BAYESIAN ANALYSIS USING MCMC METHODS OF RECORD VALUES BASED ON A NEW GENERALISED RAYLEIGH DISTRIBUTION ROBERT G. AYKROYD, M. A. W. MAHMOUD, AND HASSAN M. ALJOHANI Abstract. In this paper, we extend the Rayleigh distribution to create a gen- eralised 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 assump- tions of gamma prior distributions on model parameters. Point estimates and confidence intervals based on maximum likelihood estimation are com- puted. 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 com- pared with the maximum likelihood estimators using simulated data showing dramatic superiority of the Bayesian approach. 1. Introduction The standard Rayleigh distribution (SRay) is useful in life testing experiments, as its failure rate is a linear function of time. This distribution was originally introduced by Lord Rayleigh [21, 22] in connection with a problem in the field of acoustics. [18] derived the SRay distribution as the probability distribution of the distance from the origin to a point (X 1 ,X 2 ,...,X n ) in n-dimensional Euclidean space, where the X i ’s are independent and identically distributed N (0,θ) random variables. [6] demonstrated the importance of this distribution in communication engineering and [19] noted that some types of electro-vacuum devices have the fea- ture that their rate of ageing changes with time. [12] presented a brief account of the history and properties of this distribution, with other aspects of this distribu- tion discussed in [17]. [14] computed the modified maximum likelihood estimator for the scale parameter of the SRay distribution from doubly censored samples and [5] calculated the maximum likelihood estimator for the one parameter standard Rayleigh distribution based on Type-II censoring. [25] wrote the posterior den- sity of the hazard function and also developed Bayesian interval estimates for the one parameter of the standard Rayleigh distribution. [3] obtained the maximum 2000 Mathematics Subject Classification. 62N0, 62F15. Key words and phrases. Bayesian inference; maximum likelihood; life testing; Metropolis- Hastings methods; standard Rayleigh distribution; reliability. 1 Stochastic Modeling and Applications Vol.21, No. 2 (December, 2017), 49-66 49