MENU
| Título | A general near-exact distribution theory for the most common likelihood ratio test statstics used in Multivariate Statistics |
| Publication Type | Unpublished |
| Year of Publication | 2008 |
| Authors | Marques FJ, Coelho CA, Arnold BC |
| Series Title | Preprint |
| Palavras-chave | Generalized Integer Gamma distribution, Generalized Near-Integer Gamma distribution, independence test, mixtures, sphericity test, Wilks Lambda statistic |
| Abstract | The aim of this paper is to show that while all the exact distributions of the most common likelihood ratio test (l.r.t.) statistics, that is, the ones used to test the independence of several sets of variables, the equality of several variance-covariance matrices, sphericity and the equality of several mean vectors, may be expressed as the distribution of the product of independent Beta random variables or the product of a given number of independent random variables whose logarithm has a Gamma distribution times a given number of a given number of independent Beta random variables, near-exact distributions for theis logarithms may all be expressed as generalized Near-Integer Gamma distributions or mixtures of these distributions, whose rate parameters associated with the integer shape parameters, for samples of size n, all have the form (n-j)/n for j = 2,...,p, where for the first three statistics p is the number of variables involved, while for the fourth one it is the sum of the number of variables involved with the number of mean vectors being tested. What is interesting is that the similitudes among these statistics are somehow even more notorious interms of near-exact distributions than in terms of exact distributions. Then all the l.r.t. statistics that may be built as the product of these basic statistics also inherit a similar stucture for their near-exact distributions. To illustrate this fact, an application is done to the l.r.t. statistic to test the equality of several multivariate Normal distributions. |
| URL | http://www.dm.fct.unl.pt/sites/www.dm.fct.unl.pt/files/preprints/2008/5_08.pdf |