Simulated maximum likelihood using tilted importance sampling

This paper develops the important distinction between tilted and simple importance sampling as methods for simulating likelihood functions for use in simulated maximum likelihood. It is shown that tilted importance sampling removes a lower bound to simulation error for given importance sample size that is inherent in simulated maximum likelihood using simple importance sampling, the main method for simulating likelihood functions in the statistics literature. In addition, a new importance sampling technique, generalized Laplace importance sampling, easily combined with tilted importance sampling, is introduced. A number of applications and Monte Carlo experiments demonstrate the power and applicability of the methods. As an example, simulated maximum likelihood estimates from the infamous salamander mating model from McCullagh and Nelder (1989) can be found to easily satisfactory precision with an importance sample size of 100.