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A class of tests for exponentiality based on a continuum of moment conditions

Simos G. Meintanis (2009)

Kybernetika

The empirical moment process is utilized to construct a family of tests for the null hypothesis that a random variable is exponentially distributed. The tests are consistent against the 'new better than used in expectation' (NBUE) class of alternatives. Consistency is shown and the limit null distribution of the test statistic is derived, while efficiency results are also provided. The finite-sample properties of the proposed procedure in comparison to more standard procedures are investigated via...

A graph-based estimator of the number of clusters

Gérard Biau, Benoît Cadre, Bruno Pelletier (2007)

ESAIM: Probability and Statistics

Assessing the number of clusters of a statistical population is one of the essential issues of unsupervised learning. Given n independent observations X1,...,Xn drawn from an unknown multivariate probability density f, we propose a new approach to estimate the number of connected components, or clusters, of the t-level set ( t ) = { x : f ( x ) t } . The basic idea is to form a rough skeleton of the set ( t ) using any preliminary estimator of f, and to count the number of connected components of the resulting graph. Under...

A note on robust estimation in logistic regression model

Tadeusz Bednarski (2016)

Discussiones Mathematicae Probability and Statistics

Computationally attractive Fisher consistent robust estimation methods based on adaptive explanatory variables trimming are proposed for the logistic regression model. Results of a Monte Carlo experiment and a real data analysis show its good behavior for moderate sample sizes. The method is applicable when some distributional information about explanatory variables is available.

A note on the adaptive estimation of the differential entropy by wavelet methods

Christophe Chesneau, Fabien Navarro, Oana Silvia Serea (2017)

Commentationes Mathematicae Universitatis Carolinae

In this note we consider the estimation of the differential entropy of a probability density function. We propose a new adaptive estimator based on a plug-in approach and wavelet methods. Under the mean 𝕃 p error, p 1 , this estimator attains fast rates of convergence for a wide class of functions. We present simulation results in order to support our theoretical findings.

A note on the rate of convergence of local polynomial estimators in regression models

Friedrich Liese, Ingo Steinke (2001)

Kybernetika

Local polynomials are used to construct estimators for the value m ( x 0 ) of the regression function m and the values of the derivatives D γ m ( x 0 ) in a general class of nonparametric regression models. The covariables are allowed to be random or non-random. Only asymptotic conditions on the average distribution of the covariables are used as smoothness of the experimental design. This smoothness condition is discussed in detail. The optimal stochastic rate of convergence of the estimators is established. The results...

A scale-space approach with wavelets to singularity estimation

Jérémie Bigot (2005)

ESAIM: Probability and Statistics

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales. In...

A scale-space approach with wavelets to singularity estimation

Jérémie Bigot (2010)

ESAIM: Probability and Statistics

This paper is concerned with the problem of determining the typical features of a curve when it is observed with noise. It has been shown that one can characterize the Lipschitz singularities of a signal by following the propagation across scales of the modulus maxima of its continuous wavelet transform. A nonparametric approach, based on appropriate thresholding of the empirical wavelet coefficients, is proposed to estimate the wavelet maxima of a signal observed with noise at various scales....

A second order approximation for the inverse of the distribution function of the sample mean

Jorge M. Arevalillo (2001)

Kybernetika

The classical quantile approximation for the sample mean, based on the central limit theorem, has been proved to fail when the sample size is small and we approach the tail of the distribution. In this paper we will develop a second order approximation formula for the quantile which improves the classical one under heavy tails underlying distributions, and performs very accurately in the upper tail of the distribution even for relatively small samples.

Adaptive Dantzig density estimation

K. Bertin, E. Le Pennec, V. Rivoirard (2011)

Annales de l'I.H.P. Probabilités et statistiques

The aim of this paper is to build an estimate of an unknown density as a linear combination of functions of a dictionary. Inspired by Candès and Tao’s approach, we propose a minimization of the ℓ1-norm of the coefficients in the linear combination under an adaptive Dantzig constraint coming from sharp concentration inequalities. This allows to consider a wide class of dictionaries. Under local or global structure assumptions, oracle inequalities are derived. These theoretical results are transposed...

Adaptive density estimation for clustering with gaussian mixtures

C. Maugis-Rabusseau, B. Michel (2013)

ESAIM: Probability and Statistics

Gaussian mixture models are widely used to study clustering problems. These model-based clustering methods require an accurate estimation of the unknown data density by Gaussian mixtures. In Maugis and Michel (2009), a penalized maximum likelihood estimator is proposed for automatically selecting the number of mixture components. In the present paper, a collection of univariate densities whose logarithm is locally β-Hölder with moment and tail conditions are considered. We show that this penalized...

Adaptive density estimation under weak dependence

Irène Gannaz, Olivier Wintenberger (2010)

ESAIM: Probability and Statistics

Assume that (Xt)t∈Z is a real valued time series admitting a common marginal density f with respect to Lebesgue's measure. [Donoho et al. Ann. Stat.24 (1996) 508–539] propose near-minimax estimators f ^ n based on thresholding wavelets to estimate f on a compact set in an independent and identically distributed setting. The aim of the present work is to extend these results to general weak dependent contexts. Weak dependence assumptions are expressed as decreasing bounds of covariance terms and are...

Adaptive estimation of a density function using beta kernels

Karine Bertin, Nicolas Klutchnikoff (2014)

ESAIM: Probability and Statistics

In this paper we are interested in the estimation of a density − defined on a compact interval of ℝ− from n independent and identically distributed observations. In order to avoid boundary effect, beta kernel estimators are used and we propose a procedure (inspired by Lepski’s method) in order to select the bandwidth. Our procedure is proved to be adaptive in an asymptotically minimax framework. Our estimator is compared with both the cross-validation algorithm and the oracle estimator using simulated...

Adaptive estimation of a quadratic functional of a density by model selection

Béatrice Laurent (2005)

ESAIM: Probability and Statistics

We consider the problem of estimating the integral of the square of a density f from the observation of a n sample. Our method to estimate f 2 ( x ) d x is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for U -statistics of order 2 due to Houdré and Reynaud.

Adaptive estimation of a quadratic functional of a density by model selection

Béatrice Laurent (2010)

ESAIM: Probability and Statistics

We consider the problem of estimating the integral of the square of a density f from the observation of a n sample. Our method to estimate f 2 ( x ) d x is based on model selection via some penalized criterion. We prove that our estimator achieves the adaptive rates established by Efroimovich and Low on classes of smooth functions. A key point of the proof is an exponential inequality for U-statistics of order 2 due to Houdré and Reynaud.

Adaptive goodness-of-fit testing from indirect observations

Cristina Butucea, Catherine Matias, Christophe Pouet (2009)

Annales de l'I.H.P. Probabilités et statistiques

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H0: f=f0, where the alternative H1is expressed with respect to 𝕃 2 -norm (i.e. has the form ψ n - 2 f - f 0 2 2 𝒞 ). Our procedure is adaptive with respect to the unknown smoothness parameterτ of f. Different testing rates (ψn) are obtained according to whether f0 is polynomially...

Adaptive tests for periodic signal detection with applications to laser vibrometry

Magalie Fromont, Céline Lévy-leduc (2006)

ESAIM: Probability and Statistics

Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l2-distance between the signal and zero so that the detection...

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