A One-Sample Test for Normality with Kernel Methods

We propose a new one-sample test for normality in a Reproducing Kernel Hilbert Space (RKHS). Namely, we test the null-hypothesis of belonging to a given family of Gaussian distributions. Hence our procedure may be applied either to test data for normality or to test parameters (mean and covariance) if data are assumed Gaussian. Our test is based on the same principle as the MMD (Maximum Mean Discrepancy) which is usually used for two-sample tests such as homogeneity or independence testing. Our method makes use of a special kind of parametric bootstrap (typical of goodness-of-fit tests) which is computationally more efficient than standard parametric bootstrap. Moreover, an upper bound for the Type-II error highlights the dependence on influential quantities. Experiments illustrate the practical improvement allowed by our test in high-dimensional settings where common normality tests are known to fail. We also consider an application to covariance rank selection through a sequential procedure

Tests based on characterizations, and their efficiencies: a survey

A survey of goodness-of-fit and symmetry tests based on the characterization properties of distributions is presented. This approach became popular in recent years. In most cases the test statistics are functionals of $U$-empirical processes. The limiting distributions and large deviations of new statistics under the null hypothesis are described. Their local Bahadur efficiency for various parametric alternatives is calculated and compared with each other as well as with diverse previously known tests. We also describe new directions of possible research in this domain.Comment: Open access in Acta et Commentationes Universitatis Tartuensis de Mathematic

Nonparametric checks for single-index models

In this paper we study goodness-of-fit testing of single-index models. The large sample behavior of certain score-type test statistics is investigated. As a by-product, we obtain asymptotically distribution-free maximin tests for a large class of local alternatives. Furthermore, characteristic function based goodness-of-fit tests are proposed which are omnibus and able to detect peak alternatives. Simulation results indicate that the approximation through the limit distribution is acceptable already for moderate sample sizes. Applications to two real data sets are illustrated.Comment: Published at http://dx.doi.org/10.1214/009053605000000020 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

An evaluation of planarity of the spatial QRS loop by three dimensional vectorcardiography: its emergence and loss

Aims: To objectively characterize and mathematically justify the observation that vectorcardiographic QRS loops in normal individuals are more planar than those from patients with ST elevation myocardial infarction (STEMI). Methods: Vectorcardiograms (VCGs) were constructed from three simultaneously recorded quasi-orthogonal leads, I, aVF and V2 (sampled at 1000 samples/s). The planarity of these QRS loops was determined by fitting a surface to each loop. Goodness of fit was expressed in numerical terms. Results: 15 healthy individuals aged 35–65 years (73% male) and 15 patients aged 45–70 years (80% male) with diagnosed acute STEMI were recruited. The spatial-QRS loop was found to lie in a plane in normal controls. In STEMI patients, this planarity was lost. Calculation of goodness of fit supported these visual observations. Conclusions: The degree of planarity of the VCG loop can differentiate healthy individuals from patients with STEMI. This observation is compatible with our basic understanding of the electrophysiology of the human heart

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page