Asymptotic Optimality of Three Stage Design for Estimating Product of Means with Applications in Reliability Estimation and Risk Assessment


One-parameter exponential family
first-order optimality
three-stage sampling scheme
sequential design
system reliability
risk assessment
bayesian estimation

How to Cite

Xing Song, & Kamel Rekab. (2016). Asymptotic Optimality of Three Stage Design for Estimating Product of Means with Applications in Reliability Estimation and Risk Assessment. Journal of Advances in Applied & Computational Mathematics, 3(1), 8–19.


The one-parameter exponential family is a practically convenient and widely used unified family of distributions, which contains both discrete and continuous distributions that can be used for practical modelling, such as the Binomial, Beta, Normal, etc. The problem of estimating product of means has been explored for independent populations from one-parameter exponential family in a general sense, with a three-stage sampling design proposed and proven to be first-order efficient. The purpose of this paper is to apply the theoretical results to specific applications and to provide practical guidance on implementing the proposed sequential design. One popular application problem of interest is to estimate the system reliability, for which a Beta-Binomial model will be adopted. The other practical problem, which is often encountered in environmental study, is risk assessment and a Normal-Normal model will be used for the case.


Benkamra Z, et al. Nearly second order three-stage design for estimating a product of several Bernoulli proportions. J Stat Plan Inference 2015; 167: 90-101.

Benkamra Z, Terbeche M and Tlemcani M. Bayesian sequential estimation of the reliability of a parallel-series system. Appl Math Comput 2013; 219(23): 10842-10852.

Billinton R, et al. Reliability Evaluation of Engineering System, Concepts and Techniques. Springer: New York 1983.

Guo H, Honecker S, Mettas A and Ogden D. Reliability Estimation for On-Shot System with Zero Components test Failures. Proceedings to the Reliability and Maintainability Symposium (RAMS) – San Jose 2010. IEEE: pp 1-7.

Hardwick J and Stout Q. Determining optimal few-stage allocation procedures. Comput Sci Stat 1995; 27: 342-346.

Harkwick J and Stout QF. Optimal few-stage designs. J Stat Plan Inference 2002; 104: 121-145.

Kenneth VD, John SD and Kenneth JS. Incorporating a geometric mean formula into CPI. Mon Labor Rev 1998; October: 2-7.

Lehmann EL. Theory of Point Estimation. Wiley: New York 1983.

Littlewood B and Wright D. Some conservative stopping rules for the operational testing of safety critical software. IEEE Trans Softw Eng 1997; 23(11): 673-683.

Rekab K. A Nearly Optimal Two Stage Procedure. Comm Stat – Theory Meth 1992; 12(1): 197-201.

Rekab K. A Sampling Scheme for Estimating the Reliability of Series System. IEEE. Trans Rel 1992; 42: 287-291.

Rekab K and Cheng Y. An accelerated sequential sampling scheme for estimating the reliability of a n-parallel network system. Int J Rel Appl 2013; 14(2): 71-78.

Rekab K and Thompson H and Wu W. A multistage sequential test allocation for software reliability estimation. IEEE Trans Rel 2013; 62(2): 424-433.

Rekab K and Song X. First-Order Asymptotic Efficiency in Sequential Designs for Estimating product of means in the Exponential Family case. Manuscript submitted for publication, 2016.

Rekab K and Li Y. Bayesian Estimation of the Product of Two Proportions, Stoch. Anal Appl 1994; 12(3): 369-377.

Sun D and Ye K. Reference Prior Bayesian Analysis for Normal Mean Products. J Am Stat Assoc Theory Meth 1995; 90(430).

Woodroofe M and Hardwick J. Sequential allocation for an estimation problem with ethical cost. Ann Stat 1991; 18: 1358-1367.

Yfantis E and Flatman GT. On Confidence Interval for the Product of Three Means of Three Normally Distributed Populations. J Chemom1991; 5: 309-319.