You can obtain maximum likelihood estimates of the parameters by partially differentiating the likelihood of the sample (and not the p.d.f.) and each time, equating to zero.

This might help you out:

Article Maximum Likelihood Identification of Wiener Models with a Li...

You can introduce joint distribution to pdf. Then differentiate with respect to the parameters.

Parameters that maximize the log likelihood function are the best then.

Thanks for your suggestions. I'll go through it.