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Say I have some data $(x_{1},\ldots,x_{n})$ which I believe to be drawn from some distribution $\nu_{\theta}(x)$. I'm moderately familiar with estimation techniques for $\theta$ when I have some functional form for $\nu_{\theta}(x)$.

However I'm wondering if someone can give me an introductory reference (or at least the best words to google) for when all I can do is simulate from $\nu_{\theta}(x)$ and so want to do inference with respect to the estimated distribution $\hat \nu_{\theta}(x)$.

My naive approach would be to construct an empirical cdf from my data $\hat{F}(x)$ and then use the KS metric to determine the closest distribution. More specifically, I let \begin{equation} \hat\theta=\text{argmin}_{\theta}\sup_{x}|\hat\nu_{\theta}(x)-\hat{F}(x)| \end{equation} but then what are the statistical properties of this estimator?

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When $\nu_\theta$ is the true cdf, this method is called minimum distance estimation and there are consistency results in the literature. Since you can approximate $\nu_\theta$ with an arbitrary precision by simulating enough points, the above results extend to this case.

One famous paper in this literature is Berkson's 1980 "Minimum Chi-Square, not Maximum Likelihood!", but I do not find the arguments there completely compelling towards giving up maximum likelihood estimation. Maximum likelihood is most often enjoying asymptotic efficiency, which other methods may fail to meet.

Note that, when considering a Bayesian approach to the problem, this method pertains to approximate Bayesian inference (ABC).

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    $\begingroup$ Thanks for your answer. Do you know if there is a 'best' or 'standard' approach? I've come across maximum simulated likelihood which I think would also work (and be cheaper) but I'd like to know what the standard solution is. $\endgroup$ – Michael Jan 5 '17 at 11:20

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