The method of simulated quantiles

We introduce an inference method based on quantiles matching, which is useful for situations where the density function does not have a closed form –but it is simple to simulate– and/or moments do not exist. Functions of theoretical quantiles, which depend on the parameters of the assumed probability law, are matched with sample quantiles, which depend on observations. Since the theoretical quantiles may not be available analytically, the optimization is based on simulations. We illustrate the method with the estimation of α-stable distributions. A thorough Monte Carlo study and an illustration to 22 financial indexes show the usefulness of the method.