A method is described for estimating telephone handset nonlinearity by matching the spectral magnitude of the distorted signal to the output of a nonlinear channel model, driven by an undistorted reference. This "magnitude-only" representation allows the model to directly match unwanted speech formants that arise over nonlinear channels and that are a potential source of degradation in speaker and speech recognition algorithms. As such, the method is particularly suited to algorithms that use only spectral magnitude information. The distortion model consists of a memoryless nonlinearity sandwiched between two finite-length linear filters. Nonlinearities considered include arbitrary finite-order polynomials and parametric sigmoidal functionals derived from a carbon-button handset model. Minimization of a mean-squared spectral magnitude distance with respect to model parameters relies on iterative estimation via a gradient descent technique. Initial work has demonstrated the importance of addressing handset nonlinearity, in addition to linear distortion, in speaker recognition over telephone channels. A nonlinear handset "mapping" applied to training or testing data to reduce mismatch between different types of handset microphone outputs, improves speaker verification performance relative to linear compensation only. Finally, a method is proposed to merge the mapper strategy with a method of likelihood score normalization (hnorm) for further mismatch reduction and speaker verification performance improvement.