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M. M

Publications and source records attributed to M. M.

5 recordsLinked to original sources
arXiv2025

Multimodal Distributions for Circular Axial Data

The family of circular distributions based on non-negative trigonometric sums (NNTS), developed by Fern\'andez-Dur\'an (2004), is highly flexible for modeling datasets exhibiting multimodality and/or skewness. In this article, we extend the NNTS family to axial data by identifying conditions under which the original NNTS family is suitable for modeling undirected vectors. Since the estimation is performed using maximum likelihood, likelihood ratio tests are developed for characteristics of the density function such as uniformity and symmetry, as well as to compare different axial populations through homogeneity tests. The proposed methodology is applied to real datasets involving orientations of rocks, animals, and plants.

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arXiv2023

Testing the Regular Variation Model for Multivariate Extremes with Flexible Circular and Spherical Distributions

The regular variation model for multivariate extremes decomposes the joint distribution of the extremes in polar coordinates in terms of the angles and the norm of the random vector as the product of two independent densities: the angular (spectral) measure and the density of the norm. The support of the angular measure is the surface of a unit hypersphere and the density of the norm corresponds to a Pareto density. The dependence structure is determined by the angular measure on the hypersphere, and directions with high probability characterize the dependence structure among the elements of the random vector of extreme values. Previous applications of the regular variation model have not considered a probabilistic model for the angular density and no statistical tests were applied. In this paper, circular and spherical distributions based on nonnegative trigonometric sums are considered flexible probabilistic models for the spectral measure that allows the application of statistical tests to make inferences about the dependence structure among extreme values. The proposed methodology is applied to real datasets from finance.

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arXiv2023

Multivariate Nonnegative Trigonometric Sums Distributions for High-Dimensional Multivariate Circular Data

Fern\'andez-Dur\'an and Gregorio-Dom\'inguez (2014) defined a family of probability distributions for a vector of circular random variables by considering multiple nonnegative trigonometric sums. These distributions are highly flexible and can present numerous modes and skewness. Several operations on these multivariate distributions were translated into operations on the vector of parameters; for instance, marginalization involves calculating the eigenvectors and eigenvalues of a matrix, and independence among subsets of the vector of circular variables translates to a Kronecker product of the corresponding subsets of the vector of parameters. Furthermore, it was demonstrated that the family of multivariate circular distributions based on nonnegative trigonometric sums is closed under marginalization and conditioning, that is, the marginal and conditional densities of any order are also members of the family. The derivation of marginal and conditional densities from the joint multivariate density is important when applying this model in practice to real datasets. A goodness-of-fit test based on the characteristic function and an alternative parameter estimation algorithm for high-dimensional circular data was presented and applied to a real dataset on the daily times of occurrence of maxima and minima of prices in financial markets.

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arXiv2023

Test of Bivariate Independence Based on Angular Probability Integral Transform with Emphasis on Circular-Circular and Circular-Linear Data

The probability integral transform (PIT) of a continuous random variable $X$ with distribution function $F_X$ is a uniformly distributed random variable $U=F_X(X)$. We define the angular probability integral transform (APIT) as $\theta_U = 2 \pi U = 2 \pi F_{X}(X)$, which corresponds to a uniformly distributed angle on the unit circle. For circular (angular) random variables, the sum of absolutely continuous independent circular uniform random variables is a circular uniform random variable, that is, the circular uniform distribution is closed under summation, and it is a stable continuous distribution on the unit circle. If we consider the sum (difference) of the angular probability integral transforms of two random variables, $X_1$ and $X_2$, and test for the circular uniformity of their sum (difference), this is equivalent to the test of independence of the original variables. In this study, we used a flexible family of nonnegative trigonometric sums (NNTS) circular distributions, which include the uniform circular distribution as a member of the family, to evaluate the power of the proposed independence test; we complete this evaluation by generating samples from NNTS alternative distributions that may be at a closer proximity with respect to the circular uniform null distribution.

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