Generalized gamma distribution based on the Bayesian approach with application to investment modelling

Abstract

The Generalized Gamma Distribution (GGD) is one of the most popular distributions in analyzing real lifetime datasets. Estimating the parameters of a high dimensional probability distribution is challenging due to the complexities associated with the resulting objectives function. When traditional estimation techniques fail due to complexity in the model objectives function, other powerful computational approaches are employed. In this study, a Bayesian approach to Generalized Gamma Distribution (GGD) based on Markov Chain Monte-Carlo (MCMC) has been employed to estimate model parameters. This study considers the Bayesian approach to GGD parameters using the Adaptive Rejection Metropolis Sampling (ARMS) technique of random variable generation within the Gibbs sampler. The MCMC approach has been used for estimating the multi-dimensional objectives function distribution. The results of the ARMS were compared to the existing Simulated annealing (SA) algorithm and Method of Moment (MM) based on modified internal rate of return data (MIRR). The performances of various derived estimators were recorded using the Markov chain Monte Carlo simulation technique for different sample sizes. The study reveals that ARMS's performance is marginally better than the existing SA and MA approaches. The efficiency of ARMS does not require a larger sample size as the SA does, in the case of simulated data. The performances of ARMS and SA are similar comparing them to the MM as an initial assumption in the case of real MIRR data. However, ARMS gives an acceptable estimated parameter for the different sample sizes due to its ability to evaluate the conditional distributions easily and sample from them exactly.

Keywords

• modified internal rate of return
• markov chain monte-carlo
• maximum likelihood
• simulated annealing algorithm
• method of moment