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I have a question concerning the the fitting of a distribution model to some experimental data. The experimental data are pixel intensity values recorded from a detector and I am modeling the generative physical process describing how the light intensity varies at the detector. I am not able to write down a closed form for the distribution in light intensity at the detector resulting from the physical process but I am able to sample from the physical model. When fitting distribution models in the past I have usually been able to calculate the likelihood of the data given the distribution model and apply a simple maximum likelihood procedure to fit the model. However, in this case I only have access to a sample of data drawn from the model. I assume that the procedure I should follow will be based around a non-linear optimisation of the model parameters (5 in this case) with respect to a goodness-of-fit function that compares the empirical and predicted sample distributions, for example, something like the Komolgorov-Smirnoff, or Anderson-Darling statistic. I'd like to know whether I am on the right track here and if there are any favoured search methods that are best to apply in this context. Thanks for any help. Best wishes.

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You are on the right track. In this context, you have to perform a nonlinear regression analysis. To estimate the 5 parameters, several optimisation algorithms can be used (e.g. sequential quadratic programming, Levenberg-Marquardt method, steepest descent method). You can choose it among the options of the statistical software you use. One of the most used is the Levenberg-Marquardt, but it should be highlighted that it is valid only for unconstrained models. After estimating patameters, the goodness of fit can be evaluated by different methods, such as assessment of residuals, checking independence of errors, checking normality of error distributions (e.g. using one of the two tests that you cited), visual assessment, linearization, and so on.

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  • $\begingroup$ Hi Anatoly, Thanks for your reply. My concern in this case is that gradient based optimisation methods may struggle because the error surface derives from samples drawn from a generative model that contains several stochastic elements. This means that under repeat sampling the error function will vary even when the parameter values are held constant. In my experience the noisy gradient information deriving from these samples tends to inhibit convergence. For this reason I was wondering if a population based stochastic optimisation method might be more appropriate, best wishes, John. $\endgroup$ – John May 27 '14 at 8:59
  • $\begingroup$ I agree, if your information is considerably noisy, a population based stochastic optimisation method could be an appropriate choice. $\endgroup$ – Anatoly May 27 '14 at 21:12

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