The problem of statistical learning deals with the estimation of the Bayes: g ∗ = argmin g∈G EP `(g(X), Y). In this paper, we study this problem when we dea ..."
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Abstract: Let (X, Y) ∈ X × Y be a random couple with unknown distribution P. Let G be a class of measurable functions and ` a loss function. The problem of statistical learning deals with the estimation of the Bayes: g ∗ = argmin g∈G EP `(g(X), Y). In this paper, we study this problem when we deal with a contaminated sample (Z1, Y1),..., (Zn, Yn) of i.i.d. indirect observations. Each input Zi, i = 1,..., n is distributed from a density Af, where A is a known compact linear operator and f is the density of the direct input X. We derive fast rates of convergence for empirical risk minimizers based on regularization methods, such as deconvolution kernel density estimators or spectral cutoff. These results are comparable to the existing fast rates in Koltchinskii [2006] for the direct case. It gives some insights into the effect of indirect measurements in the presence of fast rates of convergence. 1.
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.
Nonparametric goodnessof fit testing in quantum homodyne tomography with noisy data
 ELECTRONIC JOURNAL OF STATISTICS
, 2008
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