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Non Informative Priors for the Stress-Strength Reliability in the Generalized Augmented Inverse Gaussian Distribution

Volume 13, Number 1, January 2017 - Paper 4 - pp. 45-62
DOI: 10.23940/ijpe.17.01.p4.4562


Department of Statistics, Ramanujan School of Mathematical Sciences, Pondicherry University, Puducherry-605 014, India

(Received on September 30, 2016, revised on November 25, 2016)


In this paper the Bayesian and classical estimation of augmented strength reliability under Augmentation Strategy Plan (ASP) have been considered. The Augmentation Strategy Plan (ASP) is suggested for enhancing the strength of failed equipment. The Bayes estimation is carried out by assuming the model parameters are random variable and having non-informative type of priors (uniform and Jeffery’s priors) under two different loss functions, viz. squared error loss function (SELF) and Linex loss function (LLF). We assume that the Inverse Gaussian stress (Y) is subjected to equipment having Inverse Gaussian strength (X) and are independent to each other. A comparative study between ML and Bayesian estimators have been carried out on the basis of mean square errors (MSE) and absolute biases. The Markov Chain Monte Carlo method of approximations has been applied to draw posterior expectations under both the loss functions. The MSE and absolute biases are calculated with 1000 replications.


References: 28

[1].          Alam, S.N. and Roohi. On Augmenting Exponential Strength-Reliability. IAPQR Transactions, 2002; 27: 111-117.

[2].          Banerjee, A. K. and G. K. Bhattacharyya. Bayesian results for the inverse Gaussian distribution with an application. Technometrics, 1979; 21(2): 247-251.

[3].          Basu, A. P. and N Ebrahimi. On the reliability of stochastic systems. Statistics & Probability Letters, 1983; 1(5): 265-267.

[4].          Basu, S. and R. T. Lingham. Bayesian estimation of system reliability in Brownian stress-strength models. Annals of the Institute of Statistical Mathematics, 2003; 55(1): 7-19.

[5].          Berger, J.O. Statistical Decision Theory and Bayesian Analysis. Springer-Verlag: NY; 1985.

[6].          Brooks, S. Markov chain Monte Carlo method and its application. Journal of the royal statistical society: series D (the Statistician). 1998; 47(1): 69-100.

[7].          Chandra, N. and S. Sen. Augmented Strength Reliability of Equipment under Gamma Distribution. J of Statist. Theory and Applications. 2014; 13: 212-221.

[8].          Chandra, N. and V.K Rathaur. Augmented Strategy Plans for Enhancing Strength Reliability of an Equipment under Inverse     Gaussian Distribution. J. Math. Engg. Sci. Aerospace: Special Issue on Reliability and Dependability Modeling Analysis for Complex Systems. 2015(a); 6: 233-243.

[9].          Chandra, N. and V.K Rathaur. Augmenting Exponential Stress-Strength Reliability for a coherent system. Proceedings of National Seminar on Statistical Methods and Data Analysis, published by Abhiruchi Prakashana, Mysore. 2015(b):25-34.

[10].       Chandra, N. and V.K Rathaur. Augmented Gamma Strength Reliability Models for Series and Parallel Coherent System. Proceedings of National Conference on Emerging Trends in Statistical Research: Issue and Challenges, Narosa Publication, New Delhi, India, 2015(c):43-54.

[11].       Chhikara, R.S. and J.L. Folks. Estimation of the Inverse Gaussian distribution function. J. Amer. Statis. Assoc. 1974; 69: 250-254.

[12].       Chhikara, R.S. and J.L. Folks. Statistical distributions related to the inverse Gaussian. Communications in Statistics. 1975; 4: 1081-1091.

[13].       Chhikara, R.S. and J.L. Optimum test procedures for the mean of first passage time distributions in Brownian motion with positive drift. Technometrics. 1976; 18:189-193.

[14].       Chhikara, R.S. and J.L. The Inverse Gaussian distribution as a Lifetime Model. Technometrics. 1977; 19(4): 461-468.

[15].       Ebrahimi, N. and T. Ramallingam. Estimation of system reliability in Brownian stress-strength models based on sample paths. Annals of the Institute of Statistical Mathematics. 1993; 45(1): 9-19.

[16].       Efron, B. Logistic regression, survival analysis and the Kaplan-meier curve. J. Ame. Stat. Asso. 1988; 83: 414-425.

[17].       Hastings, W.K. Monte Carlo sampling methods using Markov chains and their applications. Biometrika. 1970; 57(1): 97-109.

[18].       Jeffreys, H. The Theory of Probability. 3rd ed, Oxford University Press, New York, NY.(1998).

[19].       Johnson, N. L., S. Kotz and N Balakrishnan. Continuous univariate distributions. (2nd ed.), vol.1 John Wiley & Sons. New York, 163: 1994.

[20].       Makkar, P., P.K. Srivastava, R.S. Singh and S.K. Upadhyay. Bayesian survival analysis of head and neck cancer data using lognormal model. Communications in Statistics-Theory and Methods. 2014; 43: 392-407.

[21].       Nadas, A. Best tests for zero drift based on first passage times in Brownian motion. Technometrics. 1973; 15: 125-132.

[22].       Padgett, W. J. and L.J. Wei. Estimation for the three-parameter inverse Gaussian distribution. Communications in statistics-theory and methods. 1979; 8(2): 129-137.

[23].       Pandey, B.N. and P. Bandyopadhyay. Bayesian Estimation of Inverse Gaussian Distribution. Int. J. Agricult. Stat. Sci.      2013; 9(2): 373-386.

[24].       Sarhan, A.M., B. Smith and D.C. Hamilton. Estimation of P(Y <X) for a Two-parameter Bathtub Shaped Failure Rate Distribution. Int. J. of Statist. Prob. 2015; 4: 33-45.

[25].       Sharma,V.K., S.K. Singh, U. Singh and V. Agiwal. The inverse Lindley distribution: a stress-strength reliability model with application to head and neck cancer data. J. of Indust. and Product. Engg. 2015; 32: 162–173.

[26].       Sherif, Y.S. and M. L. Smith. First-passage time distribution of Brownian motion as a reliability model. IEEE Transactions. 1980; 29(5): 425-426.

[27].       Tweedie, M. C. K. Statistical Properties of Inverse Gaussian distribution I, Annals of Mathematics & Statistics. 1957(a); 28: 362-77.

[28].       Tweedie, M. C. K. Statistical properties of inverse Gaussian distributions II. Annals of Mathematical Statistics. 1957(b); 28(3): 696-705.


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