78 research outputs found
The Stein hull
International audienceWe are interested in the statistical linear inverse problem , where denotes a compact operator and a stochastic noise. In this setting, the risk hull point of view provides interesting tools for the construction of adaptive estimators. It sheds light on the processes governing the behaviour of linear estimators. In this paper, we investigate the link between some threshold estimators and this risk hull point of view. The penalized blockwise Stein rule plays a central role in this study. In particular, this estimator may be considered as a risk hull minimization method, provided the penalty is well-chosen. Using this perspective, we study the properties of the threshold and propose an admissible range for the penalty leading to accu- rate results. We eventually propose a penalty close to the lower bound of this range
Sharp template estimation in a shifted curves model
This paper considers the problem of adaptive estimation of a template in a
randomly shifted curve model. Using the Fourier transform of the data, we show
that this problem can be transformed into a stochastic linear inverse problem.
Our aim is to approach the estimator that has the smallest risk on the true
template over a finite set of linear estimators defined in the Fourier domain.
Based on the principle of unbiased empirical risk minimization, we derive a
nonasymptotic oracle inequality in the case where the law of the random shifts
is known. This inequality can then be used to obtain adaptive results on
Sobolev spaces as the number of observed curves tend to infinity. Some
numerical experiments are given to illustrate the performances of our approach
Classification with the nearest neighbor rule in general finite dimensional spaces: necessary and sufficient conditions
Given an -sample of random vectors whose
joint law is unknown, the long-standing problem of supervised classification
aims to \textit{optimally} predict the label of a given a new observation
. In this context, the nearest neighbor rule is a popular flexible and
intuitive method in non-parametric situations.
Even if this algorithm is commonly used in the machine learning and
statistics communities, less is known about its prediction ability in general
finite dimensional spaces, especially when the support of the density of the
observations is . This paper is devoted to the study of the
statistical properties of the nearest neighbor rule in various situations. In
particular, attention is paid to the marginal law of , as well as the
smoothness and margin properties of the \textit{regression function} . We identify two necessary and sufficient conditions to
obtain uniform consistency rates of classification and to derive sharp
estimates in the case of the nearest neighbor rule. Some numerical experiments
are proposed at the end of the paper to help illustrate the discussion.Comment: 53 Pages, 3 figure
Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
This paper considers the problem of adaptive estimation of a non-homogeneous
intensity function from the observation of n independent Poisson processes
having a common intensity that is randomly shifted for each observed
trajectory. We show that estimating this intensity is a deconvolution problem
for which the density of the random shifts plays the role of the convolution
operator. In an asymptotic setting where the number n of observed trajectories
tends to infinity, we derive upper and lower bounds for the minimax quadratic
risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used
to derive an adaptive estimator of the intensity. The proposed estimator is
shown to achieve a near-minimax rate of convergence. This rate depends both on
the smoothness of the intensity function and the density of the random shifts,
which makes a connection between the classical deconvolution problem in
nonparametric statistics and the estimation of a mean intensity from the
observations of independent Poisson processes
Intensity estimation of non-homogeneous Poisson processes from shifted trajectories
In this paper, we consider the problem of estimating nonparametrically a mean pattern intensity λ from the observation of n independent and non-homogeneous Poisson processes N1,âŠ,Nn on the interval [0,1]. This problem arises when data (counts) are collected independently from n individuals according to similar Poisson processes. We show that estimating this intensity is a deconvolution problem for which the density of the random shifts plays the role of the convolution operator. In an asymptotic setting where the number n of observed trajectories tends to infinity, we derive upper and lower bounds for the minimax quadratic risk over Besov balls. Non-linear thresholding in a Meyer wavelet basis is used to derive an adaptive estimator of the intensity. The proposed estimator is shown to achieve a near-minimax rate of convergence. This rate depends both on the smoothness of the intensity function and the density of the random shifts, which makes a connection between the classical deconvolution problem in nonparametric statistics and the estimation of a mean intensity from the observations of independent Poisson processes
Noisy classification with boundary assumptions
We address the problem of classification when data are collected from two samples with measurement errors. This problem turns to be an inverse problem and requires a specific treatment. In this context, we investigate the minimax rates of convergence using both a margin assumption, and a smoothness condition on the boundary of the set associated to the Bayes classifier. We establish lower and upper bounds (based on a deconvolution classifier) on these rates
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