Vitesse de convergence presque sûre de l'estimateur à noyau du mode

Leclerc, Joséphine; Pierre-Loti-Viaud, Daniel

doi:10.1016/s0764-4442(00)01688-8

Cited by 8 publications

(3 citation statements)

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“…In the i.i.d. case, various works such as Parzen (1962), Konakov (1973), Romano (1988) and Leclerc and Pierre-Loti-Viaud (2000) establish consistency properties of the approach under regularity assumptions. While yielding much insight into the problem, the estimator θn is hard to implement in the practice where the argmax is usually computed over a nite grid.…”

Section: Introductionmentioning

confidence: 99%

Consistency of the simple mode of a density for spatial processes

Younso

2022

Journal of Nonparametric Statistics

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We are concerned with estimating the mode of a density of a spatial process by kernel method under some dependency conditions. A simple estimate of the mode based only on data is considered. The approach consists of maximizing the density kernel estimate on data. An optimal rate of uniform consistency of the density estimate will be established. Strong consistency of the simple estimate of the mode with the rate of consistency will be investigated when data are spatially dependent by some general mixing conditions.

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Section: Introductionmentioning

confidence: 99%

Consistency of the simple mode of a density for spatial processes

Younso

2022

Journal of Nonparametric Statistics

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“…, X n with density f . This problem has been studied by many authors, see for example Parzen [8], Konakov [5], Samanta [13], Devroye [2], Romano [10], Vieu [15], Leclerc and Pierre-Loti-Viaud [6], Mokkadem and Pelletier [7] and the references therein. Mostly, the estimateθ n of θ is defined as any maximizer of f n , i.e., where f n is a kernel density estimate Rosenblatt [11], Parzen [8], Devroye [3] .…”

Section: Introductionmentioning

confidence: 99%

On the asymptotic properties of a simple estimate of the Mode

2004

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Abstract.We consider an estimate of the mode θ of a multivariate probability density f with support in R d using a kernel estimate fn drawn from a sample X1, . . . , Xn. The estimate θn is defined as any x in {X1, . . . , Xn} such that fn(x) = maxi=1,...,n fn(Xi). It is shown that θn behaves asymptotically as any maximizerθn of fn. More precisely, we prove that for any sequence (rn) n≥1 of positive real numbers such that rn → ∞ and r d n log n/n → 0, one has rn θn −θn → 0 in probability. The asymptotic normality of θn follows without further work.Mathematics Subject Classification. 62G05.

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“…In the univariate framework, upper bounds of the almost sure convergence rate of θ n are given in Eddy [5], Vieu [30] and Leclerc and Pierre-Loti-Viaud [14]. The exact strong convergence rate of the univariate kernel mode estimator is given in Mokkadem and Pelletier [16] who proved the following law of the iterated logarithm:…”

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confidence: 99%

The law of the iterated logarithm for the multivariate kernel mode estimator

Mokkadem

Pelletier

2003

ESAIM: PS

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Abstract. Let θ be the mode of a probability density and θn its kernel estimator. In the case θ is nondegenerate, we first specify the weak convergence rate of the multivariate kernel mode estimator by stating the central limit theorem for θn − θ. Then, we obtain a multivariate law of the iterated logarithm for the kernel mode estimator by proving that, with probability one, the limit set of the sequence θn − θ suitably normalized is an ellipsoid. We also give a law of the iterated logarithm for the l p norms, p ∈ [1, ∞], of θn − θ. Finally, we consider the case θ is degenerate and give the exact weak and strong convergence rate of θn − θ in the univariate framework.Mathematics Subject Classification. 62G05, 62G20, 60F05, 60F15.

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Vitesse de convergence presque sûre de l'estimateur à noyau du mode

Cited by 8 publications

References 0 publications

Consistency of the simple mode of a density for spatial processes

Consistency of the simple mode of a density for spatial processes

On the asymptotic properties of a simple estimate of the Mode

The law of the iterated logarithm for the multivariate kernel mode estimator

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