# A simple approach to maximum likelihood estimation for a model with no closed-form solution

I would like to estimate the best fitting parameters of a parametric model, $f(\theta)$, that does not have a closed-form solution.

There are $n$ i.i.d. environmental observations and the aim is to find a parameter vector that produces output from $f(\theta)$ that is maximally consistent with this set of observations.

I would like to take a maximum likelihood approach to find such a parameter vector. That is, find ${\operatorname{arg\,max}}\ \ell(\theta\,;\,x_1,\ldots,x_n)$.

Is it sensible to use simulation of the output from the parametric model to calculate the likelihood? For example, taking a large number replicates from $f(\theta)$, an empirical distribution could be generated that would permit calculation of $\mathcal{L}(\theta\,;\,x_1,\ldots,x_n) =P(x_{i} | \theta)\times P(x_{2} | \theta)\times\dots\times P(x_{n} | \theta)$. This could be done for instances by calculation of the proportion of model observations that fall within $m$ equally spaced bins. $P(x_{i} | \theta)$ is just the proportion of the bin to which $x_{i}$ corresponds.

I've looked at a paper that seems to address this issue. They use kernel methods to estimate the unknown density and then proceed to do the maximum likelihood once a density has been estimated. My question is, am I missing something critical by not applying kernel methods and just using the empirical density? Does my approach make much sense at all?

• Could you explicit what you mean by a "parametric model that does not have a closed-form solution"? If the model is completely defined, and can be simulated for a given value of the parameter $\theta$, econometrics techniques like the simulated method of moments or indirect inference can be used. – Xi'an Jan 16 '15 at 17:12
• Thanks for the links. The model is a stochastic simulation of Markov processes roughly based on that which is described in (Gillespie, 1978). It is 'parametric' in the sense that parameters supplied to the model defines the behaviour of the Markov process. Both criteria you suggest seem to apply. – skleene Jan 16 '15 at 18:34