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Introduction to Nonparametric Estimation

Add to basket. Weak Convergence and Empirical Processes A. Functional Data Analysis J. Theory of Statistics Mark J. Regression Modeling Strategies Jr. Principal Component Analysis Ian T. Nonlinear Time Series Jianqing Fan.

Back cover copy Methods of nonparametric estimation are located at the core of modern statistical science. The aim of this book is to give a short but mathematically self-contained introduction to the theory of nonparametric estimation. The emphasis is on the construction of optimal estimators; therefore the concepts of minimax optimality and adaptivity, as well as the oracle approach, occupy the central place in the book.

This is a concise text developed from lecture notes and ready to be used for a course on the graduate level. The main idea is to introduce the fundamental concepts of the theory while maintaining the exposition suitable for a first approach in the field. Therefore, the results are not always given in the most general form but rather under assumptions that lead to shorter or more elegant proofs.

The book has three chapters.

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Chapter 1 presents basic nonparametric regression and density estimators and analyzes their properties. Chapter 2 is devoted to a detailed treatment of minimax lower bounds. Chapter 3 develops more advanced topics: Pinsker's theorem, oracle inequalities, Stein shrinkage, and sharp minimax adaptivity. Table of contents Nonparametric estimators. Review Text From the reviews: "The book is meant to be an introduction to the rich theory of nonparametric estimation through some simple models and examples. The detailed proofs given in the book will help the interested reader to understand the subject better.

This well written book will be welcomed by all those interested in learning the presented concepts. The author should be complimented for a good treatise with detailed proofs of several important results in nonparametric estimation theory. A short and rigorous introduction to minimax results for estimators of densities and regression functions from independent observations. Each of the three chapters ends with a section containing detailed biographical notes and a section with exercises complementing and illustrating the main results.

A survey of more recent work on nonparametric least squares estimators can 1. These estimators have nice MISE properties for the regression model with random design where the study of local polynomial estimators is more involved and needs additional assumptions.

For the connection between Tikhonov regularization and spline smoothing we refer to the books by Eubank and Wahba An analysis of the convergence rates of spline estimators can be found, for example, in Speckman , and Golubev and Nussbaum Johnstone Stein developed a method of unbiased estimation of the risk for a rather general class of estimators in Gaussian shift models cf.

Section 3. The Cp -criterion is due to Mallows There is a whole family of closely related criteria. Akaike We mention here two particular cases of the AIC. In the context of Section 1. Then the AIC coincides with the Cp -criterion. Note that the paper of Akaike does not mention this fact. Polyak and Tsybakov They can be compared with the Cp -criterion 1. These criteria can be extended to general linear estimators of regression. For example, in the notation of Section 1. The importance of these models is motivated by the equivalence arguments that were, however, not properly formalized until the late s.

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Le Cam and Yang Brown and Low give a rigorous proof of the equivalence of nonparametric regression and Gaussian white noise models. An extension covering the multivariate case and random design regression was recently obtained by Reiss More recent references on the equivalence of experiments are Brown et al.

Find the maximum order of the Silverman kernel. Show Exercise 1. Consider the nonparametric regression model under Assumption A. Prove that the minimization problem 1. The upper bounds established in Chapter 1 imply that there exists a constant C 2. Any sequence belonging to the class of equivalent sequences can be taken as an optimal rate.

Traditionally, the power sequences are convenient for use, e. It follows from 2. Conclusion: In order to obtain lower bounds as in 2. When no ambiguity is caused, dP0a dP0a we will use a short notation for the Radon—Nikodym derivative X. In this case, the random variable dP0 X dP1 is called the likelihood ratio. In other words, the closer P0 is to P1 , the greater is the lower bound given in Theorem 2. Observe that this lower bound is not always sharp. In another extreme case, the measures P0 and P1 are mutually singular and Theorem 2.

The choice of the hypotheses is indeed very important, as illustrated by the following example. Example 2. It follows that the observations Y1 ,.

A Short Course on Nonparametric Curve Estimation

Indeed, we will see later cf. Corollary 2. This explains the symbolic notation on the right hand side of 2. The following properties are straightforward. Indeed, these properties follow from the next lemma. Lemma 2.

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The following lemma establishes a link between the total variation distance and the Hellinger distance. The equality in 2. In order to prove the second inequality in 2. It can be improved as stated in the following lemma. From 2. This yields 2.

Introduction to Nonparametric Estimation by Alexandre B. Tsybakov

Inequality 2. A nontrivial extension to larger Kullback divergences is obtained using the following lemma. From Lemmas 2. This condition is quite general but not always easy to check. Therefore, other bounds on the minimax probability of error for two hypotheses are often used, based on the distances or divergences between P0 and P1.

Some of them are given in the following theorem. Theorem 2. Next, Lemma 2. From i and Lemma 2. The idea of the proof of Theorem 2. It is often easier to check the condition on the Kullback divergence than 2. That is why the Hellinger version is more convenient in certain cases.

An example is given in Exercise 2. By Corollary 2. We will also suppose in this section that Assumption LP2 of Chapter 1 holds. By the general scheme of Section 2. In order to obtain 2. Observe that Pj the distribution of Y1 ,. Then, by 2. By part iii of Theorem 2. This yields the following result. Exercise 2. Apply now the Kullback version of Theorem 2. Proposition 2. Then 2. We now derive analogs of Theorem 2. We apply Proposition 2.

Again, we apply Proposition 2.