Testing for Homogeneity in Mixture Models

Statistical models of unobserved heterogeneity are typically formalized as mixtures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as $C(\alpha)$ tests, as in Neyman (1959), and shown to be locally, asymptotically optimal. These $C(\alpha)$ tests will be contrasted with a new approach to likelihood ratio testing for general mixture models. The latter tests are based on estimation of general nonparametric mixing distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator. Recent developments in convex optimization have dramatically improved upon earlier EM methods for computation of these estimators, and recent results on the large sample behavior of likelihood ratios involving such estimators yield a tractable form of asymptotic inference. Improvement in computation efficiency also facilitates the use of a bootstrap methods to determine critical values that are shown to work better than the asymptotic critical values in finite samples. Consistency of the bootstrap procedure is also formally established. We compare performance of the two approaches identifying circumstances in which each is preferred

Distribution-Free Tests of Independence in High Dimensions

We consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and propose tests that control the type I error in the high-dimensional setting where $d>n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives, and outperform competitors in simulations, especially when $d$ is large.Comment: to appear in Biometrik

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

Evidence of non-Markovian behavior in the process of bank rating migrations

This paper estimates transition matrices for the ratings on financial insti-tutions, using an unusually informative data set. We show that the process of rating migration exhibits significant non-Markovian behavior, in the sense that the transition intensities are affected by macroeconomic and bank spe- cific variables. We illustrate how the use of a continuous time framework may improve the estimation of the transition probabilities. However, the time homogeneity assumption, frequently done in economic applications, does not hold, even for short time intervals. Thus, the information provided by migrations alone is not enough to forecast the future behavior of ratings. The stage of the business cycle should be taken into account, and individual characteristics of banks must be considered as well.Financial institutions; macroeconomic variables; capitaliza- tion; supervision; transition intensities. Classification JEL: C4; E44; G21; G23; G38.

Characterization of the frequency of extreme events by the Generalized Pareto Distribution

Based on recent results in extreme value theory, we use a new technique for the statistical estimation of distribution tails. Specifically, we use the Gnedenko-Pickands-Balkema-de Haan theorem, which gives a natural limit law for peak-over-threshold values in the form of the Generalized Pareto Distribution (GPD). Useful in finance, insurance, hydrology, we investigate here the earthquake energy distribution described by the Gutenberg-Richter seismic moment-frequency law and analyze shallow earthquakes (depth h < 70 km) in the Harvard catalog over the period 1977-2000 in 18 seismic zones. The whole GPD is found to approximate the tails of the seismic moment distributions quite well above moment-magnitudes larger than mW=5.3 and no statistically significant regional difference is found for subduction and transform seismic zones. We confirm that the b-value is very different in mid-ocean ridges compared to other zones (b=1.50=B10.09 versus b=1.00=B10.05 corresponding to a power law exponent close to 1 versus 2/3) with a very high statistical confidence. We propose a physical mechanism for this, contrasting slow healing ruptures in mid-ocean ridges with fast healing ruptures in other zones. Deviations from the GPD at the very end of the tail are detected in the sample containing earthquakes from all major subduction zones (sample size of 4985 events). We propose a new statistical test of significance of such deviations based on the bootstrap method. The number of events deviating from the tails of GPD in the studied data sets (15-20 at most) is not sufficient for determining the functional form of those deviations. Thus, it is practically impossible to give preference to one of the previously suggested parametric families describing the ends of tails of seismic moment distributions.Comment: pdf document of 21 pages + 2 tables + 20 figures (ps format) + one file giving the regionalizatio