Sampling-Assisted non-parametric Inference of Intractable Models

The likelihood function plays a fundamental role in most statistical analysis. Nevertheless, when the likelihood is too complex to compute in traditional ways, indirect inference or Approximate Bayesian Computation (ABC) is often adopted. This paper proposes a new approach to estimate the intractable likelihood function based on the Hermite polynomial expansion. Based on the estimated likelihood function, a maximum likelihood estimate and its large sample properties are achieved. Comparing to other existing methods, the proposed method provides more accurate estimation and are justified by asymptotic theories. Moreover, it does not require extra auxiliary models, which are necessary for indirect methods and are often not easy to find. The proposed method also has a very close link to ABC in the sense that it also simulates many pseudo data sets; however it does not need to compare the pseudo data with the raw data as what ABC does. Therefore, the proposed method does not suffer from the drawbacks of ABC; for example ABC always requires rich experience for choosing a proper value of the distance function, data summary statistics and threshold for acceptance rate.