The consistency and asymptotic normality of the spatial sign covariance matrix with unknown location are shown. Simulations illustrate the different asymptotic behavior when using the mean and the spatial median as location estimator.2010 MSC: 62H12, 62G20, 62H11

We propose elliptical graphical models based on conditional uncorrelatedness as a generalization of Gaussian graphical models by letting the population distribution be elliptical instead of normal, allowing the fitting of data with arbitrarily heavy tails. We study the class of proportionally affine equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling, leading to a new class of partial correlation estimators and analogues of the classical deviance test. General expressions for the asymptotic variance of partial correlation estimators, unconstrained and under decomposable models, are given, and the asymptotic chi square approximation of the pseudo-deviance test statistic is proved. The feasibility of our approach is demonstrated by a simulation study, using, among others, Tyler's scatter estimator, which is distribution-free within the elliptical model. Our approach provides a robustification of Gaussian graphical modelling. The latter is likelihood-based and known to be very sensitive to model misspecification and outlying observations.

A new robust correlation estimator based on the spatial sign covariance matrix (SSCM) is proposed. We derive its asymptotic distribution and influence function at elliptical distributions. Finite sample and robustness properties are studied and compared to other robust correlation estimators by means of numerical simulations.

We gather several results on the eigenvalues of the spatial sign covariance matrix of an elliptical distribution. It is shown that the eigenvalues are a one-to-one function of the eigenvalues of the shape matrix and that they are closer together than the latter. We further provide a one-dimensional integral representation of the eigenvalues, which facilitates their numerical computation.

The asymptotic relative efficiency of the mean deviation with respect to the standard deviation is 88% at the normal distribution. In his seminal 1960 paper A survey of sampling from contaminated distributions, J. W. Tukey points out that, if the normal distribution is contaminated by a small -fraction of a normal distribution with three times the standard deviation, the mean deviation is more efficient than the standard deviationalready for < 1%. This came as a surprise to most statisticians at the time, and the publication is today considered as one of the main pioneering works in the development of robust statistics. In the present article, we examine the efficiency of the mean deviation and Gini's mean difference (the mean of all pairwise distances). The latter is known to have an asymptotic relative efficiency of 98% at the normal distribution. Our findings support the viewpoint that Gini's mean difference combines the advantages of the mean deviation and the standard deviation. We also answer the question, what percentage of contamination in Tukey's 1:3 normal mixture model renders Gini's mean difference more efficient than the standard deviation. MSC: 62G35, 62G05, 62G20Date: May 21, 2014.

Bayesian estimation has become very popular. However, run time of Bayesian models is often unsatisfactorily high. In this illustration, we show how to reduce run time by (a) integrating out nuisance model parameters and by (b) reformulating the model based on covariances and means. The core concept is to use the sample scatter matrix which is in our case Wishart distributed with the model-implied covariance matrix as the scale matrix. To illustrate this approach, we choose the popular multi-level null (intercept-only) model, provide a step-by-step instruction on how to implement this model in a multipurpose Bayesian software, and show how structural equation modeling techniques can be employed to bypass mathematically challenging derivations. A simulation study showed that run time is considerably reduced and an empirical example illustrates our approach. Further, we show how the JAGS sampling progress can be monitored and stopped automatically when convergence and precision criteria are reached.

Music performance anxiety (MPA) is considered a social anxiety disorder (SAD). Recent conceptualizations, however, challenge existing MPA definitions, distinguishing MPA from SAD. In this study, we aim to provide a systematic analysis of MPA interdependencies to other anxiety disorders through graphical modeling and cluster analysis. Participants were 82 music students ( Mage = 23.5 years, SD = 3.4 years; 69.5% women) with the majority being vocal (30.5%), string (24.4%), or piano (19.5%) students. MPA was measured using the German version of the Kenny Music Performance Anxiety Inventory (K-MPAI). All participants were tested for anxiety-related symptoms using the disorder-specific anxiety measures of the Diagnostic and Statistical Manual of Mental Disorders (5th ed., DSM-5), including agoraphobia (AG), generalized anxiety disorder (GAD), panic disorder (PD), separation anxiety disorder (SEP), specific phobia (SP), SAD, and illness anxiety disorder (ILL). We found no evidence of MPA being primarily connected to SAD, finding GAD acted as a full mediator between MPA and any other anxiety type. Our graphical model remained unchanged considering severe cases of MPA only (K-MPAI ⩾ 105). By means of cluster analysis, we identified two participant sub-groups of differing anxiety profiles. Participants with pathological anxiety consistently showed more severe MPA. Our findings suggest that GAD is the strongest predictor for MPA among all major DSM-5 anxiety types.

Abstract. We study subsampling estimators for the limit varianceof partial sums of a stationary stochastic process (X k ) k≥1 . We establish L2-consistency of a non-overlapping block resampling method. Our results apply to processes that can be represented as functionals of strongly mixing processes. Motivated by recent applications to rank tests, we also study estimators for the series Var(F (X1))+2, where F is the distribution function of X1. Simulations illustrate the usefulness of the proposed estimators and of a mean squared error optimal rule for the choice of the block length.