Joachim Engel

Die Grenzen des Wachstums

In Kap. 2 verfolgten wir einen Ansatz zur Modellbildung, bei dem der Funktionstyp schon von vornherein vorgegeben war.¨Uberlegungen bzw. etablierte Theorien aus dem Sachkontext gaben Anlass, einen bestimmten Typus funktionaler Abhangigkeit von vornherein anzunehmen. Die Kurvenanpassung bestand dann lediglich darin, passende Parameter zu finden, so dass sich die Modellfunktion moglichst gut den Daten anpasst. In manchen Anwendungssituationen hat man allerdings keinen sachbezogenen Anlass, a priori eine parametrisierte Funktionenklasse wie z.B. die Menge aller Parabeln oder aller Exponentialfunktionen etc. anzunehmen, man kann jedoch durch sachanalytische ¨Uberlegungen bestimmte Annahmen uber das lokale Anderungsverhalten des untersuchten Vorganges begrunden.

Local Polynomial Regression: Optimal Kernels and Asymptotic Minimax Efficiency

We consider local polynomial fitting for estimating a regression function and its derivatives nonparametrically. This method possesses many nice features, among which automatic adaptation to the boundary and adaptation to various designs. A first contribution of this paper is the derivation of an optimal kernel for local polynomial regression, revealing that there is a universal optimal weighting scheme. Fan (1993, Ann. Statist., 21, 196-216) showed that the univariate local linear regression estimator is the best linear smoother, meaning that it attains the asymptotic linear minimax risk. Moreover, this smoother has high minimax risk. We show that this property also holds for the multivariate local linear regression estimator. In the univariate case we investigate minimax efficiency of local polynomial regression estimators, and find that the asymptotic minimax efficiency for commonly-used orders of fit is 100% among the class of all linear smoothers. Further, we quantify the loss in efficiency when going beyond this class.

Fast Algorithms for Nonparametric Curve Estimation

Abstract Naive implementations of local polynomial fits and kernel estimators require almost O(n 2) operations. In this article two fast O(n) algorithms for nonparametric local polynomial fitting are presented. They are based on updating normal equations. Numerical stability is guaranteed by controlling ill-conditioned situations for small bandwidths and data-tuned restarting of the updating procedure. Restarting at every output point results in a moderately fast but highly stable O(n 7/5) algorithm. Applicability of algorithms is evaluated for estimation of regression curves and their derivatives. The idea is also applied to kernel estimators of regression curves and densities.

Effects of finger counting on numerical development - the opposing views of neurocognition and mathematics education

Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or “finger sense,” enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed.

Recent approaches to estimating Engel curves

Classical approaches of estimating cross-section Engel curves are based on parametric models. However, misspecification of a parametric model implies that information of structural nature might be masked. An alternative avoiding problems related to predetermined functional relations is the nonparametric approach. This paper surveys recent advances of nonparametric statistics in their relevance to estimating cross-section Engel curves.

An iterative bandwidth selector for kernel estimation of densities and their derivatives

A bandwidth selection rule which proved to be useful and effective for nonparametric kernel regression is modified to be suitable for estimation of a density and its derivatives. Various versions of the rule are considered. Theoretical properties are derived. A simulation study compares its finite-sample behavior with that of other bandwidth selectors.

The multiresolution histogram

We introduce a new method for locally adaptive histogram construction that doesn’t resort to a standard distribution and is easy to implement: the multiresolution histogram. It is based on aL2 analysis of the mean integrated squared error with Haar wavelets and hence can be associated with a multiresolution analysis of the sample space.

Correlation and Regression in the Training of Teachers

Although the notion of functional dependence of two variables is fundamental to school mathematics, teachers often are not trained to analyse statistical dependencies. Many teachers’ thinking about bivariate data is shaped by the deterministic concept of a mathematical function. Statistical data, however, usually do not perfectly fit a deterministic model but are characterised by variation around a possible trend. Therefore, understanding regression and correlation requires, apart from basic knowledge about functions, an appreciation of the role of variation. In this chapter, common errors and fallacies related to the concepts of correlation and regression are revisited and suggestions on how teachers may overcome some of these difficulties are provided.

Density estimation with Haar series

Rate of convergence for density estimators based on Haar series are derived under very mild condition: the unknown density has to be of bounded variation. These estimators are histograms on dyadic intervals.

From Data to Functions: Connecting Modelling Competencies and Statistical Literacy

Over the last two decades, research on learning and teaching mathematical applications greatly advanced our understanding of the processes involved in mathematical modelling. However, the vast majority of examples and concepts developed so far barely include a key source of information: data. Numerical information generated from measurements of the quantities involved is used neither at the validation nor at the modelling step. We adopt a data-oriented approach. In the context of modelling functional relationships, we look at the relationship between modelling competencies and statistical literacy and provide empirical evidence that proficiency in these areas can be jointly improved.