49,295 research outputs found
Estimation of multivariate probit models by exact maximum likelihood
In this paper, we develop a new numerical method to estimate a multivariate probit model. To this end, we derive a new decomposition of normal multivariate integrals that has two appealing properties. First, the decomposition may be written as the sum of normal multivariate integrals, in which the highest dimension of the integrands is reduced relative to the initial problem. Second, the domains of integration are bounded and delimited by the correlation coefficients. Application of a Gauss-Legendre quadrature rule to the exact likelihood function of lower dimension allows for a major reduction of computing time while simultaneously obtaining consistent and efficient estimates for both the slope and the scale parameters. A Monte Carlo study shows that the finite sample and asymptotic properties of our method compare extremely favorably to the maximum simulated likelihood estimator in terms of both bias and root mean squared error.Multivariate Probit Model, Simulated and Full Information Maximum Likelihood, Multivariate Normal Distribution, Simulations
Non-Parametric Approximations for Anisotropy Estimation in Two-dimensional Differentiable Gaussian Random Fields
Spatially referenced data often have autocovariance functions with elliptical
isolevel contours, a property known as geometric anisotropy. The anisotropy
parameters include the tilt of the ellipse (orientation angle) with respect to
a reference axis and the aspect ratio of the principal correlation lengths.
Since these parameters are unknown a priori, sample estimates are needed to
define suitable spatial models for the interpolation of incomplete data. The
distribution of the anisotropy statistics is determined by a non-Gaussian
sampling joint probability density. By means of analytical calculations, we
derive an explicit expression for the joint probability density function of the
anisotropy statistics for Gaussian, stationary and differentiable random
fields. Based on this expression, we obtain an approximate joint density which
we use to formulate a statistical test for isotropy. The approximate joint
density is independent of the autocovariance function and provides conservative
probability and confidence regions for the anisotropy parameters. We validate
the theoretical analysis by means of simulations using synthetic data, and we
illustrate the detection of anisotropy changes with a case study involving
background radiation exposure data. The approximate joint density provides (i)
a stand-alone approximate estimate of the anisotropy statistics distribution
(ii) informed initial values for maximum likelihood estimation, and (iii) a
useful prior for Bayesian anisotropy inference.Comment: 39 pages; 8 figure
Cleaning large correlation matrices: tools from random matrix theory
This review covers recent results concerning the estimation of large
covariance matrices using tools from Random Matrix Theory (RMT). We introduce
several RMT methods and analytical techniques, such as the Replica formalism
and Free Probability, with an emphasis on the Marchenko-Pastur equation that
provides information on the resolvent of multiplicatively corrupted noisy
matrices. Special care is devoted to the statistics of the eigenvectors of the
empirical correlation matrix, which turn out to be crucial for many
applications. We show in particular how these results can be used to build
consistent "Rotationally Invariant" estimators (RIE) for large correlation
matrices when there is no prior on the structure of the underlying process. The
last part of this review is dedicated to some real-world applications within
financial markets as a case in point. We establish empirically the efficacy of
the RIE framework, which is found to be superior in this case to all previously
proposed methods. The case of additively (rather than multiplicatively)
corrupted noisy matrices is also dealt with in a special Appendix. Several open
problems and interesting technical developments are discussed throughout the
paper.Comment: 165 pages, article submitted to Physics Report
The modal decomposition of aperture fields in detection and estimation of incoherent objects
Modal aperture field decomposition for optical detection and radiance estimation on incoherent object
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