Quantile-Based Inference for Tempered Stable Distributions

If the closed-form formula for the probability density function is not available, implementing the maximum likelihood estimation is challenging. We introduce a simple, fast, and accurate way for the estimation of numerous distributions that belong to the class of tempered stable probability distributions. Estimation is based on the Method of Simulated Quantiles (Dominicy and Veredas (2013)). MSQ consists of matching empirical and theoretical functions of quantiles that are informative about the parameters of interest. In the Monte Carlo study we show that MSQ is significantly faster than Maximum Likelihood and the estimates are almost as precise as MLE. A Value at Risk study using 13 years of daily returns from 21 world-wide market indexes shows that MSQ estimates provide as good risk assessments as with MLE