Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution

In this work, we develop Bayesian analysis based on the Jeffreys prior for the hyperbolic family of distributions. It is usually difficult to estimate the four parameters in this class: to be reliable the maximum likelihood estimator typically requires large sample sizes of the order of thousands of observations. Moreover, improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. Here, we show through a simulation study that Bayesian methods based on Jeffreys prior provide reliable point and interval estimators. Moreover, this simulation study shows that for the absolute loss function Bayesian estimators compare favorably to maximum likelihood estimators. Finally, we illustrate with an application to real data that our methodology allows for parameter estimation with remarkable good properties even for a small sample size.