arXiv subjects

Cristina Butucea

Publications and source records attributed to Cristina Butucea.

47 records · Page 3Linked to original sources

arXiv2011

Semiparametric mixtures of symmetric distributions

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 M-estimators of these parameters based on a Fourier approach, and prove that they are square root consistent under mild regularity conditions. Their finite-sample properties are illustrated by a Monte Carlo study and a benchmark real dataset is also studied with our method.

math.ST↗

arXiv2010

Quantum U-statistics

The notion of a $U$-statistic for an $n$-tuple of identical quantum systems is introduced in analogy to the classical (commutative) case: given a selfadjoint `kernel' $K$ acting on $(\mathbb{C}^{d})^{\otimes r}$ with $r<n$, we define the symmetric operator $U_{n}= {n \choose r} \sum_βK^{(β)}$ with $K^{(β)}$ being the kernel acting on the subset $β$ of $\{1,\dots ,n\}$. If the systems are prepared in the i.i.d state $ρ^{\otimes n}$ it is shown that the sequence of properly normalised $U$-statistics converges in moments to a linear combination of Hermite polynomials in canonical variables of a CCR algebra defined through the Quantum Central Limit Theorem. In the special cases of non-degenerate kernels and kernels of order $2$ it is shown that the convergence holds in the stronger distribution sense. Two types of applications in quantum statistics are described: testing beyond the two simple hypotheses scenario, and quantum metrology with interacting hamiltonians.

quant-ph↗

arXiv2009

Quadratic functional estimation in inverse problems

We consider in this paper a Gaussian sequence model of observations $Y_i$, $i\geq 1$ having mean (or signal) $θ_i$ and variance $σ_i$ 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 $\sum_{i\geq 1}θ_i^2$ when the signal belongs to ellipsoids of both finite smoothness functions (polynomial weights $i^α$, $α>0$) and infinite smoothness (exponential weights $e^{βi^r}$, $β>0$, $0 γ+1/4$ 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 $α\leq γ+1/4$, we compute non parametric upper bounds of the risk of and we presume also that the constant is asymptotically sharp.

math.ST↗

arXiv2008

Adaptivity in convolution models with partially known noise distribution

We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density $f$ and some partially known noise density $g$. In this work, $g$ is assumed exponentially smooth with stable law having unknown self-similarity index $s$. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities $f$, with smoothness parameter $β$. In this context, we first provide a consistent estimation procedure for $s$. This estimator is then plugged-into three different procedures: estimation of the unknown density $f$, of the functional $\int f^2$ and goodness-of-fit test of the hypothesis $H_0:f=f_0$, where the alternative $H_1$ is expressed with respect to $\mathbb {L}_2$-norm (i.e. has the form $ψ_n^{-2}\|f-f_0\|_2^2\ge \mathcal{C}$). These procedures are adaptive with respect to both $s$ and $β$ and attain the rates which are known optimal for known values of $s$ and $β$. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of $s$ is illustrated on synthetic data.

math.ST↗

arXiv2008

State estimation in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter $0 < η\leq 1$, we propose two estimators of a quantum state whose density matrix elements $ρ_{m,n}$ decrease like $e^{-B(m+n)^{r/ 2}}$, for fixed known $B>0$ and $0<r\leq 2$. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for $0<η\leq 1/2$), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in $\mathbb{L}_2$ risk.

math.ST↗

arXiv2008

New $M$-estimators in semi-parametric regression with errors in variables

In the regression model with errors in variables, we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f_{θ^0}(X)+ξ$ and $Z=X+ε$ involving independent and unobserved random variables $X,ξ,ε$ plus a regression function $f_{θ^0}$, known up to a finite dimensional $θ^0$. The common densities of the $X_i$'s and of the $ξ_i$'s are unknown, whereas the distribution of $ε$ is completely known. We aim at estimating the parameter $θ^0$ by using the observations $(Y_1,Z_1),...,(Y_n,Z_n)$. We propose an estimation procedure based on the least square criterion $\tilde{S}_{θ^0,g}(θ)=\m athbb{E}_{θ^0,g}[((Y-f_θ(X))^2w(X)]$ where $w$ is a weight function to be chosen. We propose an estimator and derive an upper bound for its risk that depends on the smoothness of the errors density $p_ε$ and on the smoothness properties of $w(x)f_θ(x)$. Furthermore, we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of $w$ in the case of nonlinear regression functions which are smooth on pieces allowing to gain in the order of the rate of convergence, up to the parametric rate in some cases. We also consider extensions of the estimation procedure, in particular, when a choice of $w_θ$ depending on $θ$ would be more appropriate.

math.ST↗

arXiv2007

Goodness-of-fit testing and quadratic functional estimation from indirect observations

We consider the convolution model where i.i.d. random variables $X_i$ having unknown density $f$ are observed with additive i.i.d. noise, independent of the $X$'s. We assume that the density $f$ belongs to either a Sobolev class or a class of supersmooth functions. The noise distribution is known and its characteristic function decays either polynomially or exponentially asymptotically. We consider the problem of goodness-of-fit testing in the convolution model. We prove upper bounds for the risk of a test statistic derived from a kernel estimator of the quadratic functional $\int f^2$ based on indirect observations. When the unknown density is smoother enough than the noise density, we prove that this estimator is $n^{-1/2}$ consistent, asymptotically normal and efficient (for the variance we compute). Otherwise, we give nonparametric upper bounds for the risk of the same estimator. We give an approach unifying the proof of nonparametric minimax lower bounds for both problems. We establish them for Sobolev densities and for supersmooth densities less smooth than exponential noise. In the two setups we obtain exact testing constants associated with the asymptotic minimax rates.

math.ST↗

arXiv2007

Functional approach for excess mass estimation in the density model

We consider a multivariate density model where we estimate the excess mass of the unknown probability density $f$ at a given level $ν>0$ from $n$ i.i.d. observed random variables. This problem has several applications such as multimodality testing, density contour clustering, anomaly detection, classification and so on. For the first time in the literature we estimate the excess mass as an integrated functional of the unknown density $f$. We suggest an estimator and evaluate its rate of convergence, when $f$ belongs to general Besov smoothness classes, for several risk measures. A particular care is devoted to implementation and numerical study of the studied procedure. It appears that our procedure improves the plug-in estimator of the excess mass.

math.ST↗

arXiv2007

Minimax and adaptive estimation of the Wigner function in quantum homodyne tomography with noisy data

We estimate the quantum state of a light beam from results of quantum homodyne measurements performed on identically prepared quantum systems. The state is represented through the Wigner function, a generalized probability density on $\mathbb{R}^2$ which may take negative values and must respect intrinsic positivity constraints imposed by quantum physics. The effect of the losses due to detection inefficiencies, which are always present in a real experiment, is the addition to the tomographic data of independent Gaussian noise. We construct a kernel estimator for the Wigner function, prove that it is minimax efficient for the pointwise risk over a class of infinitely differentiable functions, and implement it for numerical results. We construct adaptive estimators, that is, which do not depend on the smoothness parameters, and prove that in some setups they attain the minimax rates for the corresponding smoothness class.

math.ST↗

arXiv2007

Adaptive procedures in convolution models with known or partially known noise distribution

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known or partially known noise density g. In this paper, we focus on statistical procedures, which are adaptive with respect to the smoothness parameter tau of unknown density f, and also (in some cases) to some unknown parameter of the noise density g. In a first part, we assume that g is known and polynomially smooth. We provide goodness-of-fit procedures for the test H_0:f=f_0, where the alternative H_1 is expressed with respect to L_2-norm. Our adaptive (w.r.t tau) procedure behaves differently according to whether f_0 is polynomially or exponentially smooth. A payment for adaptation is noted in both cases and for computing this, we provide a non-uniform Berry-Esseen type theorem for degenerate U-statistics. In the first case we prove that the payment for adaptation is optimal (thus unavoidable). In a second part, we study a wider framework: a semiparametric model, where g is exponentially smooth and stable, and its self-similarity index s is unknown. In order to ensure identifiability, we restrict our attention to polynomially smooth, Sobolev-type densities f. In this context, we provide a consistent estimation procedure for s. This estimator is then plugged-into three different procedures: estimation of the unknown density f, of the functional \int f^2 and test of the hypothesis H_0. These procedures are adaptive with respect to both s and tau and attain the rates which are known optimal for known values of s and tau. As a by-product, when the noise is known and exponentially smooth our testing procedure is adaptive for testing Sobolev-type densities.

math.ST↗

arXiv2004

Sharp optimality for density deconvolution with dominating bias

We consider estimation of the common probability density $f$ of i.i.d. random variables $X_i$ that are observed with an additive i.i.d. noise. We assume that the unknown density $f$ belongs to a class $\mathcal{A}$ of densities whose characteristic function is described by the exponent $\exp(-α|u|^r)$ as $|u|\to \infty$, where $α>0$, $r>0$. The noise density is supposed to be known and such that its characteristic function decays as $\exp(-β|u|^s)$, as $|u| \to \infty$, where $β>0$, $s>0$. Assuming that $r<s$, we suggest a kernel type estimator that is optimal in sharp asymptotical minimax sense on $\mathcal{A}$ simultaneously under the pointwise and the $\mathbb{L}_2$-risks. The variance of the estimators turns out to be asymptotically negligible w.r.t. its squared bias. For $r<s/2$ we construct a sharp adaptive estimator of $f$. We discuss some effects of dominating bias, such as superefficiency of minimax estimators.

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