Approximate Maximum Likelihood Estimation
Bertl et al. arXived last July a paper on a maximum likelihood estimator based on an alternative to ABC techniques. And to indirect inference. (One of the authors in et al. is Andreas Futschik whom I visited last year in Linz.) Paper that I only spotted when gathering references for a reading list on ABC… The method is related to the “original ABC paper” of Diggle and Gratton (1984) which, parallel to Rubin (1984), contains in retrospect the idea of ABC methods. The starting point is stochastic approximation, namely the optimisation of a function of a parameter θ when written as an expectation of a random variable Y, E[Y|θ], as in the Kiefer-Wolfowitz algorithm. However, in the case of the likelihood function, there is rarely an unbiased estimator and the authors propose instead to use a kernel density estimator of the density of the summary statistic. This means that, at each iteration of the Kiefer-Wolfowitz algorithm, two sets of observations and hence of summary statistics are simulated and two kernel density estimates derived, both to be applied to the observed summary. The sequences underlying the Kiefer-Wolfowitz algorithm are taken from (the excellent optimisation book of) Spall (2003). Along with on-the-go adaptation and convergence test.
The theoretical difficulty in this extension is however that the kernel density estimator is not unbiased and thus that, rigorously speaking, the validation of the Kiefer-Wolfowitz algorithm does not apply here. On the practical side, the need for multiple starting points and multiple simulations of pseudo-samples may induce considerable time overload. Especially if bootstrap is used to evaluate the precision of the MLE approximation. Besides normal and M/G/1 queue examples, the authors illustrate the approach on a population genetic dataset of Borneo and Sumatra orang-utans. With 5 parameters and 28 summary statistics. Which thus means using a kernel density estimator in dimension 28, a rather perilous adventure..!