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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?

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  • $\begingroup$ What $\chi^2$ test you have in mind? $(\theta-\theta_{ML})(Var(\theta))^{-1}(\theta-\theta_{ML})$? $\endgroup$
    – mpiktas
    Commented Aug 30, 2012 at 6:07
  • $\begingroup$ Yes. One problem is that I don't know $Var(\theta)$ so I would have to estimate it from the data. $\endgroup$
    – Arrigo Benedetti
    Commented Aug 31, 2012 at 6:09

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There are three excellent books on nonlinear regression that are at the level of your interest. I don't know if they look at goodness of fit tests for the model though. Anyway one is Gallant

and a second one is Bates and Watts

the third is Seber and Wild

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