Displaying similar documents to “The law of the iterated logarithm for the multivariate kernel mode estimator”

Towards a universally consistent estimator of the Minkowski content

Antonio Cuevas, Ricardo Fraiman, László Györfi (2013)

ESAIM: Probability and Statistics

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We deal with a subject in the interplay between nonparametric statistics and geometric measure theory. The measure () of the boundary of a set  ⊂ ℝ (with  ≥ 2) can be formally defined, a simple limit, by the so-called Minkowski content. We study the estimation of () from a sample of random points inside and outside . The sample design assumes that, for each sample point, we know (without error) whether or not that point belongs to . Under this design we...

Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory

Elena Di Bernardino, Thomas Laloë, Véronique Maume-Deschamps, Clémentine Prieur (2013)

ESAIM: Probability and Statistics

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This paper deals with the problem of estimating the level sets () =  {() ≥ }, with  ∈ (0,1), of an unknown distribution function on ℝ . A plug-in approach is followed. That is, given a consistent estimator of , we estimate () by () =  { () ≥ }. In our setting, non-compactness property is required for the level sets to estimate. We state consistency results with respect to the Hausdorff distance and the volume of the symmetric...

Model selection and estimation of a component in additive regression

Xavier Gendre (2014)

ESAIM: Probability and Statistics

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Let  ∈ ℝ be a random vector with mean and covariance matrix where is some known  × -matrix. We construct a statistical procedure to estimate as well as under moment condition on or Gaussian hypothesis. Both cases are developed for known or unknown . Our approach is free from any prior assumption on and is based on non-asymptotic model selection methods....

Nonparametric estimation of the derivatives of the stationary density for stationary processes

Emeline Schmisser (2013)

ESAIM: Probability and Statistics

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In this article, our aim is to estimate the successive derivatives of the stationary density of a strictly stationary and -mixing process (). This process is observed at discrete times  = 0 . The sampling interval can be fixed or small. We use a penalized least-square approach to compute adaptive estimators. If the derivative belongs to the Besov space B 2 , ∞ α B 2 , ∞ α , then our estimator converges at rate (). Then we consider a diffusion...

Expansions for Repeated Integrals of Products with Applications to the Multivariate Normal

Christopher S. Withers, Saralees Nadarajah (2011)

ESAIM: Probability and Statistics

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We extend Leibniz' rule for repeated derivatives of a product to multivariate integrals of a product. As an application we obtain expansions for ( < < ) for ~ (0,) and for repeated integrals of the density of . When > 0 in the expansion for ( < ) reduces to one given by [H. Ruben B 68 (1964) 3–11]. in terms of the moments of (0, ). This is shown to be a special case of an expansion...

Estimation in autoregressive model with measurement error

Jérôme Dedecker, Adeline Samson, Marie-Luce Taupin (2014)

ESAIM: Probability and Statistics

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Consider an autoregressive model with measurement error: we observe = + , where the unobserved is a stationary solution of the autoregressive equation = ( ) + . The regression function is known up to a finite dimensional parameter to be estimated. The distributions of and are unknown and...

Adaptive non-asymptotic confidence balls in density estimation

Matthieu Lerasle (2012)

ESAIM: Probability and Statistics

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We build confidence balls for the common density of a real valued sample . We use resampling methods to estimate the projection of onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all  ≥ 2 and the balls are adaptive over a collection of linear spaces.

Adaptive non-asymptotic confidence balls in density estimation

Matthieu Lerasle (2012)

ESAIM: Probability and Statistics

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We build confidence balls for the common density of a real valued sample . We use resampling methods to estimate the projection of onto finite dimensional linear spaces and a model selection procedure to choose an optimal approximation space. The covering property is ensured for all  ≥ 2 and the balls are adaptive over a collection of linear spaces.

Cramér type moderate deviations for Studentized U-statistics

Tze Leng Lai, Qi-Man Shao, Qiying Wang (2011)

ESAIM: Probability and Statistics

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Let be a Studentized U-statistic. It is proved that a Cramér type moderate deviation ( ≥ )/(1 − Φ()) → 1 holds uniformly in ∈ [0, ( )) when the kernel satisfies some regular conditions.

Penalization versus Goldenshluger − Lepski strategies in warped bases regression

Gaëlle Chagny (2013)

ESAIM: Probability and Statistics

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This paper deals with the problem of estimating a regression function , in a random design framework. We build and study two adaptive estimators based on model selection, applied with warped bases. We start with a collection of finite dimensional linear spaces, spanned by orthonormal bases. Instead of expanding directly the target function on these bases, we rather consider the expansion of  =  ∘ , where is the cumulative distribution function of the design, following...

Survival probabilities of autoregressive processes

Christoph Baumgarten (2014)

ESAIM: Probability and Statistics

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Given an autoregressive process of order (  =   + ··· +   +  where the random variables , ,... are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a constant barrier up to time (survival or persistence probability). Depending on the coefficients ,...,...

Means in complete manifolds: uniqueness and approximation

Marc Arnaudon, Laurent Miclo (2014)

ESAIM: Probability and Statistics

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Let be a complete Riemannian manifold,  ∈ ℕ and  ≥ 1. We prove that almost everywhere on  = ( ,, ) ∈  for Lebesgue measure in , the measure μ ( x ) = N ∑ k = 1 N x k μ ( x ) = 1 N ∑ k = 1 N δ x k has a unique–mean (). As a consequence, if  = ( ,, ) is a -valued random variable with absolutely continuous law, then almost surely (()) has a unique –mean. In particular if ( ...

Hydrodynamic limit of a d-dimensional exclusion process with conductances

Fábio Júlio Valentim (2012)

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

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Fix a polynomial of the form () = + ∑2≤≤    =1 with (1) gt; 0. We prove that the evolution, on the diffusive scale, of the empirical density of exclusion processes on 𝕋 d , with conductances given by special class of functions, is described by the unique weak solution of the non-linear parabolic partial differential equation = ∑    ...

Universality in the bulk of the spectrum for complex sample covariance matrices

Sandrine Péché (2012)

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

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We consider complex sample covariance matrices = (1/)* where is a × random matrix with i.i.d. entries , 1 ≤ ≤ , 1 ≤ ≤ , with distribution . Under some regularity and decay assumptions on , we prove universality of some local eigenvalue statistics in the bulk of the spectrum in the limit where → ∞ and lim→∞ / = for any real number ∈ (0, ∞).

Upper large deviations for maximal flows through a tilted cylinder

Marie Theret (2014)

ESAIM: Probability and Statistics

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We consider the standard first passage percolation model in ℤ for  ≥ 2 and we study the maximal flow from the upper half part to the lower half part (respectively from the top to the bottom) of a cylinder whose basis is a hyperrectangle of sidelength proportional to and whose height is () for a certain height function . We denote this maximal flow by (respectively ). We emphasize the fact that the cylinder may be tilted. We look at the probability that...

Pointwise constrained radially increasing minimizers in the quasi-scalar calculus of variations

Luís Balsa Bicho, António Ornelas (2014)

ESAIM: Control, Optimisation and Calculus of Variations

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We prove of vector minimizers () =  (||) to multiple integrals ∫ ((), |()|)  on a  ⊂ ℝ, among the Sobolev functions (·) in + (, ℝ), using a  : ℝ×ℝ → [0,∞] with (·) and . Besides such basic hypotheses, (·,·) is assumed to satisfy also...