voigtinference: Exact likelihood calculus and conditional attribution for the Voigt profile

Abstract

Fast, accurate algorithms for the Faddeeva function w(z), and hence the Voigt profile K(x,a), have existed for four decades, and analytic first derivatives are available in some implementations. What the established libraries reviewed here have not provided is the full likelihood calculus of the normalized Voigt distribution: applications still commonly resort to pseudo-Voigt approximations, finite-difference derivatives, or numerical convolution for parameter inference. Because w'(z) = -2z w(z) + 2i/sqrt(pi), every derivative of the Voigt log-likelihood is an algebraic function of K and the dispersion part L(x,a) = Im w(z), from the single complex evaluation that delivers the profile. This yields the score and Hessian in closed form, and the expected Fisher information by one-dimensional quadrature of an analytic integrand; for fixed interior widths sigma, gamma > 0, the MLE of the center and both widths is consistent and asymptotically normal at rate sqrt(n), despite the distribution having no finite mean or variance, so conventional likelihood-based standard errors apply. The conditional mean of the Gaussian component given an observation is (y - mu) - gamma L/K: a redescending function that attributes moderate deviations to the Gaussian (Doppler/resolution) component and extreme ones to the Lorentzian tail. The package voigtinference (Python, NumPy/SciPy, with a cross-validated Julia companion) supplies the toolkit: score, full parameter Hessian, expected information, Newton-based unbinned maximum likelihood with boundary diagnostics, conditional component moments, and evaluation validated against high-precision references at extreme width ratios. It applies directly to unbinned non-relativistic, constant-width Breit-Wigner x Gaussian resonance fits and supplies analytic Jacobians for line-shape refinement. Companion paper: arXiv:2605.01665.