Obtaining the exact and near-exact distributions of the likelihood ratio statistic to test circular symmetry through the use of characteristic functions

In this paper the authors show how through the use of the characteristic function of the negative logarithm of the likelihood ratio test (l.r.t.) statistic to test circular symmetry it is possible to obtain highly manageable expressions for the exact distribution of such statistic, when the number of variables, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation>, is odd, and highly manageable and accurate approximations for an even <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation>. For the case of an even <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$p$$</EquationSource> </InlineEquation>, two kinds of near-exact distributions are developed for the l.r.t. statistic which correspond, for the logarithm of the l.r.t. statistic, to a Generalized Near-Integer Gamma distribution or finite mixtures of these distributions. Numerical studies conducted in order to assess the quality of these new approximations show their impressive performance, namely when compared with the only available asymptotic distribution in the literature. Copyright Springer-Verlag Berlin Heidelberg 2013