The block sphericity test-exact and near-exact distributions for the likelihood ratio statistic

Using a suitable decomposition of the null hypothesis of the test of sphericity for $k$ blocks of $p_i$ variables, into a sequence of conditionally independent null hypotheses we show that it is possible to obtain the expression of the likelihood ratio test statistic, the expression for the $h$-null moment and the characteristic function of the logarithm of the likelihood ratio test statistic. The exact distribution of the logarithm of the likelihood ratio test statistic is obtained under the form of the sum of a Generalized Integer Gamma distribution with the sum of a 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 on 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 respective 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.