Estimation des densit�s: risque minimax

Bretagnolle, Jean; Huber, Catherine

doi:10.1007/bf00535278

Cited by 175 publications

(32 citation statements)

References 16 publications

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“…Accuracy of the set estimates will be measured by the distances d f g (G G 0 ) a n d d 4 . This is what we assume in the sequel, without deriving explicitly the restrictions on these parameters.…”

Section: The Resultsmentioning

confidence: 99%

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Smooth discrimination analysis

Mammen¹,

Tsybakov²

1999

Ann. Statist.

263

260

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Discriminant analysis for two d a t a s e t s i n IR d with probability densities f and g can be based on the estimation of the set G = fx : f(x) g(x)g.We consider applications where it is appropriate to assume that the region G has a smooth boundary. In particular, this assumption makes sense if discriminant analysis is used as a data analytic tool. We discuss optimal rates for estimation of G. 1991 AMS: primary 62G05 , secondary 62G20

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Section: The Resultsmentioning

confidence: 99%

“…We use Assouad's lemma see Bretagnolle and Huber (1979) and Assouad (1983)]. For our purposes it will be more convenient to apply the version of this lemma stated in Korostelev and Tsybakov (1993), which is adapted to the problem of estimation of sets.…”

mentioning

confidence: 99%

Smooth discrimination analysis

Mammen¹,

Tsybakov²

1999

Ann. Statist.

263

260

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“…This is certainly the best rate of convergence in probability implied by the results in Wahba [13], Bretagnolle and Huber [4], and Dacunha-Castelle [6], pages 85-86. Finally, using Corollary 6 in conjunction with Sobolov's inequality (see Agmon [2], page 32) we obtain COROLLARY 7.…”

Section: Under Assumptions A2 and B If R And Q Are As In Theorem 5 mentioning

confidence: 83%

“…(b) Under Assumption A2, r is the unique minimizer of (2) among all functions r satisfying (3) and ( 4 ) r = r PROOF. We have by (1),…”

Section: ~' Df~=n-15~' (X~)'=lmentioning

confidence: 96%

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Cox

1985

Ann Inst Stat Math

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SummaryGiven a random sample of size n from a density f0 on the real line satisfying certain regularity conditions, we propose a nonparametric estimator for rThe estimate is the minimizer of a quadratic functional of the form ~J(r [r162where ~>0 is a smoothing parameter, J(.) is a roughness penalty, and F, is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which is related to spline functions. A more complete study of the case J(r f [d~r is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness limit (~--~c~). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem is simple in comparison with the analogous penalized likelihood estimators.

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“…The associated kernel was defined for the one dimensional case in Bretagnolle and Huber (1979) and for the multivariate case in Holmstro m and Klemela (1992). The following lemma gives an expansion of convolution.…”

Section: The Bias Of the Estimatorsmentioning

confidence: 99%

Estimation of Densities and Derivatives of Densities with Directional Data

Klemelä

2000

Journal of Multivariate Analysis

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Estimating the density function of a random vector taking values on the d-dimensional unit sphere is considered. Also the estimation of the Laplacian of the density and estimation of other types of derivatives is considered. Fast convergence rate theory is developed for pointwise, L 1 , and L 2 error, extending some results of Hall, Watson and Cabrera (1987). It is also proved that asymptotically the plug-in method is as good as using the asymptotically optimal deterministic smoothing parameter sequence. Academic PressAMS 1991 subject classifications: 62G07, 62H11.

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Estimation des densit�s: risque minimax

Cited by 175 publications

References 16 publications

Smooth discrimination analysis

Smooth discrimination analysis

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Estimation of Densities and Derivatives of Densities with Directional Data

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