By Alexandre B. Tsybakov

It is a concise textual content constructed from lecture notes and able to be used for a path at the graduate point. the most thought is to introduce the basic ideas of the speculation whereas preserving the exposition appropriate for a primary technique within the box. accordingly, the implications will not be continually given within the such a lot normal shape yet particularly less than assumptions that result in shorter or extra stylish proofs. The e-book has 3 chapters. bankruptcy 1 provides simple nonparametric regression and density estimators and analyzes their houses. bankruptcy 2 is dedicated to a close therapy of minimax decrease bounds. bankruptcy three develops extra complex issues: Pinsker’s theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity. This e-book may be valuable for researchers and grad scholars attracted to theoretical elements of smoothing suggestions. Many vital and worthy effects on optimum and adaptive estimation are supplied. As one of many major mathematical statisticians operating in nonparametrics, the writer is an expert at the topic.

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Additional resources for Introduction to Nonparametric Estimation (Springer Series in Statistics)

Example text

K0 (ω) = min(1, max(K(ω), 0)). Clearly, K0 (ω) ≤ K(ω) , 1 − K0 (ω) ≤ 1 − K(ω) , ∀ ω ∈ R. 48) Since K ∈ L2 (R), we get that K0 ∈ L2 (R). Therefore, there exists a function K0 ∈ L2 (R) with the Fourier transform K0 . Since K is symmetric, the Fourier transforms K and K0 are real-valued, so that K0 is also symmetric. 48) and the fact that φ(ω) ≤ 1 for any characteristic function φ, we get Jn (K, h, φ) − Jn (K0 , h, φ) 1 2 2 2 = φ(ω) dω 1 − K(hω) − 1 − K0 (hω) 2π 1 2 2 2 dω (1 − φ(ω) ) K(hω) − K0 (hω) + n ≥ 0.

It is sufficient to prove that for all v ∈ R +1 41 satisfying v = 0 we have v T Bv > 0. Clearly, v T Bv = (v T U (u))2 K(u)du ≥ 0. If there exists v = 0 such that [v T U (u)]2 K(u) du = 0, then v T U (u) = 0 for almost all u on the set {u : K(u) > 0}, which has a positive Lebesgue measure by assumption of the lemma. But the function u → v T U (u) is a polynomial of degree ≤ which cannot be equal to zero except for a finite number of points. Thus, we come to a contradiction showing that [v T U (u)]2 K(u) du = 0 is impossible for v = 0.

Nonparametric regression with random design Let (X, Y ) be a pair of real-valued random variables such that E|Y | < ∞. The function f : R → R defined by f (x) = E(Y |X = x) is called the regression function of Y on X. Suppose that we have a sample (X1 , Y1 ), . . d. pairs of random variables having the same distribution as (X, Y ). We would like to estimate the function f from the data (X1 , Y1 ), . . , (Xn , Yn ). The nonparametric approach only assumes that f ∈ F, where F is a given nonparametric class.

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