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 absolutely continuous Annals of Statistics asymptotic b₁ Bosq bound of Theorem Butzer and Nessel Cauchy-Schwarz inequality Chapter classes of densities compact support concludes the proof constant cos(ix defined Deheuvels density estimate density f detector Devroye distribution dominated convergence theorem example finite follows Fourier series ƒ and g H₁ Hermite series estimate histogram estimate inequality K₁ L₁ error L₂ Lebesgue Lemma 22 Let f lim inf lim sup lower bound Mathematical Statistics minimax nonnegative obtain optimal orthogonal series orthogonal series estimate orthonormal system P₁ parameter pointwise convergence positive numbers probability density proof of Lemma proof of Theorem random variables rate of convergence sample satisfying Section sequence singular integral estimate Sm(f Sm(ƒ smoothing factor term transformed triangular density trigonometric series estimate uniform upper bound values X₁ Y₁