Main Article Content

Abstract

In accelerated life testing researcher generally use a life stress relationship between life characteristic and stress to estimate the parameters of failure time distributions at use condition which is just a re-parameterization of original parameters but from statistical point of view it is easy and reasonable to deal with original parameters of the distribution directly instead of developing inference for the parameters of the life stress relationship. So, an attempt is made here to estimate the parameters of Burr Type X life distribution directly in accelerated life testing by assuming that the lifetimes at increasing stress levels forms a geometric process. A mathematical model for the analysis of constant stress accelerated life testing for type-I censored data is developed and the estimates of parameters are obtained by using the maximum likelihood method. Also a Fisher information matrix is constructed in order to get the asymptotic variance and interval estimates of the parameters. Lastly, a simulation study is performed to illustrate the statistical properties of the parameters and the confidence intervals.

Keywords

Type-I censored sample Maximum Likelihood Estimation Reliability Function Fisher information Matrix Confidence Intervals Simulation Study

Article Details

Author Biographies

Ahmadur Rahman, Department of Statistics and O.R., Aligarh Muslim University, Aligarh-202002, India

Department of Statistics and O.R., Aligarh Muslim University, Aligarh

Tabassum Naz Sindhu, Department of Statistics, Quaid-i-Azam University 45320Islamabad 44000, Pakistan

She is working as visiting faculty in International University Islamabad, Quaid -i Azam University and Fast University ISB.

How to Cite
Rahman, A., Sindhu, T. N., Lone, S. A., & Kamal, M. (2020). Statistical Inference for Burr Type X Distribution using Geometric Process in Accelerated Life Testing Design for Time censored data. Pakistan Journal of Statistics and Operation Research, 16(3), 577-586. https://doi.org/10.18187/pjsor.v16i3.2252

References

  1. Ahmad, N., Islam, A., Kumar, R. and Tuteja, R. K. (1994). Optimal Design of Accelerated Life Test Plans Under Periodic Inspection and Type I Censoring: The Case of Rayleigh Failure Law. South African Statistical Journal, 28, 27-35.
  2. Ahmad, N. and Islam, A. (1996). Optimal accelerated life test designs for Burr type XII distributions under periodic inspection and type I censoring. Naval Research Logistics, 43, 1049-1077.
  3. Ahmad, K. E., Fakhry, M. E. and Jaheen, Z. F. (1997). Empirical Bayes estimation of P(Y < X) and characterization of Burr-type X model. Journal of Statistical Planning and Inference, 64, 297-308.
  4. Ahmad, N., Islam, A. and Salam, A. (2006). Analysis of optimal accelerated life test plans for periodic inspection: The case of Exponentiated Weibull failure model. International Journal of Quality & Reliability Management, 23(8), 1019-1046.
  5. Ahmad, N. (2010). Designing Accelerated Life Tests for Generalized Exponential Distribution with Log-linear Model. International Journal of Reliability and Safety, 4 (2/3), 238-264.
  6. Burr, I. W. (1942). Cumulative frequency distribution. Annals of Mathematical Statistics, 13, 215-232.
  7. Braun, W. J., Li, W. and Zhao, Y. Q. (2005). Properties of the geometric and related processes. Naval Research Logistics, 52 (7), 607–616.
  8. Chen, W., Gao, L., Liu, J., Qian, P. and Pan, J. (2012). Optimal design of multiple stress constant accelerated life test plan on non-rectangle test region. Chinese Journal of Mechanical Engineering, 25 (6), 1231-1237. http://dx.doi.org/10.3901/CJME.2012.06.1231
  9. Huang, S. (2011). Statistical inference in accelerated life testing with geometric process model, Master’s thesis, San Diego State University.
  10. Islam, A. and Ahmad, N. (1994). Optimal design of accelerated life test plan for Weibull distribution under periodic inspection and Type I censoring. Microelectronics Reliability, 34 (9), 1459- 1468.
  11. Jaheen, Z. F. (1996). Empirical Bayes estimation of the reliability and failure rate functions of the Burr type X failure model. Journal of Applied Statistical Sciences, 3, 281-288.
  12. Kamal, M., Zarrin, S. and Islam, A. (2013). Accelerated life testing design using geometric process for Pareto distribution. International Journal of Advanced Statistics and Probability, 1 (2), 25-31.
  13. Lam, Y. (1988). Geometric Process and Replacement Problem. Acta Mathematicae Applicatae Sinica, 4 (4), 366-377.
  14. Lam, Y. (2007). The Geometric Process and Its Application. World Scientific Publishing Co. Pvt. Ltd., Singapore.
  15. Lone, S. A., Rahman, A. and Islam, A. (2016). A Study of Accelerated Life Testing Design using Geometric Process for Generalized Exponential Distribution using Time Constraint. International Journal of Engineering Science & Research Technology, 5(2).
  16. Pan, Z., Balakrishnan, N. and Sun, Q. (2011.) Bivariate constant-stress accelerated degradation model and inference. Communications in Statistics-Simulation and Computation, 40 (2), 247–257. http://dx.doi.org/10.1080/03610918.2010.534227
  17. Rodriguez, R. N. (1977). A guide to Burr Type XII distributions. Biometrika, 64, 129-134.
  18. Raqab, M. Z. (1998). Order statistics from the Burr type X model. Computers Mathematics and Applications, 36, 111-120.
  19. Rahman, A., Lone, S. A. and Alam, I. (2016). Application of Geometric Process for Generalized Exponential Distribution in Accelerated Life Testing with Complete data. International Journal of Scientific & Engineering Research, 7 (4).
  20. Sartawi, H. A. and Abu-Salih, M. S. (1991). Bayes prediction bounds for the Burr type X model. Communications in Statistics - Theory and Methods, 20, 2307-2330.
  21. Sindhu, T. N., Hussain, Z. and Aslam, M. (2016). A simulation study of parameters for the censored shifted Gompertz mixture distribution: A Bayesian approach. Journal of Statistics and Management Systems 19 (3), 423-450.
  22. Tabassum, N.S., Riaz, M., Aslam, M. and Ahmed, Z. (2016). A Study of Cumulative Quantity Control Chart for a Mixture of Rayleigh Model under a Bayesian Framework. Revista Colombiana de Estadística 39 (2), 185-204.
  23. Tabassum, N.S. and Hussain, Z. (2018). Mixture of two generalized inverted exponential distributions with censored sample: properties and estimation. Statistica Applicata - Italian Journal of Applied Statistics, 30 (3), 373-391.
  24. Sindhu, T. N., Hussain, Z. and Aslam, M. (2019). Parameter and reliability estimation of inverted Maxwell mixture model. Journal of Statistics and Management Systems, 22 (3), 459-493.
  25. Surles, J. G. and Padgett, W. J. (2001). Inference for reliability and stress-strength for a scaled Burr Type X distribution. Lifetime Data Analysis, 7, 187-200.
  26. Wingo, D. R. (1993). Maximum likelihood methods for fitting the Burr Type XII distribution to multiply (progressively) censored life test data. Metrika, 40, 203-210.
  27. Watkins, A. J. and John, A. M. (2008). On constant stress accelerated life tests terminated by Type II censoring at one of the stress levels. Journal of Statistical Planning and Inference, 138 (3), 768-786. http://dx.doi.org/10.1016/j.bbr.2011.03.031
  28. Yang, G. B. (1994). Optimum constant-stress accelerated life-test plans. IEEE Transactions on Reliability, 43 (4), 575-581.
  29. Zhou, K., Shi, Y. M. and Sun, T. Y. (2012). Reliability Analysis for Accelerated Life-Test with Progressive Hybrid Censored Data Using Geometric Process. Journal of Physical Sciences, 16, 133-143.