Estimating functions in indirect inference

There are models for which the evaluation of the likelihood is infeasible in practice. For these models the Metropolis–Hastings acceptance probability cannot be easily computed. This is the case, for instance, when only departure times from a G/G/1 queue are observed and inference on the arrival and service distributions are required. Indirect inference is a method to estimate a parameter (theta) in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of (beta). First an auxiliary model is chosen whose parameter (theta) can be directly estimated. Next, the parameters in the auxiliary model are estimated for the original data, leading to an estimate . The parameter (beta) is also estimated by using several sampled data sets, simulated from the original model for different values of the original parameter. Finally, the parameter which leads to the best match to is chosen as the indirect inference estimate. We analyse which properties an auxiliary model should have to give satisfactory indirect inference. We look at the situation where the data are summarized in a vector statistic T, and the auxiliary model is chosen so that inference on (beta) is drawn from T only. Under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of T and componentwise inversely proportional to the square of the derivative, with respect to (beta), of the expected value of T. We discuss how these results can be used in selecting good estimating functions. We apply our findings to the queuing problem.