We formulate a correlation-function theory of the multiphonon absorption due to nonlinear electric moments in anharmonic crystals, at frequencies ω far above the reststrahl ω0 employing the method of cumulants. In contrast to previous treatments, direct expansion of the moment and anharmonicity potential in powers of displacements is avoided; we thus obtain expressions containing various classes of phonon processes summed to infinite order. The frequency and temperature dependence of the absorption coefficient α is calculated for various approximations and simplified limits, including: the harmonic limit, for which computations with Debye and Einstein models are carried out; the quadratic anharmonic approximation to the cumulant, for which computations are carried out in the noninteracting-cell picture; and the single-particle model, within which "exact" computations are pursued. The results indicate exponentiallike frequency dependences for α for all cases, with enhancement and broadening of α for large anharmonicity, behavior similar to that predicted for anharmonicity absorption stemming from linear moments alone. Application of the single-particle model to cubic diatomic crystals demonstrates significant effects of anharmonicity on α, especially for ωω0>>1. Moreover, for covalent crystals nonlinear moment contributions to α dominate, while for highly ionic crystals the linear term can exceed or compete with the nonlinear ones for small to intermediate values of ωω0.