A goodness-of-fit test for multivariate logistic distributions

We propose a novel goodness-of-fit test for the multivariate logistic distribution, leveraging the properties of its characteristic function. The test statistic is constructed by measuring a weighted L2-distance between the empirical characteristic function of the standardized data and the theoretical characteristic function of the standard multivariate logistic distribution. The approach is grounded in a radial Fourier transform, expressed via the Hankel transform, and yields an explicit, closed-form expression involving three integrals. We derive the asymptotic distribution of the test statistic under both the null hypothesis and contiguous alternatives, and prove its affine invariance. Critical values are determined using Monte Carlo simulation. Through extensive simulations, we demonstrate that the proposed test has excellent control of Type I error and exhibits superior power relative to existing methods, including the multivariate Kolmogorov–Smirnov and energy tests, especially in higher dimensions or under heavy-tailed alternatives. Applications to real data further underscore the test's practical utility and robustness. MATLAB and R implementations are made publicly available.