Drifting Together or Falling Apart? The Empirics of Regional Economic Growth in Post-Unification Germany

The objective of this paper is to address the question of convergence across German districts in the first decade after German unification by drawing out and emphasising some stylised facts of regional per capita income dynamics. We achieve this by employing non-parametric techniques which focus on the evolution of the entire cross-sectional income distribution. In particular, we follow a distributional approach to convergence based on kernel density estimation and implement a number of tests to establish the statistical significance of our findings. This paper finds that the relative income distribution appears to be stratifying into a trimodal/bimodal distribution.regional economic growth, Germany, convergence clubs, density estimation, modality tests

Local Component Analysis

Kernel density estimation, a.k.a. Parzen windows, is a popular density estimation method, which can be used for outlier detection or clustering. With multivariate data, its performance is heavily reliant on the metric used within the kernel. Most earlier work has focused on learning only the bandwidth of the kernel (i.e., a scalar multiplicative factor). In this paper, we propose to learn a full Euclidean metric through an expectation-minimization (EM) procedure, which can be seen as an unsupervised counterpart to neighbourhood component analysis (NCA). In order to avoid overfitting with a fully nonparametric density estimator in high dimensions, we also consider a semi-parametric Gaussian-Parzen density model, where some of the variables are modelled through a jointly Gaussian density, while others are modelled through Parzen windows. For these two models, EM leads to simple closed-form updates based on matrix inversions and eigenvalue decompositions. We show empirically that our method leads to density estimators with higher test-likelihoods than natural competing methods, and that the metrics may be used within most unsupervised learning techniques that rely on such metrics, such as spectral clustering or manifold learning methods. Finally, we present a stochastic approximation scheme which allows for the use of this method in a large-scale setting

Model-based Methods of Classification: Using the mclust Software in Chemometrics

Due to recent advances in methods and software for model-based clustering, and to the interpretability of the results, clustering procedures based on probability models are increasingly preferred over heuristic methods. The clustering process estimates a model for the data that allows for overlapping clusters, producing a probabilistic clustering that quantifies the uncertainty of observations belonging to components of the mixture. The resulting clustering model can also be used for some other important problems in multivariate analysis, including density estimation and discriminant analysis. Examples of the use of model-based clustering and classification techniques in chemometric studies include multivariate image analysis, magnetic resonance imaging, microarray image segmentation, statistical process control, and food authenticity. We review model-based clustering and related methods for density estimation and discriminant analysis, and show how the R package mclust can be applied in each instance.

Least squares fitting the three-parameter inverse Weibull density

The inverse Weibull model was developed by Erto [10]. In practice, the unknown parameters of the appropriate inverse Weibull density are not known and must be estimated from a random sample. Estimation of its parameters has been approached in the literature by various techniques, because a standard maximum likelihood estimate does not exist. To estimate the unknown parameters of the three-parameter inverse Weibull density we will use a combination of nonparametric and parametric methods. The idea consists of using two steps: in the first step we calculate an initial nonparametric density estimate which needs to be as good as possible, and in the second step we apply the nonlinear least squares method to estimate the unknown parameters. As a main result, a theorem on the existence of the least squares estimate is obtained, as well as its generalization in the $l_p$ norm ($1leq p<infty$). Some simulations are given to show that our approach is satisfactory if the initial density is of good enough quality