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Johannes T. N. Krebs

Publications and source records attributed to Johannes T. N. Krebs.

10 recordsLinked to original sources
arXiv2018

The autoregression bootstrap for kernel estimates of smooth nonlinear functional time series

Functional times series have become an integral part of both functional data and time series analysis. This paper deals with the functional autoregressive model of order 1 and the autoregression bootstrap for smooth functions. The regression operator is estimated in the framework developed by Ferraty and Vieu [2004] and Ferraty et al. [2007] which is here extended to the double functional case under an assumption of stationary ergodic data which dates back to Laib and Louani [2010]. The main result of this article is the characterization of the asymptotic consistency of the bootstrapped regression operator.

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arXiv2018

Non-parametric Regression for Spatially Dependent Data with Wavelets

We study non-parametric regression estimates for random fields. The data satisfies certain strong mixing conditions and is defined on the regular $N$-dimensional lattice structure. We show consistency and obtain rates of convergence. The rates are optimal modulo a logarithmic factor in some cases. As an application, we estimate the regression function with multidimensional wavelets which are not necessarily isotropic. We simulate random fields on planar graphs with the concept of concliques (cf. Kaiser et al. [2012]) in numerical examples of the estimation procedure.

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arXiv2018

The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions

We consider the double functional nonparametric regression model $Y=r(X)+ε$, where the response variable $Y$ is Hilbert space-valued and the covariate $X$ takes values in a pseudometric space. The data satisfy an ergodicity criterion which dates back to Laib and Louani (2010) and are arranged in a triangular array. So our model also applies to samples obtained from spatial processes, e.g., stationary random fields indexed by the regular lattice $\mathbb{Z}^N$ for some $N\in\mathbb{N}_+$. We consider a kernel estimator of the Nadaraya--Watson type for the regression operator $r$ and study its limiting law which is a Gaussian operator on the Hilbert space. Moreover, we investigate both a naive and a wild bootstrap procedure in the double functional setting and demonstrate their asymptotic validity. This is quite useful as building confidence sets based on an asymptotic Gaussian distribution is often difficult.

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arXiv2017

Nonparametric Density Estimation for Spatial Data with Wavelets

Nonparametric density estimators are studied for $d$-dimensional, strongly spatial mixing data which is defined on a general $N$-dimensional lattice structure. We consider linear and nonlinear hard thresholded wavelet estimators which are derived from a $d$-dimensional multiresolution analysis. We give sufficient criteria for the consistency of these estimators and derive rates of convergence in $L^{p'}$ for $p'\in [1,\infty)$. For this reason, we study density functions which are elements of a $d$-dimensional Besov space $B^s_{p,q}(\mathbb{R}^d)$. We also verify the analytic correctness of our results in numerical simulations.

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arXiv2017

A Bernstein Inequality For Spatial Lattice Processes

In this article we present a Bernstein inequality for sums of random variables which are defined on a spatial lattice structure. The inequality can be used to derive concentration inequalities. It can be useful to obtain consistency properties for nonparametric estimators of conditional expectation functions.

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arXiv2017

A Note on Exponential Inequalities in Hilbert Spaces for Spatial Processes with Applications to the Functional Kernel Regression Model

In this manuscript we present exponential inequalities for spatial lattice processes which take values in a separable Hilbert space and satisfy certain dependence conditions. We consider two types of dependence: spatial data under $α$-mixing conditions and spatial data which satisfies a weak dependence condition introduced by Dedecker and Prieur [2005]. We demonstrate their usefulness in the functional kernel regression model of Ferraty and Vieu [2004] where we study uniform consistency properties of the estimated regression operator on increasing subsets of the underlying function space.

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arXiv2017

A Bernstein Inequality For Exponentially Growing Graphs

In this article we present a Bernstein inequality for sums of random variables which are defined on a graphical network whose nodes grow at an exponential rate. The inequality can be used to derive concentration inequalities in highly-connected networks. It can be useful to obtain consistency properties for nonparametric estimators of conditional expectation functions which are derived from such networks.

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arXiv2017

Orthogonal Series Estimates on Strong Spatial Mixing Data

We study a nonparametric regression model for sample data which is defined on an $N$-dimensional lattice structure and which is assumed to be strong spatial mixing: we use design adapted multidimensional Haar wavelets which form an orthonormal system w.r.t. the empirical measure of the sample data. For such orthonormal systems, we consider a nonparametric hard thresholding estimator. We give sufficient criteria for the consistency of this estimator and we derive rates of convergence. The theorems reveal that our estimator is able to adapt to the local smoothness of the underlying regression function and the design distribution. We illustrate our results with simulated examples.

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arXiv2017

A Large Deviation Inequality for $β$-mixing Time Series and its Applications to the Functional Kernel Regression Model

We give a new large deviation inequality for sums of random variables of the form $Z_k = f(X_k,X_t)$ for $k,t\in \mathbb{N}$, $t$ fixed, where the underlying process $X$ is $β$-mixing. The inequality can be used to derive concentration inequalities. We demonstrate its usefulness in the functional kernel regression model of Ferraty et al. (2007) where we study the consistency of dynamic forecasts.

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