The Lower Bound of Density Estimation for Biased Data in Sobolev Spaces

2013 ◽  
Vol 765-767 ◽  
pp. 744-748
Author(s):  
Jin Ru Wang ◽  
Yuan Zhou
Keyword(s):  
Lower Bound ◽  
Density Estimation ◽  
Sobolev Spaces ◽  
Convergence Rates ◽  
Estimation Problem ◽  
Independent And Identically Distributed

In this paper, we consider the density estimation problem from independent and identically distributed (i.i.d.) biased observations. We study the lower bound of convergence rates of density estimation over Sobolev spaces WrN(NN+) under the Lp risk by using Fanos lemma.

scholarly journals On the Baire Generic Validity of the t-Multifractal Formalism in Besov and Sobolev Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-14
Author(s):  
Moez Ben Abid ◽  
Mourad Ben Slimane ◽  
Ines Ben Omrane ◽  
Borhen Halouani
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Besov Space ◽  
Sobolev Spaces ◽  
Wavelet Coefficients ◽  
Multifractal Formalism ◽  
Spectrum Of A Function ◽  
Set Of Points

The t-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce the t-spectrum of a function f from the knowledge of the (p,t)-oscillation exponent of f. The t-spectrum is the Hausdorff dimension of the set of points where f has a given value of pointwise Lt regularity. The (p,t)-oscillation exponent is measured by determining to which oscillation spaces Op,ts (defined in terms of wavelet coefficients) f belongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the (p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of the t-multifractal formalism.

Nonparametric Least Squares Mixture Density Estimation

2013 ◽  
Vol 63 (2) ◽  
Author(s):  
Chew-Seng Chee
Keyword(s):  
Least Squares ◽  
Density Estimation ◽  
Least Squares Method ◽  
Bandwidth Selection ◽  
Estimation Problem ◽  
Density Estimator ◽  
Mixing Distribution ◽  
Mixture Density ◽  
Integrated Squared Error ◽  
Nonparametric Mixture Model

In this paper, we consider using nonparametric mixtures for density estimation. The mixture density estimation problem simply reduces to the problem of estimating a mixing distribution in the nonparametric mixture model. We focus on the least squares method for mixture density estimation problem. In a simulation experiment, the performance of the least squares mixture density estimator (MDE) and the kernel density estimator (KDE) is assessed by the mean integrated squared error. The performance improvement of MDE over KDE for some common densities is achieved by using cross-validation method for bandwidth selection.

Sup-norm convergence rates for Lévy density estimation

Extremes ◽  
2016 ◽  
Vol 19 (3) ◽  
pp. 371-403 ◽  
Author(s):  
Valentin Konakov ◽  
Vladimir Panov
Keyword(s):  
Density Estimation ◽  
Convergence Rates ◽  
Norm Convergence ◽  
Lévy Density

scholarly journals BLOCK THRESHOLDING FOR A DENSITY ESTIMATION PROBLEM WITH A CHANGE-POINT

2009 ◽  
Vol 02 (04) ◽  
pp. 545-555 ◽  
Author(s):  
Christophe Chesneau
Keyword(s):  
Density Estimation ◽  
Change Point ◽  
General Framework ◽  
Rates Of Convergence ◽  
Estimation Problem ◽  
Adaptive Estimator ◽  
Function Classes ◽  
Wide Range ◽  
Optimal Rates Of Convergence ◽  
Block Thresholding

We consider a density estimation problem with a change-point. The contribution of the paper is theoretical: we develop an adaptive estimator based on wavelet block thresholding and we evaluate these performances via the minimax approach under the 𝕃p risk with p ≥ 1 over a wide range of function classes: the Besov classes, [Formula: see text] (with no particular restriction on the parameters π and r). Under this general framework, we prove that it attains near optimal rates of convergence.

A pair of complementary theorems on convergence rates in the law of large numbers

1967 ◽  
Vol 63 (1) ◽  
pp. 73-82 ◽  
Author(s):  
C. C. Heyde ◽  
V. K. Rohatgi
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

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