  The frequency dependence of third-harmonic medium amplitude oscillatory shear
(MAOS) modulus $G_{33}^{*}(\omega)$ provides insight into material behavior and
microstructure in the asymptotically nonlinear regime. Motivated by the
difficulty in the measurement of MAOS moduli, we propose a test for data
validation based on nonlinear Kramers-Kronig relations. We extend the approach
used to assess the consistency of linear viscoelastic data by expressing the
real and imaginary parts of $G_{33}^{*}(\omega)$ as a linear combination of
Maxwell elements: the functional form for the MAOS kernels is inspired by
time-strain separability (TSS). We propose a statistical fitting technique
called the SMEL test, which works well on a broad range of materials and models
including those that do not obey TSS. It successfully copes with experimental
data that are noisy, or confined to a limited frequency range. When Maxwell
modes obtained from the SMEL test are used to predict the first-harmonic MAOS
modulus $G_{31}^{*}$, it is possible to identify the range of timescales over
which a material exhibits TSS.
