Correction on Moments of minors of Wishart matrices

Correction on Moments of minors of Wishart matrices by M. Drton and A. Goia (Ann. Statist. 36 (2008) 2261-2283), arXiv:math/0604488Comment: Published in at http://dx.doi.org/10.1214/12-AOS988 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Classification methods for Hilbert data based on surrogate density

An unsupervised and a supervised classification approaches for Hilbert random curves are studied. Both rest on the use of a surrogate of the probability density which is defined, in a distribution-free mixture context, from an asymptotic factorization of the small-ball probability. That surrogate density is estimated by a kernel approach from the principal components of the data. The focus is on the illustration of the classification algorithms and the computational implications, with particular attention to the tuning of the parameters involved. Some asymptotic results are sketched. Applications on simulated and real datasets show how the proposed methods work.Comment: 33 pages, 11 figures, 6 table

Exploring the total positivity of yields correlations

We test the plausibility of the total positivity assumption of interest rates changes recently introduced in order to justify the presence of shift, slope and curvature for yield curves. To this aim, we introduce and discuss a test of total positivity of order for covariance and correlation matrices. The explicit expressions of the test statistics are given for Gaussian samples and an extension to a distribution-free framework is made via a bootstrap method. After exploring with simulation the robustness of such tests, we show using real data how it is realistic to assume that correlation matrices of interest rates changes are totally positive of order two. Conclusions on total positivity of order three are more controversial

Some Insights About the Small Ball Probability Factorization for Hilbert Random Elements

Asymptotic factorizations for the small\u2013ball probability (SmBP) of a Hilbert valued random element X are rigorously established and discussed. In particular, given the first d principal components (PCs) and as the radius \u3b5 of the ball tends to zero, the SmBP is asymptotically proportional to (a) the joint density of the first d PCs, (b) the volume of the d\u2013dimensional ball with radius \u3b5, and (c) a correction factor weighting the use of a truncated version of the process expansion. Moreover, under suitable assumptions on the spectrum of the covariance operator of X and as d diverges to infinity when \u3b5 vanishes, some simplifications occur. In particular, the SmBP factorizes asymptotically as the product of the joint density of the first d PCs and a pure volume parameter. All the provided factorizations allow to define a surrogate intensity of the SmBP that, in some cases, leads to a genuine intensity. To operationalize the stated results, a non\u2013parametric estimator for the surrogate intensity is introduced and it is proved that the use of estimated PCs, instead of the true ones, does not affect the rate of convergence. Finally, as an illustration, simulations in controlled frameworks are provided

Describing the Concentration of Income Populations by Functional Principal Component Analysis on Lorenz Curves

Lorenz curves are widely used in economic studies (inequality, poverty, differentiation, etc.). From a model point of view, such curves can be seen as constrained functional data for which functional principal component analysis (FPCA) could be defined. Although statistically consistent, performing FPCA using the original data can lead to a suboptimal analysis from a mathematical and interpretation point of view. In fact, the family of Lorenz curves lacks very basic (e.g., vectorial) structures and, hence, must be treated with ad hoc methods. This work aims to provide a rigorous mathematical framework via an embedding approach to define a coherent FPCA for Lorenz curves. This approach is used to explore a functional dataset from the Bank of Italy income survey

Modeling functional data: a test procedure

The paper deals with a test procedure able to state the compatibility of observed data with a reference model, by using an estimate of the volumetric part in the small-ball probability factorization which plays the role of a real complexity index. As a preliminary by-product we state some asymptotics for a new estimator of the complexity index. A suitable test statistic is derived and, referring to the U-statistics theory, its asymptotic null distribution is obtained. A study of level and power of the test for finite sample sizes and a comparison with a competitor are carried out by Monte Carlo simulations. The test procedure is performed over a financial time series

Evaluating the complexity of some families of functional data

In this paper we study the complexity of functional data set by means of a two steps approach. The first step considers a new graphical tool for assessing to which family the data belong: the main aim is to detect whether a sample comes from a monomial or an exponential family. This first tool is based on a nonparametric kNN estimation of Small Ball Probability. Once the family is specified, the second step consists in evaluating the extent of complexity by estimating some specific indexes related to the assigned family. It turns out that the developed methodology is fully free from assumptions on model, distribution as well as dominating measure. This large flexibility ensures the wide applicability of the methodology. Computational issues are carried out by means of simulations and finally the method is applied to analyze some financial real curves dataset