By Alexandre B. Tsybakov

Methods of nonparametric estimation can be found on the center of recent statistical technological know-how. the purpose of this ebook is to provide a quick yet mathematically self-contained advent to the idea of nonparametric estimation. The emphasis is at the development of optimum estimators; accordingly the recommendations of minimax optimality and adaptivity, in addition to the oracle strategy, occupy the primary position within the book.

This is a concise textual content constructed from lecture notes and able to be used for a direction at the graduate point. the most notion is to introduce the basic recommendations of the speculation whereas keeping the exposition compatible for a primary process within the box. for that reason, the consequences usually are not continually given within the such a lot basic shape yet fairly lower than assumptions that bring about shorter or extra dependent proofs.

The ebook has 3 chapters. bankruptcy 1 provides easy nonparametric regression and density estimators and analyzes their houses. bankruptcy 2 is dedicated to a close therapy of minimax reduce bounds. bankruptcy three develops extra complicated issues: Pinsker's theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity.

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**Example text**

The functions CV (h) (solid line), MISE(h) − p2 (dashed line) and their minimizers hCV , hid . 58) K(hω)dω −2K(hω) + K 2 (hω) 1 − = 1 n 2 4πK(0) , nh where φn is the empirical characteristic function and we have used that, by the inverse Fourier transform, K(ω)dω = 2πK(0). 5 Nonparametric regression. The Nadaraya–Watson estimator = 2π 1 − 1 n MISE(h) − p2 31 . ˜ Therefore, the functions h → Ep (J(h)) and h → MISE(h) have the same minimizers. In the same spirit as above we now approximate the unknown minimizers of MISE(·) by ˜ = arg min J(h).

More generally, the Fourier transform is deﬁned in a standard way for any g ∈ L2 (R) using the fact that L1 (R) ∩ L2 (R) is dense in L2 (R). 33) is true for any g ∈ L2 (R). For example, if K is the sinc kernel, a version of its Fourier transform has the form F[K](ω) = I(|ω| ≤ 1). The Fourier transform of g ∈ L2 (R) is deﬁned up to an arbitrary modiﬁcation on a set of Lebesgue measure zero. This will not be further recalled, in particular, all equalities between Fourier transforms will be understood in the almost everywhere sense.

N. 7 Projection estimators 53 n ∗∗ Q(Xi )Wni (x) = Q(x) i=1 for all x ∈ [0, 1]. 7 and from the deﬁnition of Wni . The next result gives the bias and the squared risk of the estimators θˆj . 87) satisfy (i) E(θˆj ) = θj + αj , (ii) E[(θˆj − θj )2 ] = σξ2 /n + αj2 , where αj = 1 n 1 ≤ j ≤ n − 1, n 1 f (i/n)ϕj (i/n) − f (x)ϕj (x)dx. 0 i=1 Proof. We have 1 θˆj = n n Yi ϕj (i/n) = i=1 Therefore E(θˆj ) = 1 n 1 n n n f (i/n)ϕj (i/n) + i=1 ξi ϕj (i/n) . i=1 n f (i/n)ϕj (i/n) = αj + θj . i=1 Then E[(θˆj − θj )2 ] = E[(θˆj − E(θˆj ))2 ] + (E(θˆj ) − θj )2 = E[(θˆj − E(θˆj ))2 ] + αj2 .