Adaptive Estimation Of Linear Functionals Research Articles

On Adaptive Estimation of Linear Functionals from Observations against White Noise

We consider the problem of adaptive estimation of a linear functional of an unknown multivariate vector from its observations against white Gaussian noise. As a family of estimators for the functional, we use those generated by projection estimators of the unknown vector, and the main problem is to select the best estimator in this family. The goal of the paper is to explain and mathematically justify a simple statistical idea used in adaptive (i.e., observation-based) choice of the best estimator of a linear functional from a given family of estimators. We also discuss generalizations of the considered statistical model and the proposed estimation method, which allow to cover a broad class of statistical problems.

Optimal adaptive estimation of linear functionals under sparsity

We consider the problem of estimation of a linear functional in the Gaussian sequence model where the unknown vector θ ∈ R^d belongs to a class of s-sparse vectors with unknown s. We suggest an adaptive estimator achieving a non-asymptotic rate of convergence that differs from the minimax rate at most by a logarithmic factor. We also show that this optimal adaptive rate cannot be improved when s is unknown. Furthermore, we address the issue of simultaneous adaptation to s and to the variance σ^2 of the noise. We suggest an estimator that achieves the optimal adaptive rate when both s and σ^2 are unknown.

Adaptive estimation of linear functionals in functional linear models

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of randomfunctions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski’s method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as theminimizer of a stochastic penalized contrast function imitating Lepski’smethod among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers pointwise estimation as well as the estimation of local averages of the slope parameter.

Nonparametric adaptive estimation of linear functionals for low frequency observed L evy processes

For a L evy process X having nite variation on compact sets and nite rst moments, ( dx) = x ( dx) is a nite signed measure which completely describes the jump dynamics. We construct kernel estimators for linear functionals of and provide rates of convergence under regularity assumptions. Moreover, we consider adaptive estimation via model selection and propose a new strategy for the data driven choice of the smoothing parameter.

Adaptive estimation of linear functionals by model selection

We propose an estimation procedure for linear functionals based on Gaussian model selection techniques. We show that the procedure is adaptive, and we give a non asymptotic oracle inequality for the risk of the selected estimator with respect to the $\mathbb{L}_{p}$ loss. An application to the problem of estimating a signal or its rth derivative at a given point is developed and minimax rates are proved to hold uniformly over Besov balls. We also apply our non asymptotic oracle inequality to the estimation of the mean of the signal on an interval with length depending on the noise level. Simulations are included to illustrate the performances of the procedure for the estimation of a function at a given point. Our method provides a pointwise adaptive estimator.

Adaptive estimation of linear functionals in the convolution model and applications

We consider the model $Z_i=X_i+\varepsilon_i$, for i.i.d. $X_i$’s and $\varepsilon_i$’s and independent sequences $(X_i)_{i∈ℕ}$ and $(\varepsilon_i)_{i∈ℕ}$. The density $f_\varepsilon$ of $\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by $g$, is unknown. Our aim is to estimate linear functionals of $g, 〈ψ, g〉$ for a known function $ψ$. We propose a general estimator of $〈ψ, g〉$ and study the rate of convergence of its quadratic risk as a function of the smoothness of $g, f_\varepsilon$ and $ψ$. Different contexts with dependent data, such as stochastic volatility and AutoRegressive Conditionally Heteroskedastic models, are also considered. An estimator which is adaptive to the smoothness of unknown $g$ is then proposed, following a method studied by Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give upper bounds and asymptotic lower bounds of the quadratic risk of this estimator. The results are applied to adaptive pointwise deconvolution, in which context losses in the adaptive rates are shown to be optimal in the minimax sense. They are also applied in the context of the stochastic volatility model.

On adaptive estimation of linear functionals

Adaptive estimation of linear functionals over a collection of parameter spaces is considered. A between-class modulus of continuity, a geometric quantity, is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single parameter space. A general construction of optimally adaptive estimators based on an ordered modulus of continuity is given. The results are complemented by several illustrative examples.

Adaptive estimation of linear functionals under different performance measures

Lower bounds are given for probabilistic error subject to a mean squared error constraint. Consequences for the expected length of variable length confidence intervals centred on adaptive estimators are given. It is shown that in many contexts centring confidence intervals on adaptive estimators must lead either to poor coverage probability or unnecessarily long intervals.

Sharp Adaptive Estimation of Linear Functionals

We consider estimation of a linear functional $T(f)$ where $f$ is an unknown function observed in Gaussian white noise.We find asymptotically sharp adaptive estimators on various scales of smoothness classes in multidimensional situations. The results allow evaluating explicitly the effect of dimension and treating general scales of classes. Furthermore, we establish a connection between sharp adaptation and optimal recovery. Namely, we propose a scheme that reduces the construction of sharp adaptive estimators on a scale of functional classes to a solution of the corresponding optimization problem.

Adaptive estimation of linear functionals in Hilbert scales from indirect white noise observations

We consider adaptive estimating the value of a linear functional from indirect white noise observations. For a flexible approach, the problem is embedded in an abstract Hilbert scale. We develop an adaptive estimator that is rate optimal within a logarithmic factor simultaneously over a wide collection of balls in the Hilbert scale. It is shown that the proposed estimator has the best possible adaptive properties for a wide range of linear functionals. The case of discretized indirect white noise observations is studied, and the adaptive estimator in this setting is developed.

Asymptotically Minimax Adaptive Estimation. I: Upper Bounds. Optimally Adaptive Estimates

Asymptotically Minimax Adaptive Estimation. I: Upper Bounds. Optimally Adaptive Estimates