MENU
| Título | The exact and near-exact distributions of the likelihood ratio statistic for the block sphericity test |
| Publication Type | Unpublished |
| Year of Publication | 2010 |
| Authors | Marques FJ, Coelho CA |
| Series Title | Preprint |
| Palavras-chave | asymptotic distributions, block, Generalized Integer Gamma distribution, mixtures, sphericity test |
| Abstract | Using a suitable decomposition of the null hypothesis of the sphericity test for several blocks of variables, into a sequence of conditionally independent null hypotheses, we show that it is possible to obtain the expressions for the likelihood ratio test statistic, for its h-th null moment and for the characteristic function of its logarithm. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained in the form of a sum of a Generalized Integer Gamma distribution with the sum of a given number of independent Logbeta distributions, taking the form of a single Generalized Integer Gamma distribution when each set of variables has two variables. The development of near-exact distributions arises, from the previous decomposition of the null hypothesis and from the consequent induced factorization of the characteristic function, as a natural and practical way to approximate the exact distribution of the test statistic. A measure based on the exact and approximating characteristic functions, which gives an upper bound on the distance between the corresponding distribution functions, is used to assess the quality of the near-exact distributions proposed and to compare them with an asymptotic approximation based on Box's method. |
| URL | http://www.dm.fct.unl.pt/sites/www.dm.fct.unl.pt/files/preprints/2010/22_10.pdf |