In this paper, we extend the Rayleigh distribution to create a generalised Rayleigh distribution which is more flexible than the standard. The general properties of the new distribution are derived and investigated, with properties of more standard distributions, such as the exponential, standard Rayleigh and the Weibull, appearing as special cases. Further, we consider maximum likelihood estimation and Bayesian inference under the assumptions of gamma prior distributions on model parameters. Point estimates and confidence intervals based on maximum likelihood estimation are computed. The main challenge, however, is that the Bayesian estimators cannot easily be found and hence, Markov chain Monte Carlo (MCMC) techniques are proposed to generate samples from the posterior distributions leading to approximate posterior inference. The approximate Bayes estimators are compared with the maximum likelihood estimators using simulated data showing dramatic superiority of the Bayesian approach
