Estimating functions in indirect inference

• Published in: Journal of the Royal Statistical Society. Series B, Statistical methodology Statistical methodology, 2004-01, Vol.66 (2), p.447-462
• Main Authors: Heggland, Knut, Frigessi, Arnoldo
• Format: Article
• Language: English
• Subjects: Asymptotic variance; Covariance matrices; Datasets; Distribution theory; Estimating functions; Estimation; Estimators; Evaluation; Exact sciences and technology; G/G/1 queue; General topics; Implicit statistical models; Indirect inference; Inference; Mathematical independent variables; Mathematics; Maximum likelihood estimation; Modeling; Nonparametric inference; Parametric inference; Parametric models; Probability; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Simulated likelihood; Simulations; Statistical methods; Statistical variance; Statistics; Transactional data
• ISSN: 1369-7412; 1467-9868
• DOI: 10.1111/j.1369-7412.2003.05341.x

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 θ in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of θ. First an auxiliary model is chosen whose parameter β can be directly estimated. Next, the parameters in the auxiliary model are estimated for the original data, leading to an estimate β̂. The parameter β is also estimated by using several sampled data sets, simulated from the original model for different values of the original parameter θ. Finally, the parameter θ which leads to the best match to β̂ 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 β 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 θ, 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.