Statistical significance of deviation from a complex-valued model

I have complex-valued data. At each one of about 100 linearly spaced x values, I have a corresponding measurement of a complex quantity with well-defined Gaussian uncertainties on both the real and complex parts. For the simplest applicable physical model, the data should be described by a function of the form

\( f(x)=Ae^{i\theta} \)

Here is an example of data and a corresponding model of the type described above, where the red / blue points correspond to the real / imaginary parts of the complex numbers:


Now I want to assess the statistical significance of deviations of the data from models of that type. For a scalar-valued function, I'd just do the simple thing and use Pearson's \( \chi^2 \) test. There must surely be analogous for complex-valued functions, but I'm having trouble finding out what that is. For my problem, the errors on the real and imaginary parts of a given data point are approximately equal, but can vary systematically in magnitude with x.

If there is anyone that can point me in the right direction it would be much appreciated!
Last edited: