FFT has unexpected DC component
The first and the obvious algorithm for the solution of this problem is the EM algorithm. This algoritm is getting worse in the number of parameters. It is also too much dependent on the initial conditions.

The drawbacks of this algorithm can be circumvented by parallel computing. This means you have $N$ EM algorithms run simultaneously which have different initialization points. The final algoritm takes all parameters estimated by N processors and based on the wellness of the fit gives the final decicion.

Another method is to use the central moments until Mth order. When written in a parametric form each xth order moment gives a linear/non-linear combination of parameters which are to be estimated. From here one can construct a test. Of course the moments should also be estimated from the data.

Another method would be to employ a better search algorithn. That is called particle swarm optimization. It will run similar to EM. However it is a stochastic algorithm have different characteristics than EM such as to be able to deal with more number of parameters. It is a kind of team optimization algorithm.

Thats all in my mind at the moment. Good luck with the experiments.