Carlos Coelho

Universidade Nova de Lisboa, Mathematics, Faculty Member

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Carlos A. Coelho hold a Ph.D from the University of Michigan and his research interests are in Multivariate Analysis, namely Likelihood ratio tests for elaborate covariance structures and for equality of mean vectors with elaborate covariance structures as well as likelihood ratio tests for high-dimensionality situations.
Address: Lisbon, Lisboa, Portugal

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Papers by Carlos Coelho

O teste de esfericidade para várias amostras

Multivariate Analysis - A Great Tool for a Multivariate World

In this short report the author tries to introduce in simple terms some Multivariate Analysis tec... more

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A double decomposition of the test of independence of sets of variables that allows for a modeling view of this test

Near-exact distributions for the likelihood ratio statistic used to test the reality of a covariance matrix

11TH INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2013: ICNAAM 2013, 2013

ABSTRACT In a first stage the authors show how the exact distribution of the likelihood ratio sta... more ABSTRACT In a first stage the authors show how the exact distribution of the likelihood ratio statistic used to test the reality of a covariance matrix may be expressed as the distribution of the sum of two independent random variables, one with a Generalized Integer Gamma distribution and the other with the distribution of a sum of independent Logbeta random variables. From this form of the exact distribution the authors develop then a family of near-exact distributions, based on finite mixtures of Generalized Near-Integer Gamma distributions. These near-exact distributions match, by construction, some of the first exact moments and they have very manageable cumulative distribution functions, which allow for an easy computation of sharp near-exact quantiles and p-values.

Generalized Canonical Analysis

De Gruyter eBooks, Dec 31, 1987

Near-Exact Distributions for Likelihood Ratio Statistics Used in the Simultaneous Test of Conditions on Mean Vectors and Patterns of Covariance Matrices

Mathematical Problems in Engineering, 2016

The authors address likelihood ratio statistics used to test simultaneously conditions on mean ve... more The authors address likelihood ratio statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices. Tests for conditions on mean vectors, assuming or not a given structure for the covariance matrix, are quite common, since they may be easily implemented. But, on the other hand, the practical use of simultaneous tests for conditions on the mean vectors and a given pattern for the covariance matrix is usually hindered by the nonmanageability of the expressions for their exact distribution functions. The authors show the importance of being able to adequately factorize the c.f. of the logarithm of likelihood ratio statistics in order to obtain sharp and highly manageable near-exact distributions, or even the exact distribution in a highly manageable form. The tests considered are the simultaneous tests of equality or nullity of means and circularity, compound symmetry, or sphericity of the covariance matrix. Numerical studies show the high accuracy of the near-exact distributions and their adequacy for cases with very small samples and/or large number of variables. The exact and near-exact quantiles computed show how the common chi-square asymptotic approximation is highly inadequate for situations with small samples or large number of variables.

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Likelihood Ratio Test for the Hyper-Block Matrix Sphericity Covariance Structure

DOAJ (DOAJ: Directory of Open Access Journals), Jul 1, 2018

Testing simultaneously different covariance block diagonal structures – the multi-sample case

Journal of Applied Statistics, Jan 13, 2020

In this work a likelihood ratio test which allows to test simultaneously if, several covariance m... more In this work a likelihood ratio test which allows to test simultaneously if, several covariance matrices are equal and block diagonal with different specific structures in the diagonal blocks, is developed. The distribution of the likelihood ratio statistic is studied and the expression of its hth null moment are derived. In order to make the test useful in practical terms, near-exact approximations are developed for the likelihood ratio statistic. A practical application to real data set together with numerical studies and simulations are provided in order illustrate the applicability of the test and also to assess the precision of the near-exact approximations developed.

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Near-exact distributions for the likelihood ratio test statistic to test equality of several variance-covariance matrices in elliptically contoured distributions

Computational Statistics, Oct 5, 2011

Page 1. Comput Stat DOI 10.1007/s00180-011-0281-1 ORIGINAL PAPER Near-exact distributions for the... more

MathematicaⓇ, Maxima, and R Packages to Implement the Likelihood Ratio Tests and Compute the Distributions in the Previous Chapter

Lecture notes in statistics, 2019

In this chapter details are given on the packages programmed in MathematicaⓇ, Maxima, and R for t... more In this chapter details are given on the packages programmed in MathematicaⓇ, Maxima, and R for the implementation of all the tests addressed in Chap. 5. Functions and modules were developed for several tasks such as the computation of p-values and quantiles for each test and to obtain the computed value of the statistics and corresponding p-values from data stored in a data file. As such, also functions and modules used to read these data files, which may have several different internal structures, were programmed. Details are given on the use of all functions and modules, and comparisons are established among the three packages. The full content of all three packages is available on the book’s supplementary material web site.

Near-Exact Distributions for the Likelihood Ratio Test Statistic for Testing Multisample Independence—The Real and Complex Cases

Journal of statistical theory and practice, Jul 17, 2014

ABSTRACT We consider a generalization of the well-known independence of several groups of variabl... more ABSTRACT We consider a generalization of the well-known independence of several groups of variables test, which we designate by multisample independence test of several groups of variables. This new generalization is of great interest whenever we want to test if in different populations, which may follow a multivariate complex or real normal distribution, the Hermitian covariance matrices have the same structure and if there is independence between different groups of variables. We show that the test statistic has the distribution of the product of independent beta random variables; however, the explicit expressions for the probability density and cumulative distribution functions turn out to be very complicated and almost impossible to use in practice. Our objective is to use a breakthrough technique to develop near-exact distributions for the test statistic. These approximations are known to be highly accurate and easy to use, which facilitates and encourages their use in practice. Using a decomposition of the null hypothesis of the test into two null hypotheses we obtain, in a simple way, the likelihood ratio test statistic, the expression of its hth null moment, and the characteristic function of its logarithm. The decomposition of the null hypothesis also induces a factorization on the characteristic function of the logarithm of the test statistic, which enables the development of near-exact distributions. The numerical studies presented highlight the good properties of these approximations and show their great precision. Simulation studies conducted show the good power of the test proposed even for alternatives quite close to the null hypothesis. An example of application of the test is also provided.

Likelihood Ratio Tests for Equality of Mean Vectors with Circular Covariance Matrices

Contributions to statistics, 2018

While the likelihood ratio test for the equality of mean vectors, when the covariance matrices ar... more While the likelihood ratio test for the equality of mean vectors, when the covariance matrices are assumed to be only positive-definite, is a common test in multivariate analysis, similar likelihood ratio tests are not available in the literature when the covariance matrices are assumed to have some common given structure. In this compact paper the author deals with the problem of developing likelihood ratio tests for the equality of mean vectors when the covariance matrices are assumed to have a circular or circulant structure. The likelihood ratio statistic is obtained and its exact distribution is expressed in terms of products of independent Beta random variables. Then, it is shown how for some particular cases it is possible to obtain very manageable finite form expressions for the probability density and cumulative distribution functions of this distribution, while for the other cases, given the intractability of the expressions for these functions, very sharp near-exact distributions are developed. Numerical studies show the extreme closeness of these near-exact distributions to the exact distributions.

Testing the equality of several linear regression models

Computational Statistics, Dec 1, 2016

The linear regression models are widely used in different research fields, and often there is the... more The linear regression models are widely used in different research fields, and often there is the need to analyze if there are similarities between two or more different linear models or to verify if a given relation between two variables remains the same in different intervals of time, in particular in cases where small differences might make a big difference. Motivated by these problems the authors consider a test of equality of k linear regression models which is a simultaneous test of equality of slopes, intercepts and variances. In order to overcome the extreme difficulties that exist in the use of the exact distribution of the likelihood ratio test (LRT) statistic and to make this test reliable and easy to use, we propose the use of near-exact distributions to approximate the distribution of the LRT statistic, under H 0 , in the balanced case, and of new asymptotic approximations for the unbalanced case. The near-exact approximations are built by approximating one factor of an adequate factorization of the characteristic function of the logarithm of the LRT statistic and may be easily implemented. The asymptotic approximations are developed using an expansion for the ratio of gamma functions. The quality of these approximations is analyzed and confirmed. Power studies are conducted in order to better assess the performance of the test. Finally to illustrate the applicability of the test we consider a real data set of gross domestic product at market prices and final consumption expenditure in European countries and one tests the existence of similarities between countries.

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Likelihood Ratio Tests for Elaborate Covariance Structures and for MANOVA Models with Elaborate Covariance Structures—A Review

Journal of the Indian Institute of Science, Apr 17, 2022

In this paper a review is made from the primordia of the history of likelihood ratio tests for co... more In this paper a review is made from the primordia of the history of likelihood ratio tests for covariance structures and equality of mean vectors through the development of likelihood ratio tests that refer to elaborate covariance structures. Relations are established among several covariance structures, taking more elaborate ones as umbrella structures and examining then their particular cases of interest. References are made to bibliography where the corresponding likelihood ratio tests are developed and the distributions of the corresponding statistics addressed. Most of the likelihood ratio test statistics for one-way manova models where the covariance matrices have elaborate structures were developed quite recently. Also for these likelihood ratio tests a similar approach is taken. Although we start with the common test that uses unstructured covariance matrices, then we go on to consider tests with more elaborate covariance structures, and subsequently we specify them to their particular cases of interest. Some special attention is also given to the so-called Wilks Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda$$\end{document} statistics.

Testing the hypothesis of a doubly exchangeable covariance matrix for elliptically contoured distributions

RePEc: Research Papers in Economics, 2014

In this paper the authors study the problem of testing the hypothesis of a doubly exchangeable co... more In this paper the authors study the problem of testing the hypothesis of a doubly exchangeable covariance matrix for three-level multivariate observations, taken on m variables over u sites and over v time/space points. Through the decomposition of the main hypothesis into a set of three sub-hypotheses, the likelihood ratio test statistic is defined, its exact moments are determined, and its exact distribution is studied. Because this distribution is very much intricate, a very precise near-exact distribution is developed. Numerical studies conducted to evaluate the closeness between this nearexact distribution and the exact distribution show the very good performance of this approximation even for very small sample sizes. A simulation study is also conducted and two real-data examples are presented.

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Setting the Scene

Lecture notes in statistics, 2019

Although the Fox H and Meijer G functions yield very handy representations for the probability de... more Although the Fox H and Meijer G functions yield very handy representations for the probability density functions and cumulative distribution functions of several distributions and distributions of products of independent random variables, they are not computationally efficient and most of the time not even utilizable in practice because of serious difficulties found in their efficient computational implementation, even when using the most up-to-date software. This way, looking for alternative representations, namely finite form ones, is a most useful and desirable goal.

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Approximate Finite Forms for the Cases Not Covered by the Finite Representation Approach

Lecture notes in statistics, 2019

In this chapter the authors set the guidelines to approach cases not covered by the finite form r... more In this chapter the authors set the guidelines to approach cases not covered by the finite form representations studied in the book, give new Mellin inversion formulas for both the p.d.f. and the c.d.f., and develop sharp upper bounds on the difference between the exact and approximate representations for the Meijer G functions as well as for the differences between the exact and approximate p.d.f.’s and c.d.f.’s of the product of independent Beta r.v.’s.

Approximate Finite Forms for the Cases Not Covered by the Finite Representation Approach

Finite Form Representations for Meijer G and Fox H Functions, 2019

The Meijer G and Fox H Functions

Finite Form Representations for Meijer G and Fox H Functions, 2019

In this chapter besides briefly introducing the Meijer G and Fox H functions through their usual ... more In this chapter besides briefly introducing the Meijer G and Fox H functions through their usual definitions, the authors open the way to the establishment of several cases where there are finite representations for these functions which have not been previously identified and that as such are not recognized by any of the available software. These cases are related to the distribution of three extended products of independent Beta random variables, which are treated in Chap. 3.

Finite Form Representations for Extended Instances of Meijer G and Fox H Functions

Finite Form Representations for Meijer G and Fox H Functions, 2019

In this chapter the authors use the results in the three theorems in the previous chapter to obta... more In this chapter the authors use the results in the three theorems in the previous chapter to obtain the p.d.f.’s and c.d.f.’s of all the products of independent Beta r.v.’s in those theorems in terms of the Meijer G and the Fox H functions as well as their finite forms based on the EGIG (Exponentiated Generalized Integer Gamma) p.d.f. and c.d.f., this way obtaining the equivalences among several instances of the Meijer G and Fox H functions and their finite representations through the EGIG p.d.f. and c.d.f..

Carlos Coelho

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