Abstract
The power law process has been used extensively in software reliability models, reliability growth models and more generally reliable systems. In this paper we work on the Power Law Process via empirical Bayes (EB) approach. Based on a two-hyperparameter natural conjugate prior and a more generalized three-hyperparameter natural conjugate prior, which was stated in Huang and Bier (1998), we work out an empirical Bayes (EB) procedure and provide statistical inferences based on the natural conjugate priors. Given past experience about the parameters of the model, the empirical Bayes (EB) approach uses the observed data to estimate the hyperparamters of priors and then proceeds as though the prior were known.
Highlights
Bayesian inference on the power law process was studied during the past two decades
An empirical Bayes (EB) approach on the Power Law Process with natural conjugate priors has been developed in this paper
We obtained a closed form of marginal distribution of observed data in the two-hyperparameter case
Summary
Bayesian inference on the power law process was studied during the past two decades. We applied Huang’s (1998) prior to obtain Bayesian maximum likelihood estimate of a hyperparameter a, posterior means of the shape parameter β and the scale parameter η in closed forms. The natural conjugate prior distribution for the power law failure model is given by. Tn) from one system, which means we only have a random sample of size one (η, β) from the prior π(η, β|a, m, ym), our inference shall be regarded as Bayesian maximum likelihood approach. For this special case, we have the posterior distribution of (η, β) is π(η, β|t, n, m , a, ym) c−1ηn+m −1 β n+m −1[exp(−a)ymm. Variation in these estimates would lead to more variation in the estimates of function of parameters, such as intensity and reliability etc
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