On the estimation of stress strength reliability parameter of inverted exponential distribution

  • Authors

    • Sanjay Kumar Singh
    • Umesh Singh
    • Abhimanyu Yadav Banaras Hindu University
    • Pradeep Kumar Viswkarma
    2015-03-13
    https://doi.org/10.14419/ijsw.v3i1.4329
  • Inverted exponential distribution, Inverse Moment estimate and generating function, Stochastic Ordering, Renyi entropy and Bayesian inferences.

  • Abstract

    This paper aims to estimate the stress-strength reliability parameter R = P(Y < X), when X and Y are independent inverted exponential random variable. We have also discussed some fundamental properties of the considered distribution. The maximum likelihood estimator (MLE) of R and its asymptotic distribution are obtained. The Bayesian estimation of the reliability parameter has been also discussed under the assumption of independent gamma prior. Numerical integration technique is used for Bayesian computation. The proposed estimators are compared in terms of their mean squared errors through the simulation study. Two real data sets representing survival of head and neck cancer patients are fitted using the inverted exponential distribution and used to estimate the stress-strength parameters and reliability.

  • References

    1. [1] M. E. Ghitany, D. K. Al-Mutairi, and S. M. Aboukhamseen (2013), Estimation of the reliability of a stress-strength system from power lindley distributions. Communications in Statistics-Simulation and Computation (2013).

      [2] Puja Makkar, Puneet K. Srivastava, R. S. Singh, and S. K. Upadhyay (2014), Bayesian survival analysis of head and neck cancer data using log normal model. Communications in Statistics - Theory and Methods 43,392-407.

      [3] A. Renyi (1961), On measures of entropy and information, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Berkeley, pp. 547-561.

      [4] M. Shaked and J.G. Shanthikumar (1994), Stochastic orders and their applications, Boston: Academic Press.

      [5] Bhupendra Singh and Puneet Kumar Gupta (2012), Load-sharing system model and its application to the real data set. Mathematics and Computers in Simulation 82, 1615-1629.

      [6] Dey S. (2007). Inverted exponential distribution as a life distribution model from a Bayesian viewpoint. Data Sci. J. 6: 107-113.

      [7] Lin C., Duran B. S. and Lewis T. O. (1989). Inverted gamma as a life distribution. Microelectron. Reliab. 29 (4):619-626.

      [8] Prakash G. (2009). Some estimation procedures for the inverted exponential distribution. The South Pacific Journal of Natural Science, 27:71-78.

      [9] Singh S. K., Singh U. and Kumar D.(2003). Bayes estimators of the reliability function and parameter of inverted exponential distribution using informative and non-informative priors. Journal of Statistical Computation and Simulation, 83(12): 2258-2269.

      [10] Vikas Kumar Sharma, Sanjay Kumar Singh, and Umesh Singh (2014), A new upside-down bathtub shaped hazard rate model for survival data analysis. Applied Mathematics and Computation 239, 242-253.

      [11] Ali, M., Pal, M., and Woo, J. (2010). Estimation of Pr(Y < X) when X and Y belong to different distribution families. Journal of Probability and Statistical Science, 8, p. 35-48.

      [12] Kundu, D. and Gupta, R. D. (2005). Estimation of P(Y < X) for generalized exponential distribution. Metrika, 61, p. 291-308.

      [13] Awad, A.M. and Gharraf, M.K. (1986). Estimation of P(Y < X) in the Burr case: A comparative study. Communications in Statistics-Simulation 15:389-403.

      [14] Kundu, D. and Raqab, M.Z. (2009). Estimation of R = P(Y < X) for three-parameter Weibull distribution. Statistics and Probability Letters 79:1839-1846.

      [15] Kundu, D. and Gupta, R.D. (2006). Estimation of R = P[Y < X] for Weibull distributions. IEEE Transactions on Reliability 55:270-280.

      [16] Pundir P. S., Singh B. P. and Maheshwari S (2014). On hybrid censored inverted exponential distribution. International Journal of Current Research, Vol. 6, Issue, 01, pp. 4539-4544.

      [17] Singh S. K., Singh U., Vishwakarma P. K. and Yadav A. S (2014), Bayesian Reliability Estimation of Inverted Exponential Distribution Under Progressive Type-II Censored Data. J. Stat. Appl. Pro. 3, No. 3, 1-17.

      [18] McCool, J.I., (1991). Inference on P(Y < X) in the Weibull case. Communications in Statistics. Simulation and Computations 20, 129-148.

      [19] Kotz, S., Lumelskii, Y. and Pensky, M., (2003). The Stress-Strength Model and Its Generalizations. World Scientific Press, Singapore.

      [20] Sharma V. K., Singh S. K. , and Singh Umesh (2014), Estimation on System Reliability in Generalized Lindley Stress-Strength Model. J. Stat. Appl. Pro. 3, No. 1, 61-75.

      [21] Efron, B. (1988). Logistic regression, survival analysis, and the Kaplan-Meier curve. J. Amer.Statist. Assoc. 83:414-425.

      [22] Church, J.D. and Harris, B. (1970). The estimation of reliability from stress-strength relationships. Technometrics 12:49-54.

      [23] Birnbaum, Z.W. (1956). On a use of the Mann-Whitney statistic. In: Proceedings of Third Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 13-17, University of California Press, Berkeley, CA.

      [24] Birnbaum, Z.W., McCarty, R.C. (1958). A distribution-free upper confidence bound for P(Y < X), based on independent samples of X and Y . Annals of Mathematical Statistics 29:558-562.

      [25] A. Zellner (1986). A Bayesian estimation and prediction using asymmetric loss function, Journal of American Statistical Association, 81, pp. 446-451.

      [26] Zakkula Govindarajulu (1962). Theory of inverse moments, AFOSR 62-72.

      [27] David, F.N. and Johnson, N.L. (1957). Reciprocal Bernoulli and Poisson variables. Metron 18, 77-81.

      [28] Zakkula Govindarajulu (1962). The Reciprocal of the decapitated negative binomial variable. To appear in the December issue of the Journal of the Amer. Statist. Assn. 57.

  • Downloads

    Additional Files

  • How to Cite

    Singh, S. K., Singh, U., Yadav, A., & Viswkarma, P. K. (2015). On the estimation of stress strength reliability parameter of inverted exponential distribution. International Journal of Scientific World, 3(1), 98-112. https://doi.org/10.14419/ijsw.v3i1.4329

    Received date: 2015-02-10

    Accepted date: 2015-03-09

    Published date: 2015-03-13