Uniform central limit theorems for kernel density estimators

Giné, Evarist; Nickl, Richard

doi:10.1007/s00440-007-0087-9

Cited by 57 publications

(102 citation statements)

References 28 publications

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“…Under the conditions in (4.8), it can be shown that h * * ≃ (n/ log n) −1/(2s+d) , which is the optimal bandwidth for the estimation of p in sup-norm, also satisfies (4.5) and, therefore, (4.6) and (4.7) hold for h = h * * . (ii) The condition on r in (4.8) coincides with the one obtained for {1 (−∞,t] : t ∈ R} in Bickel and Ritov [6] and bounded variation and Lipschitz classes with d = 1 in Giné and Nickl [15], see Remarks 7 and 8. This condition shows that for the kernel density estimator with bandwidth h * to be optimal in the weak topology (assuming ω * ≤ 1, ω K < 1 and K satisfying the conditions in Theorem 3.2), the order of the kernel has to be chosen higher by d 2 than the usual (the usual being estimating p using the kernel density estimator in L 1 -norm).…”

supporting

confidence: 61%

“…→ f dP as n → ∞, written as P n P. In fact, if nothing is known about P, then P n is probably the most appropriate estimator to use as it is asymptotically efficient and minimax in the sense of van der Vaart [37], Theorem 25.21, Several recent works (Bickel and Ritov [6], Nickl [26], Giné and Nickl [15,[17][18][19]) have shown that many popular density estimators on X = R, such as maximum likelihood estimator, kernel density estimator and wavelet estimator satisfy (1.2) if F is P-Donskerthe Donsker classes that were considered in these works are: functions of bounded variation, {1 (−∞,t] : t ∈ R}, Hölder, Lipschitz and Sobolev classes on R. In other words, these…”

Section: Introductionmentioning

confidence: 99%

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On the optimal estimation of probability measures in weak and strong topologies

Sriperumbudur¹

2016

Bernoulli

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Given random samples drawn i.i.d. from a probability measure P (defined on say, R d ), it is well-known that the empirical estimator is an optimal estimator of P in weak topology but not even a consistent estimator of its density (if it exists) in the strong topology (induced by the total variation distance). On the other hand, various popular density estimators such as kernel and wavelet density estimators are optimal in the strong topology in the sense of achieving the minimax rate over all estimators for a Sobolev ball of densities. Recently, it has been shown in a series of papers by Giné and Nickl that these density estimators on R that are optimal in strong topology are also optimal in · F for certain choices of F such that · F metrizes the weak topology, where P F := sup{ f dP: f ∈ F}. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing F to be a unit ball in a reproducing kernel Hilbert space (say FH defined over R d ), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that · F H metrizes the weak topology and the kernel density estimator (with L 1 optimal bandwidth) estimates P at dimension independent optimal rate of n −1/2 in · F H along with providing a uniform central limit theorem for the kernel density estimator.

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supporting

confidence: 61%

Section: Introductionmentioning

confidence: 99%

On the optimal estimation of probability measures in weak and strong topologies

Sriperumbudur¹

2016

Bernoulli

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“…The proof is based on ideas from the theory of smoothed empirical processes (in particular from Giné and Nickl, 2008). The main mathematical challenges consist in dealing with envelopes of the empirical process that can be as large as 1/ √ ∆ → ∞ in the high-frequency setting, and in accommodating the presence of an n-dependent Fourier multiplier m that needs to be general enough to allow for m = FK h /ϕ ∆ .…”

Section: Numerical Examplementioning

confidence: 99%

“…We will rely on the following auxiliary result, which is a modification of Theorem 3 in Giné and Nickl (2008), which in itself goes back to fundamental ideas in Giné and Zinn (1984). It is designed to allow for maximally growing envelopes of the empirical process, which is crucial in our setting to allow for minimal conditions on ∆.…”

Section: Asymptotic Equicontinuity Of the 'Critical Term'mentioning

confidence: 99%

“…It is designed to allow for maximally growing envelopes of the empirical process, which is crucial in our setting to allow for minimal conditions on ∆. Note that indeed Condition (a) only requires M n /n 1/2 → 0 instead of the more stringent condition M n /n 1/4 → 0 which was required in Theorem 3 in Giné and Nickl (2008).…”

Section: Asymptotic Equicontinuity Of the 'Critical Term'mentioning

confidence: 99%

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High-frequency Donsker theorems for Lévy measures

Nickl

Reiß

Söhl

et al. 2015

Probab. Theory Relat. Fields

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Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov-Smirnov type tests and confidence bands.MSC 2000 subject classification: Primary: 60F05; Secondary: 60G51, 62G05

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Doubly robust inference for targeted minimum loss–based estimation in randomized trials with missing outcome data

Díaz

Laan

2017

Statistics in Medicine

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Missing outcome data is one of the principal threats to the validity of treatment effect estimates from randomized trials. The outcome distributions of participants with missing and observed data are often different, which increases the risk of bias. Causal inference methods may aid in reducing the bias and improving efficiency by incorporating baseline variables into the analysis. In particular, doubly robust estimators incorporate estimates of two nuisance parameters: the outcome regression and the missingness mechanism (i.e., the probability of missingness conditional on treatment assignment and baseline variables), to adjust for differences in the observed and unobserved groups that can be explained by observed covariates.To obtain consistent estimators of the treatment effect, one of these two nuisance parameters mechanism must be consistently estimated. Such nuisance parameters are traditionally estimated using parametric models, which generally preclude consistent estimation, particularly in moderate to high dimensions. Recent research on missing data has focused on data-adaptive * corresponding author: ild2005@med.cornell.edu 1 arXiv:1704.01538v1 [stat.ME] 5 Apr 2017 estimation of the nuisance parameters in order to achieve consistency, but the large sample properties of such estimators are poorly understood. In this article we discuss a doubly robust estimator that is consistent and asymptotically normal (CAN) under data-adaptive consistent estimation of the outcome regression or the missingness mechanism. We provide a formula for an asymptotically valid confidence interval under minimal assumptions. We show that our proposed estimator has smaller finite-sample bias compared to standard doubly robust estimators. We present a simulation study demonstrating the enhanced performance of our estimators in terms of bias, efficiency, and coverage of the confidence intervals. We present the results of an illustrative example: a randomized, double-blind phase II/III trial of antiretroviral therapy in HIV-infected persons, and provide R code implementing our proposed estimators.

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Uniform central limit theorems for kernel density estimators

Cited by 57 publications

References 28 publications

On the optimal estimation of probability measures in weak and strong topologies

On the optimal estimation of probability measures in weak and strong topologies

High-frequency Donsker theorems for Lévy measures

Doubly robust inference for targeted minimum loss–based estimation in randomized trials with missing outcome data

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