The test of exponentiality based on the mean residual life function revisited

We revisit the family of goodness-of-fit tests for exponentiality based on the mean residual life time proposed by Baringhaus & Henze (2008). We motivate the test statistic by a characterisation of Shanbhag (1970) and provide an alternative representation, which leads to simple and short proofs for the known theory and an easy to access covariance structure of the limiting Gaussian process under the null hypothesis. Explicit formulas for the eigenvalues and eigenfunctions of the operator associated with the limit covariance are derived using results on weighted Brownian bridges. In addition we provide further asymptotic theory under fixed alternatives and derive approximate Bahadur efficiencies, which provide an insight into the choice of the tuning parameter with regard to the power performance of the tests

Characterizations of non-normalized discrete probability distributions and their application in statistics

From the distributional characterizations that lie at the heart of Stein’s method we derive explicit formulae for the mass functions of discrete probability laws that identify those distributions. These identities are applied to develop tools for the solution of statistical problems. Our characterizations, and hence the applications built on them, do not require any knowledge about normalization constants of the probability laws. To demonstrate that our statistical methods are sound, we provide comparative simulation studies for the testing of fit to the Poisson distribution and for parameter estimation of the negative binomial family when both parameters are unknown. We also consider the problem of parameter estimation for discrete exponential-polynomial models which generally are non-normalized

Testing normality in any dimension by Fourier methods in a multivariate Stein equation

We study a novel class of affine-invariant and consistent tests for multivariate normality. The tests are based on a characterization of the standard d-variate normal distribution by way of the unique solution of an initial value problem connected to a partial differential equation, which is motivated by a multivariate Stein equation. The test criterion is a suitably weighted L2-statistic. We derive the limit distribution of the test statistic under the null hypothesis as well as under contiguous and fixed alternatives to normality. A consistent estimator of the limiting variance under fixed alternatives, as well as an asymptotic confidence interval of the distance of an underlying alternative with respect to the multivariate normal law, is derived. In simulation studies, we show that the tests are strong in comparison with prominent competitors and that the empirical coverage rate of the asymptotic confidence interval converges to the nominal level. We present a real data example and also outline topics for further research

Editorial: E-Learning-Strategien für die Hochschullehre

27.04.2015 | Sabine Seufert (St. Gallen), Martin Ebner, Michael Kopp (Graz) & Bettina Schlass (Amsterdam

Towards a Learning-Aware Application Guided by Hierarchical Classification of Learner Profiles

Learner profiling is a methodology that draws a parallel from user profiling. Implicit feedback is often used in recommender systems to create and adapt user profiles. In this work the implicit feedback is based on the learner's answering behaviour in the Android application UnlockYourBrain, which poses different basic mathematical questions to the learners. We introduce an analytical approach to model the learners' profile according to the learner's answering behaviour. Furthermore, similar learner's profiles are grouped together to construct a learning behaviour cluster. The choice of hierarchical clustering as a means of classification of learners' profiles derives from the observations of learners behaviour. This in turn reflects the similarities and subtle differences of learner behaviour, which are further analysed in more detail. Building awareness about the learner's behaviour is the first and necessary step for future learning-aware applications

On the theory of diamagnetism in granular superconductors

We study a highly disordered network of superconducting granules linked by weak Josephson junctions in magnetic field and develop a mean field theory for this problem. The diamagnetic response to a slow {\it variations} of magnetic field is found to be analogous to the response of a type-II superconductor with extremely strong pinning. We calculate an effective penetration depth $\lambda_g$ and critical current $j_c$ and find that both $\lambda_g^{-1}$ and $j_c$ are non-zero but are strongly suppressed by frustration.Comment: REVTEX, 12 pages, two Postscript figure