In the context of a statistical signal processing problem, I have a signal model of the form $x[n]=s[n;\theta]+w[n]$ where $w[n]\sim{\cal N}(0,\sigma^2)$ and $s[n;\theta]$ depends nonlinearly on $\theta$. $n=0,\ldots,N-1$ is the time sample index. I compute the Maximum Likelihood estimate of $\theta$ as $\theta_{\rm ML}=\underset{\theta}{\arg\min}\,\,p(x[0],x[1],\ldots,x[N-1]|\theta)$ by minimizing the log-likelihood.

Now my question is: can I use the $\chi^2$ distribution for a goodness-of-fit test in this case? Most statistics books I have seen present the $\chi^2$ test in the context of linear regression.

I am especially interested in the general multivariate case ($\boldsymbol{\theta}$ is a vector) and $\mathbf{w}[n]\sim{\cal N}(0,\mathbf{C})$. Also, can someone please recommend a good book (at the level of Casella and Berger) presenting relevant material?