Nonparametric Density Estimation: The L1 ViewWiley, 1985年1月18日 - 356 頁 The first systematic single-source examination of density estimates. It develops, from first principles, the ``natural'' theory for density estimation, L1, and shows why the classical L2 theory masks some fundamental properties of density estimates. Chapters comprehensively treat consistency, lower bounds for rates of convergence, rates of convergence in L1, the transformed kernel estimate, applications in discrimination, and estimators based on orthogonal series. All theorems are fully proven in rigorous, step-by-step detail. Additionally, the relevant recent literature is tied in with the more classic works of Parzen, Rosenblatt, and others. |
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a₁ Abou-Jaoude absolutely continuous Annals of Mathematical Annals of Statistics asymptotic b₁ Borel Borel sets bound of Theorem C₁ Chapter class of densities compact support concludes the proof defined Deheuvels densities f density estimate Devroye distribution dominated convergence theorem E(fn estimate f example Fatou's lemma finite follows Fourier series H₁ histogram estimate inequality Jensen's inequality K₁ L₁ error L₂ Lebesgue Lemma 22 Let f lim inf lim sup Lipschitz lower bound Mathematical Statistics method minimax nonnegative obtain optimal orthogonal series P₁ parameter pointwise positive numbers probability density proof of Lemma proof of Theorem random variables rate of convergence S\fn sample satisfying Section sequence singular integral estimate Sm(ƒ smoothing factor standard kernel estimate trigonometric series estimate uniform unimodal upper bound variation X₁ Y₁