Abstract
In this paper, we have illustrated the suitability of Gumbel Model for software reliability data. The model parameters are estimated using likelihood based inferential procedure: classical as well as Bayesian. The quasi Newton-Raphson algorithm is applied to obtain the maximum likelihood estimates and associated probability intervals. The Bayesian estimates of the parameters of Gumbel model are obtained using Markov Chain Monte Carlo(MCMC) simulation method in OpenBUGS(established software for Bayesian analysis using Markov Chain Monte Carlo methods). The R functions are developed to study the statistical properties, model validation and comparison tools of the model and the output analysis of MCMC samples generated from OpenBUGS. Details of applying MCMC to parameter estimation for the Gumbel model are elaborated and a real software reliability data set is considered to illustrate the methods of inference discussed in this paper.
Highlights
A frequently occurring problem in reliability analysis is model selection and related issues
Statistical properties of a recently proposed distribution is examined closer and parameter estimation using maximum likelihood as a classical approach by R functions is performed where comparison is made to Bayesian approach using OpenBUGS
The potential applicability of the Gumbel model to represent the distribution of maxima relates to extreme value theory which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type
Summary
A frequently occurring problem in reliability analysis is model selection and related issues. In reliability theory the Gumbel model is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions. The potential applicability of the Gumbel model to represent the distribution of maxima relates to extreme value theory which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type. The Gumbel model is a particular case of the generalized extreme value distribution ( known as the Fisher-Tippett distribution)[2]. It is known as the log-Weibull model and the double exponential model (which is sometimes used to refer to the Laplace model). The Gumbel model is appropriate for modeling strength, which is sometimes skewed to the left
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