Combinatorial Methods in Density Estimation

Devroye, Luc; Lugosi, Gábor

doi:10.1007/978-1-4613-0125-7

Cited by 509 publications

(680 citation statements)

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“…Although a more recent summary of Le Cam's point of view on this subject appeared in Le Cam (1997), these ideas remained widely unknown since then (with the exception of Groeneboom, 1986) and, to our knowledge, nobody (including the present author) tried to apply the method to other stochastic frameworks like the Gaussian one, that we consider here, or to extend it. Related points of view about the relationships between the minimax risk and the metric structure of the parameter space are to be found in Yatracos (1985), Yang and Barron (1999) and Devroye and Lugosi (2001).…”

Section: Some Historical Remarksmentioning

confidence: 99%

“…Such risk bounds can be found in Donoho, Johnstone, Kerkyacharian and Picard (1996). The more recent improved results of Kerkyacharian and Picard (2000) do not apply to the L 1 -loss.…”

Section: Two Illustrations With Independent Variablesmentioning

confidence: 99%

“…This result, which allows arbitrary families of partitions and requires no assumption at all on P s should be compared, for instance, with Castellan (1999 and and Devroye and Lugosi (2001). Let us now focus on the case X = [0, 1) and µ is Lebesgue measure.…”

Section: Partition Selection For Histogramsmentioning

confidence: 99%

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Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Birgé¹

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

189

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This paper is devoted to the description and study of a family of estimators, that we shall call T -estimators (T for tests), for minimax estimation and model selection. Their construction is based on former ideas about deriving estimators from some families of tests due to Le Cam (1973 and1975) and Birgé (1983, 1984a and and about complexity based model selection from Barron and Cover (1991).It is well-known that maximum likelihood estimators or, more generally, minimum contrast estimators do suffer from various weaknesses, and their penalized versions as well. In particular they are not robust and they require restrictive assumptions on both the models and the underlying parameter for the estimators to work correctly. Our method, which derives an estimator from many simultaneous tests between some probability balls in a suitable metric space, tends to solve many of these difficulties. Its robustness properties allow to deal with minimax estimation and model selection in a unified way, since bounding the minimax risk amounts to perform the method with a single, well-chosen, model. This results in simple bounds for the minimax risk solely based on some metric properties of the parameter space. Moreover the method applies to various statistical frameworks (we shall concentrate here on the i.i.d. and the Gaussian sequence settings only) and can handle essentially all types of models, linear or not, parametric and non-parametric, simultaneously. From these viewpoints, it is much more flexible than traditional methods. In particular, we shall be able to derive some adaptation results for density estimation over Besov balls that do not seem to be accessible to classical methods. The counterpart for these nice properties is the very high computational complexity of our construction which makes our estimators look more like theoretical than practical tools. 0 AMS 1991 subject classifications. Primary 62F35; secondary 62G05.

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Section: Some Historical Remarksmentioning

confidence: 99%

“…Such risk bounds can be found in Donoho, Johnstone, Kerkyacharian and Picard (1996). The more recent improved results of Kerkyacharian and Picard (2000) do not apply to the L 1 -loss.…”

Section: Two Illustrations With Independent Variablesmentioning

confidence: 99%

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Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Birgé¹

2006

Annales de l'Institut Henri Poincare (B) Probability and Statis

189

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“…Strictly speaking, statistical optimality such as in the filter procedure cannot be guaranteed anymore. We compare 3 induction algorithms to search for the best subset of features, in terms of discriminating normal controls from stroke patients: knearest neighbor, least squares support vector machines (LSSVM, Suykens et al 2002) and a Bayesian classifier with kernel density estimation (KDE), for KDE see (Devroye et al 2001). We use Gaussian kernels and use the maximum likelihood cross-validation method for kernel bandwidth estimation in KDE.…”

Section: Feature Subset Selectionmentioning

confidence: 99%

Posterior probability profiles for the automated assessment of the recovery of patients with stroke from activity of daily living tasks

Dijck

Vaerenbergh

Hulle

2009

Artificial Intelligence in Medicine

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Assessing recovery from stroke has been so far a time consuming procedure in which highly trained clinicians are required. This paper proposes a mechatronic platform which measures low forces and torques exerted by subjects. Class posterior probabilities are used as a quantitative and statistically sound tool to assess motor recovery from these force and torque measurements. The performance of the patients is expressed in terms of the posterior probability to belong to the class of normal subjects. The mechatronic platform together with the class posterior probabilities enables to automate motor recovery assessment without the need for highly trained clinicians. It is shown that the class posterior probability profiles are highly correlated, r 0.8, with the well-established Fugl-Meyer scale assessment in motor recovery. These results have been obtained through careful feature subset selection procedures in order to prune the large feature set being generated. The overall approach is general and can be applied to many other health monitoring systems where different categories (diseased vs. healthy) can be identified.

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“…(6) by changing 1 {x i / ∈C} into (ρn − fn(xi)) 1 {fn(x i )<ρn} (this is the standard hinge loss which is used in, e.g., SVMs). Similar smoothing also arises in kernel density estimation (see, e.g., [8,Chapter 9]). The modified excess mass problem writes:…”

Section: The Kernel Approach and Connection With 1-class Support Vectmentioning

confidence: 87%

Estimation of Minimum Measure Sets in Reproducing Kernel Hilbert Spaces and Applications.

Davy

Desobry

Canu

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

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Minimum measure sets (MMSs) summarize the information of a (single-class) dataset. In many situations, they can be preferred to estimated probability density functions (pdfs): they are strongly related to pdf level sets while being much easier to estimate in large dimensions.The main contribution of this paper is a theoretical connection between MMSs and one class Support Vector Machines. This justifies the use of one-class SVMs in the following applications: novelty detection (we give explicit convergence rate) and change detection.

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Combinatorial Methods in Density Estimation

Cited by 509 publications

References 0 publications

Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Model selection via testing: an alternative to (penalized) maximum likelihood estimators

Posterior probability profiles for the automated assessment of the recovery of patients with stroke from activity of daily living tasks

Estimation of Minimum Measure Sets in Reproducing Kernel Hilbert Spaces and Applications.

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