On Bayesian Estimation of the Product of Poisson Rates with Application to Reliability

ABSTRACT This article derives a non informative prior for a product of k independent Poisson rates, i.e., , thereby considering the problem of estimating the product. Based on the prior, a procedure for obtaining posterior quantiles and HPD interval of the product is suggested. The procedure provides a facile solution for both distributional specification and computational implementation, whose outputs are readily adapted for required inference summaries. A salient feature of the procedure is that it provides a unified method for estimating various types of the product, thereby solving all the existing problems (involved in estimating the product) simultaneously and satisfactorily, at least within the Bayesian framework. In the three examples provided, practical applications of the procedure to reliability are well described.

[1]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[2]  James O. Berger,et al.  Noninformative priors for inferences in exponential regression models , 1991 .

[3]  Kishor S. Trivedi,et al.  A COMPARISON OF APPROXIMATE INTERVAL ESTIMATORS FOR THE BERNOULLI PARAMETER , 1993 .

[4]  M. Ghosh,et al.  Second-order probability matching priors , 1997 .

[5]  R. Tibshirani.  Noninformative priors for one parameter of many , 1989 .

[6]  Marvin Rausand,et al.  Bayesian Reliability Analysis , 2008, System Reliability Theory.

[7]  Malay Ghosh,et al.  ON THE INVARIANCE OF NONINFORMATIVE PRIORS , 1996 .

[8]  Rahul Mukerjee,et al.  On a property of probability matching priors: matching the alternative coverage probabilities , 1999 .

[9]  B. L. Welch,et al.  On Formulae for Confidence Points Based on Integrals of Weighted Likelihoods , 1963 .

[10]  W. Michael Conklin,et al.  Monte Carlo Methods in Bayesian Computation , 2001, Technometrics.

[11]  Ming-Hui Chen,et al.  Monte Carlo Estimation of Bayesian Credible and HPD Intervals , 1999 .

[12]  Albert Madansky,et al.  APProximate Confidence Limits for the Reliability of Series and Parallel Systems , 1965 .

[13]  Bernard Harris,et al.  Hypothesis Testing and Confidence Intervals for Products and Quotients of Poisson Parameters with Applications to Reliability , 1971 .

[14]  C. Page.  Allocation Proportional to Coefficients of Variation When Estimating the Product of Parameters , 1990 .

[15]  The Reliability of Systems of Independent Parallel Components When Some Components are Repeated , 1973 .

[16]  Keying Ye,et al.  Reference Prior Bayesian Analysis for Normal Mean Products , 1995 .

[17]  Charles Stein,et al.  On the coverage probability of confidence sets based on a prior distribution , 1985 .

[18]  M. Ghosh,et al.  Some new Results on Probability Matching Priors , 2000 .

[19]  D. Cox,et al.  Parameter Orthogonality and Approximate Conditional Inference , 1987 .

[20]  John S. Greenlees,et al.  Incorporating a geometric mean formula into the CPI , 1998 .