Displaying similar documents to “Theory of classification : a survey of some recent advances”

Probabilities of discrepancy between minima of cross-validation, Vapnik bounds and true risks

Przemysław Klęsk (2010)

International Journal of Applied Mathematics and Computer Science

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Two known approaches to complexity selection are taken under consideration: n-fold cross-validation and structural risk minimization. Obviously, in either approach, a discrepancy between the indicated optimal complexity (indicated as the minimum of a generalization error estimate or a bound) and the genuine minimum of unknown true risks is possible. In the paper, this problem is posed in a novel quantitative way. We state and prove theorems demonstrating how one can calculate pessimistic...

On the Optimality of Sample-Based Estimates of the Expectation of the Empirical Minimizer

Peter L. Bartlett, Shahar Mendelson, Petra Philips (2010)

ESAIM: Probability and Statistics

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We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result due to Bartlett and Mendelson, which...

Stochastic Inverse Problem with Noisy Simulator. Application to aeronautical model

Nabil Rachdi, Jean-Claude Fort, Thierry Klein (2012)

Annales de la faculté des sciences de Toulouse Mathématiques

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Inverse problem is a current practice in engineering where the goal is to identify parameters from observed data through numerical models. These numerical models, also called Simulators, are built to represent the phenomenon making possible the inference. However, such representation can include some part of variability or commonly called uncertainty (see [4]), arising from some variables of the model. The phenomenon we study is the...

Risk bounds for mixture density estimation

Alexander Rakhlin, Dmitry Panchenko, Sayan Mukherjee (2005)

ESAIM: Probability and Statistics

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In this paper we focus on the problem of estimating a bounded density using a finite combination of densities from a given class. We consider the Maximum Likelihood Estimator (MLE) and the greedy procedure described by Li and Barron (1999) under the additional assumption of boundedness of densities. We prove an O ( 1 n ) bound on the estimation error which does not depend on the number of densities in the estimated combination. Under the boundedness assumption, this improves the bound of Li...

Information-type divergence when the likelihood ratios are bounded

Andrew Rukhin (1997)

Applicationes Mathematicae

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The so-called ϕ-divergence is an important characteristic describing "dissimilarity" of two probability distributions. Many traditional measures of separation used in mathematical statistics and information theory, some of which are mentioned in the note, correspond to particular choices of this divergence. An upper bound on a ϕ-divergence between two probability distributions is derived when the likelihood ratio is bounded. The usefulness of this sharp bound is illustrated by several...

A comparison of automatic histogram constructions

Laurie Davies, Ursula Gather, Dan Nordman, Henrike Weinert (2009)

ESAIM: Probability and Statistics

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Even for a well-trained statistician the construction of a histogram for a given real-valued data set is a difficult problem. It is even more difficult to construct a fully automatic procedure which specifies the number and widths of the bins in a satisfactory manner for a wide range of data sets. In this paper we compare several histogram construction procedures by means of a simulation study. The study includes plug-in methods, cross-validation, penalized maximum likelihood and the...