Estimation of integrated squared density derivatives

By Brian Aldershof. 1991

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Citations in the corpus

A Kernelized Stein Discrepancy for Goodness-of-fit Tests (2016, positive): “…of φ q ( x ) (which only depends on s q ) under sample { x i } ∼ p , the second term is more difﬁcult to estimate, since it depends on the score function s p ( x ) of the unknown p ( x ) , and hence requires a kernel density estimator for p ( x ) ; see Hall & Marron (1987); Birge & Massart (1995).”
Empirical estimation of entropy functionals with confidence (2010, background): “These include consistent estimators based on entropic graphs [19, 36], gap estimators [43], nearest neighbor distances [17, 26, 29, 45], kernel density plug-in estimators [1, 11, 3, 18, 4, 16], Edgeworth approximations [21], convex risk minimization [35] and orthogonal projections [25].”
Nonquadratic estimators of a quadratic functional (2005, background): “Early results on this and related problems were given in [3, 11, 15, 17, 18, 19].”
Bandwidth Selection for Recursive Kernel Density Estimators Defined by Stochastic Approximation Method (2014, unknown): “…𝐾 𝑏 satisfies assumption ( A 1) , (𝑏 𝑛 ) , and assumption ( A 2)( ii ) and 𝑓 satisfies assumption ( A 3) To minimize the MISE of ̃𝐼 1 , the bandwidth (𝑏 𝑛 ) must equal then one has × 𝑛 −4/5 + 𝑜 (𝑛 −4/5 ) , × 𝑛 −8/5 + 𝑜 (𝑛 −8/5 ) , Furthermore, to estimate 𝐼 2 , we use the…”
Density Estimation (2004, unknown): “In particular, Hall and Marron (1987a) showed that the smallest possible relative error for any data based bandwidth ĥ is ĥ ĥISE − 1= Op ( n−1/10 ) .”
Root n consistent density estimators for sums of independent random variables (2004, background): “For quadratic functionals R f(k)(x)2dx see Hall and Marron (1987) and Bickel and Ritov (1988).”
A simple adaptive estimator of the integrated square of a density (2008, background): “Nickl introduced in Hall and Marron (1987), Tn(hn) := 2 n(n− 1)hn ∑”
Multivariate plug-in bandwidth selection with unconstrained pilot bandwidth matrices (2010, methodology): “In this sense, our estimator is a multivariate generalization of the one studied in Jones and Sheather (1991) (see also Hall and Marron 1987). However, notice that this generalization to the multivariate case for general d and unconstrained bandwidth matrices is far from trivial, as Wand and Jones (”
Goodness-of-fit testing and quadratic functional estimation from indirect observations (2006, background): “In direct models, such a kernel based estimator can be found in Hall and Marron [16].”
The Nearest Neighbor Information Estimator is Adaptively Near Minimax Rate-Optimal (2017, background): “We do not attempt to survey the extensive literature in this area, but instead refer to the interested reader to, e.g., [24, 5, 12, 16, 6, 32, 37, 47, 8, 9, 54] and the references therein.”
Donsker-type theorems for nonparametric maximum likelihood estimators (2007, background): “This improves substantially upon related work by [40], and can be applied, in particular, to the integral functionals considered in [2,5,15,17,18].”
Chi-square oracle inequalities (2000, background): “See, for example Hall and Marron (1987); Bickel and Ritov (1988); Hall and Johnstone (1992) Ibragimov et al. (1986); Donoho and Nussbaum (1990); Fan (1991); Birgé and Massart (1995); Laurent (1996); Gayraud and Tribouley (1999); Laurent and Massart (1998) and the references therein.”
Estimation of integrated squared density derivatives from a contaminated sample (2002, methodology): “For the variance of the estimator, expression (A.15) in the proof of proposition 2 becomes var.θ̂r/ = R.fX/ πn2h4r+1 ∫ t4r|ϕK.t/|4 dt + 4n−1 {∫ f .2r/ X .u/ 2 fX.u/ du− θ2r } + o.n−2h−4r−1/; which coincides with lemma 3.1(a) of Hall and Marron (1987).”
Bandwidth Selection in Kernel Density Estimation (2010, unknown): “In this chapter we suggest a new data-driven bandwidth selector, which is a modified version of the direct-plug in selector from Hall and Marron (1987), [8], and Sheather and Jones (1991), [22].”
Estimating nonquadratic functionals of a density using Haar wavelets (1996, background): “Ž .skii and Ibragimov 1978 , Hall and Marron 1987 , Bickel and Ritov 1988 , Ž .”
Uniform in Bandwidth Estimation of Integral Functionals of the Density Function (2008, methodology): “…and to uniformity in bandwidth a result of Giné & Nickl (2008b) who show, in particular, that, in order to achieve the bias improvement of Bickel & Ritov (1988) over Hall & Marron (1987) for the estimation of the integral of the square of a density, it is not necessary to combine two estimators.”
On the estimation of the marginal density of a moving average process (2000, methodology): “Along the nonparametric curve estimation literature, several authors have been concerned with the estimation of this parameter, finding √ n-consistent estimators based, for instance, on the kernel method; cf. e.g., Veraverbeke (1985), Hall & Marron (1987), Jones & Sheather (1991) or Wu (1995).”