23 research outputs found
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
Mixed Needlets
The construction of needlet-type wavelets on sections of the spin line
bundles over the sphere has been recently addressed in Geller and Marinucci
(2008), and Geller et al. (2008,2009). Here we focus on an alternative proposal
for needlets on this spin line bundle, in which needlet coefficients arise from
the usual, rather than the spin, spherical harmonics, as in the previous
constructions. We label this system mixed needlets and investigate in full
their properties, including localization, the exact tight frame
characterization, reconstruction formula, decomposition of functional spaces,
and asymptotic uncorrelation in the stochastic case. We outline astrophysical
applications.Comment: 26 page
Inversion of noisy Radon transform by SVD based needlet
A linear method for inverting noisy observations of the Radon transform is
developed based on decomposition systems (needlets) with rapidly decaying
elements induced by the Radon transform SVD basis. Upper bounds of the risk of
the estimator are established in () norms for functions
with Besov space smoothness. A practical implementation of the method is given
and several examples are discussed
On high-frequency limits of -statistics in Besov spaces over compact manifolds
In this paper, quantitative bounds in high-frequency central limit theorems
are derived for Poisson based -statistics of arbitrary degree built by means
of wavelet coefficients over compact Riemannian manifolds. The wavelets
considered here are the so-called needlets, characterized by strong
concentration properties and by an exact reconstruction formula. Furthermore,
we consider Poisson point processes over the manifold such that the density
function associated to its control measure lives in a Besov space. The main
findings of this paper include new rates of convergence that depend strongly on
the degree of regularity of the control measure of the underlying Poisson point
process, providing a refined understanding of the connection between regularity
and speed of convergence in this framework.Comment: 19 page
Nonparametric estimation in random coefficients binary choice models
This paper considers random coefficients binary choice models. The main goal is to estimate the density of the random coefficients nonparametrically. This is an ill-posed inverse prob- lem characterized by an integral transform. A new density estimator for the random coefficients is developed, utilizing Fourier-Laplace series on spheres. This approach offers a clear insight on the identification problem. More importantly, it leads to a closed form estimator formula that yields a simple plug-in procedure requiring no numerical optimization. The new estimator, therefore, is easy to implement in empirical applications, while being flexible about the treatment of unobserved hetero- geneity. Extensions including treatments of non-random coefficients and models with endogeneity are discussed
Normal Approximations for Wavelet Coefficients on Spherical Poisson Fields
We compute explicit upper bounds on the distance between the law of a
multivariate Gaussian distribution and the joint law of wavelets/needlets
coefficients based on a homogeneous spherical Poisson field. In particular, we
develop some results from Peccati and Zheng (2011), based on Malliavin calculus
and Stein's methods, to assess the rate of convergence to Gaussianity for a
triangular array of needlet coefficients with growing dimensions. Our results
are motivated by astrophysical and cosmological applications, in particular
related to the search for point sources in Cosmic Rays data.Comment: 28 page
Modern Nonparametric Statistics: Going Beyond Asymptotic Minimax
During the years 1975-1990 a major emphasis in nonparametric estimation was put on computing the asymptotic minimax risk for many classes of functions. Modern statistical practice indicates some serious limitations of the asymptotic minimax approach and calls for some new ideas and methods which can cope with the numerous challenges brought to statisticians by modern sets of data
Nonparametric estimation in random coefficients binary choice models
This paper considers random coefficients binary choice models. The main goal is to estimate the density of the random coefficients nonparametrically. This is an ill-posed inverse prob- lem characterized by an integral transform. A new density estimator for the random coefficients is developed, utilizing Fourier-Laplace series on spheres. This approach offers a clear insight on the identification problem. More importantly, it leads to a closed form estimator formula that yields a simple plug-in procedure requiring no numerical optimization. The new estimator, therefore, is easy to implement in empirical applications, while being flexible about the treatment of unobserved hetero- geneity. Extensions including treatments of non-random coefficients and models with endogeneity are discussed