Likelihood-based Inference in S-distributions

Abstract

In this paper, we propose new estimation techniques in connection with the system of S-distributions. Besides “exact” maximum likelihood (ML), we propose simulated ML and a characteristic function-based procedure. The “exact” and simulated likelihoods can be used to provide numerical, MCMC-based Bayesian inferences.

Keywords:

• Bayesian inference
• Likelihood function
• MCMC
• Posterior distribution
• S-distributions

Mathematics Subject Classification:

• 62F15
• 62E17

Acknowledgments

The author wishes to thank an anonymous referee for numerous useful comments on an earlier version of the paper.

Notes

One such approach would be to replace the Fs by their empirical counterpart. After ordering the data, the estimate of the distribution function is independent of the parameters and (2) does not possess a maximum.

The simulated data are obtained using rndseed 11 in WinGauss 6.0.

We have found that even 100 simulations are enough in the sense that ML estimates are quantitatively similar compared to ML estimates obtained through a larger number of simulations.

Figure 1. Artificial data (h = 2.2, α = 10, μ = 0 at F₀ = 0.5).

The CF-based procedure is implemented using 10 equispaced values of τ, in the interval ±2.50. The results were not sensitive to these choices provided the interval is not smaller than about ±0.20.

Table 1 ML estimates from artificial data

For the random walk chain the proposal is a multivariate normal with covariance λV, where V is the covariance matrix obtained from the Hessian matrix of the log-likelihood at the ML estimates and λ is a constant adapted during the burn-in phase to guarantee an acceptance rate between 20% and 30%.