Efficient simulated maximum likelihood with an application to online retailing

Simulated maximum likelihood estimates an analytically intractable likelihood function with an empirical average based on data simulated from a suitable importance sampling distribution. In order to use simulated maximum likelihood in an efficient way, the choice of the importance sampling distribution as well as the mechanism to generate the simulated data are crucial. In this paper we develop an automated, multistage implementation of simulated maximum likelihood which, by adaptively updating the importance sampler, approximates the optimal importance sampling distribution. The proposed method also allows for a convenient incorporation of quasi-Monte Carlo methods. Quasi-Monte Carlo methods produce simulated data which can significantly increase the accuracy of the likelihood-estimate over regular Monte Carlo methods. Several examples provide evidence for the potential efficiency gain of this new method. We apply the method to a computationally challenging geostatistical model of online retailing.