Estimation of Integral Functionals of a Density

By Lucien Birgé, Pascal Massart. 1995

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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).”
Estimating Divergence Functionals and the Likelihood Ratio by Convex Risk Minimization (2008, neutral): “08 53”
MINIMAX ESTIMATION OF A FUNCTIONAL ON A STRUCTURED (2016, background): “Our second order estimators are related to work on quadratic functionals, or functionals that are well approximated by quadratic functionals, as in [8, 12, 3, 4, 14, 15, 6, 7].”
ADAPTIVE ESTIMATION OF A QUADRATIC FUNCTIONAL BY MODEL SELECTION (2001, methodology): “…interested reader will ﬁnd some details about the density framework in Laurent (1996), where simpler estimators of (cid:1) s (cid:1) 2 than those used by Bickel and Ritov are also introduced [see also Birg´e and Massart (1995) for minimax lower bounds concerning smooth functionals of the density].”
Estimating divergence functionals and the likelihood ratio by penalized convex risk minimization (2007, unknown): “work [9, 13] as well as the more recent work [3, 4, 12].”
Estimating nonquadratic functionals of a density using Haar wavelets (1996, background): “It seems more fruitful, at this stage, to extend Ž .the construction of Birge and Massart 1995 since, in addition, it applies to´ general functionals.”
Quarticity and other functionals of volatility: Efficient estimation (2012, background): “, [2, 3]), and with an asymptotic variance always smaller than if one uses the first method.”
Towards optimal doubly robust estimation of heterogeneous causal effects (2020, background): “…s/d is slower than the usual functional estimation rate n − 4 s/ (4 s + d ) , which is minimax optimal for the ATE if the co-variate density is smooth enough [Robins et al., 2009b], and also occurs for simpler functionals like the expected density [Bickel and Ritov, 1988, Birgé and Massart, 1995].”
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.”
On Estimating L 22 Divergence (2014, methodology): “Our estimator is based on a line of work studying the estimation of integral functionals of a density in the nonparametric setting [2, 3, 5, 8, 10].”
On Estimating $L_2^2$ Divergence (2014, methodology): “For example, Birge and Massart [3] establish lower bounds on estimating integral functionals of a single density, while Krishnamurthy et al.”
Semiparametric Minimax Rates (2009, background): “2007/04/13 file: minimax_final.tex date: August 6, 2009”
Semiparametric doubly robust targeted double machine learning: a review (2022, background): “Then the expected density is This functional is a staple of the classical functional estimation literature and arises in tuning parameter selection in density estimation [Bickel and Ritov, 1988, Birgé and Massart, 1995].”