Suppose one wants to obtain the maximum likelihood estimate $x^{MLE}$ of a non-random vector $x \in \mathcal{R}^{p}$ using $n$ measurements $z_1, \dots, z_n \in \mathcal{R}^k$. The observations are non-linear function of $x$ corrupted with (iid) zero-mean Gaussian noise $w_i$ with known covariance matrix $R_i$: $$ z_i = f(x) + w_i $$ I know that if $f(\cdot)$ was a linear function of $x$, then $$ S(x^{MLE}) = \sum_i (z_i - f(x^{MLE}))^T R_i^{-1} (z_i - f(x^{MLE})) $$ would be distributed according to a $\chi^2$ distribution with $nk-p$ degrees of freedom.
Now for non-linear $f(\cdot)$:
- Is $S(x^{MLE})$ distributed according to a $\chi^2$ distribution? If yes, what is its degrees of freedom?
- Let's assume $S(x^{MLE})$ is approximately distributed according to a $\chi^2$ distribution with the same degrees of freedom. Suppose we are sure about the correctness of the model: distribution of the noise, iid noise assumption, etc. Is it justified to use the $\chi^2$ test to see if a candidate solution $\tilde{x}$ can be the maximum likelihood estimate $x^{MLE}$ with certain probability by checking the probability of getting $S(\tilde{x})$?
