Klaus Ziegler

On nonparametric kernel estimation of the mode of the regression function in the random design model

In the nonparametric regression model with random design, where the regression function m is given by m(x) = {\open E}(Y\mid X = x), estimation of the location \theta ( mode ) of a unique maximum of m by the location \hat{\theta} of a maximum of the Nadaraya-Watson kernel estimator \hat{m} for the curve m is considered. Within this setting, we obtain consistency and asymptotic normality results for \hat{\theta} under very mild assumptions on m , the design density g of X and the kernel K . The bandwidths being considered in the present work are data-dependent of the type being generated by plug-in methods. The estimation of the size of the maximum is also considered as well as the estimation of a unique zero of the regression function. Applied to the estimation of the mode of a density, our methods yield some improvements on known results. As a by-product, we obtain some uniform consistency results for the (higher) derivatives of the Nadaraya-Watson estimator with a certain additional uniformity in the bandwiths. The proofs of those rely heavily on empirical process methods.

On the asymptotic normality of kernel regression estimators of the mode in the nonparametric random design model

Abstract In the nonparametric regression model with random design, where the regression function m is given by m(x)= E (Y|X=x) , estimation of the location θ ( mode ) of a unique maximum of m by the location θ of a maximum of the Nadaraya–Watson kernel estimator m for the curve m is considered. Within this setting, we obtain consistency and asymptotic normality results for θ under mild local smoothness assumptions on m and the design density g of X . The estimation of the size of the maximum is also considered as well as the estimation of a unique zero of the regression function. As a by-product, we obtain an asymptotic normality result for the Nadaraya–Watson estimator itself which improves on previous results.

A Uniform Law of Large Numbers for Set-Indexed Processes with Applications to Empirical and Partial-Sum Processes

The purpose of the present paper is to establish a uniform law of large numbers (ULLN) in form of a Mean Glivenko-Cantelli result for so-called partial-sum processes with random locations and indexed by Vapnik-Chervonenkis classes (VCC) of sets in arbitrary sample spaces. The context is as follows: Let X = (X, x) be an arbitrary measurable space, \((\eta_{nj})_{1 \leq j \leq j (n),n \in \mathbb{N}}\) be a triangular array of random elements (r.e.) in X (that is, the ηnj’s are assumed to be defined on some basic probability space (p-space) \((\Omega,A,\mathbb{P})\) with values in X such that each ηnj : Ω → X is A \(\mathfrak{X}\)-measurable), and let \((\xi _{nj})_{1 \leq j \leq j(n), n \in \mathbb{N}}\) be a triangular array of real-valued random variables (r.v.) (also defined on \((\Omega,A,\mathbb{P}))\) such that for each \(n\in \mathbb{N} (\eta_{n1},\xi_{n1}),\cdots, (\eta_{nj(n)},\xi_{nj(n)})\) is a sequence of independent but not necessarily identically distributed (i.d.) pairs of r.e.’s in \((X \times \mathbb{R},X \otimes \mathbb{B})\), where \(X \otimes \mathbb{B}\) denotes the product σ-field of x and the Borel σ-field IB in ℝ; i.e. the components within each pair need not be independent. Given a class \(C\subset X\), define a set-indexed process (of sample size \(n \in \mathbb{N})S_n =(S_n(C))_{C\in C}\) by

On bootstrapping the mode in the nonparametric regression model with random design

S_n(C):= \sum_{j \geq (n)}\;\;\;\;1_C(\eta_{nj})\xi_{nj},\;\;\;\;\;C \in C,\;\;\;\;\;\;\;\;\;\;(1.1)

On approximations to the bias of the nadaraya-watson regression estimator

where 1C denotes the indicator function of C.

On Hoffmann-Jørgensen-type Inequalities for Outer Expectations with Applications

The purpose of the present paper is to establish a functional central limit theorem (FCLT) for partial-sum processes with random locations and indexed by Vapnik-Cervonenkis classes (VCC) of sets in arbitrary sample spaces. The context is as follows: Let X = (X,X) be an arbitrary measurable space, (ηj)j∈N be a sequence of independent and identically distributed (i.i.d.) random elements (r.e.) in X with distribution v on X (that is, the η; j ’s are asumed to be defined on some basic probability space (Ω, F, P) with values in X such that each ηj: (Ω, F) → (X,X) is measurable), and let (ξnj)1≤ j ≤ j (n),n∈ℕ be a triangular array of rowwise independent (but not necessarily identically distributed) real-valued random variables (r.v.) (also defined on (Ω, F, P)) such that the whole triangular arrray is independent of the sequence (ηj)j∈ℕ. Given a class C ⊂ X, define a partial-sum process (of sample size n ∈ IN) S n = (S n(C))C∈c by

A Maximal Inequality and a Functional Central Limit Theorem for set-indexed empirical processes

S_{n}:=\sum_{j\leq j(n)}I_{C}(\eta_{j})\xi_{nj}, C\in c