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Estimation Using Censored Data from Exponentiated Burr Type XII Population

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Maximum likelihood and Bayes estimators of the parameters, survival function (SF) and hazard rate function (HRF) are obtained for the three-parameter exponentiated Burr type XII distribution when sample is available from type II censored scheme. Bayes estimators have been developed using the standard Bayes and MCMC methods under square error and LINEX loss functions, using informative type of priors for the parameters. Simulation comparison of various estimation methods is made when n = 20, 40, 60 and censored data. The Bayes estimates are found to be, generally, better than the maximum likelihood estimates against the proposed prior, in the sense of having smaller mean square errors. This is found to be true whether the data are complete or censored. Estimates improve by increasing sample size. Analysis is also carried out for real life data.

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E. AL-Hussaini and M. Hussein, "Estimation Using Censored Data from Exponentiated Burr Type XII Population,"

*Open Journal of Statistics*, Vol. 1 No. 2, 2011, pp. 33-45. doi: 10.4236/ojs.2011.12005.

[1] | I. W. Burr, “Cumulative Frequency Functions,” The Annals of Mathematical Statistics, Vol. 1, No. 2, 1942, pp. 215-232. doi:10.1214/aoms/1177731607 |

[2] | M. A. Hatke, “A Certain Cumulative Probability Function,” The Annals of Mathematical Statistics, Vol. 20, No. 3, 1949, pp. 461-463. doi:10.1214/aoms/1177730002 |

[3] | I. W. Burr, “Parameters for a General System of Distributions to Match a Grid of α3 and α4,” Communications in Statistics, Vol. 2, No. 2, 1973, pp. 1-21. doi:10.1080/03610927308827052 |

[4] | R. N. Rodriguez, “A Guide to the Burr Type XII Distributions,” Biometrika, Vol. 64, No. 1, 1977, pp. 129-134. doi:10.1093/biomet/64.1.129 |

[5] | P. R. Tadikamalla, “A Look at the Burr and Related Distributions,” International Statistical Review, Vol. 48, No. 3, 1980, pp. 337-344. doi:10.2307/1402945 |

[6] | K. Takahasi, “Note on the Multivariate Burr’s Distribution,” Annals of the Institute of Statistical Mathematics, Vol. 17, No. 1, 1965, pp. 257-260. doi:10.1007/BF02868169 |

[7] | P. Embrechts, C. Kluppelberg and T. Mikosch, “Modeling Extremal Events,” Springer-Verlag, Berlin, 1977. |

[8] | A. S. Klugman, “Loss Distributions,” Wiley Interscience, New York, 1986, pp. 31-55. |

[9] | S. W Drane, D. B. Owen and G. B. Seibetr Jr., “The Burr Distribution and Quantal Responses,” Statistical Papers, Vol. 19, No. 3, 1978, pp. 204-210. doi:10.1007/BF02932803 |

[10] | J. B. McDonald and D. O. Richards, “Model Selection: Some Generalized Distributions,” Communications in Statistics-Theory and Methods, Vol. 17, 1978, pp. 287-296. |

[11] | D. G. Morrison and D. C. Schmittlein, “Jobs, Strikes and Wars: Probability Models for Duration,” Organizational Behavior and Human Performance, Vol. 25, No. 2, 1980, pp. 224-251. doi:10.1016/0030-5073(80)90065-3 |

[12] | D. C. Schmittlein, “Some Sampling Properties of a Model for Income Distribution,” Journal of Business and Economic Statistics, Vol. 1, No. 2, 1983, pp. 147-153. doi:10.2307/1391855 |

[13] | J. B. McDonald, “Some Generalized Function for the Size Distribution of Income,” Econometrica, Vol. 52, No. 3, 1984, pp. 647-663. doi:10.2307/1913469 |

[14] | S. R. Lindsay, G. R. Wood and R. C. Woollons, “Modeling the Diameter Distribution of Forest Stands Using the Burr Distribution,” Journal of Applied Statistics, Vol. 23, No. 6, 1996, pp. 609-619. doi:10.1080/02664769623973 |

[15] | Q. Shoa, “Estimation for Hazardous Concentrations Based on NOEC Toxicity Data: An Alternative Approach,” Environmetrics, Vol. 11, No. 5, 2000, pp. 583-595. doi:10.1002/1099-095X(200009/10)11:5<583::AID-ENV456>3.0.CO;2-X |

[16] | S. D. Dubey, “Statistical Contributions to Reliability Engineerings,” Aerospace Research Laboratories, Charlottesville, 1972. |

[17] | S. D. Dubey, “Statistical Treatment of Certain Life Testing and Reliability Problems,” Aerospace Research Laboratories, Charlottesville, 1973. |

[18] | A. S. Papadopoulos, “The Burr Distribution as a Life Time Model from a Bayesian Approach,” IEEE Transactions on Reliability, Vol. 27, No. 5, 1978, 369-371. doi:10.1109/TR.1978.5220427 |

[19] | A. W. Lewis, “The Burr Distribution as a General Parametric Family in Survivor-Ship and Reliability Theory Applications,” Ph.D. Thesis, University of North Carolina, North Carolina, 1981. |

[20] | I. G. Evans and A. S. Ragab, “Bayesian Inferences Given a Type-2 Censored Sample from Burr Distribution,” Communications in Statistics-Theory and Methods, Vol. 12, No. 13, 1983, pp. 1569-1580. doi:10.1080/03610928308828551 |

[21] | G. S. Lingappaiah, “Bayesian Approach to the Estimation of Parameters in the Burr’s XII Distribution with Outliers,” Journal of Orissa Mathematical Society, Vol. 1, 1983, pp. 55-59. |

[22] | Z. F. Jaheen, “Bayesian Estimations and Predictions Based on Single Burr type XII Models and Their Finite Mixture,” Ph.D. Thesis, University of Assiut, Asyut, 1993. |

[23] | E. K. AL-Hussaini, M. A. Mousa and Z. F. Jaheen, “Estimation under the Burr type XII Failure Model: A Comparative Study,” Test, Vol. 1, No. 1, 1992, pp. 33-42. |

[24] | A. Shah and D.V. Gokhale, “On Maximum Product of Spacings (MPS) Estimation for Burr XII Distribution,” Communications in Statistics-Theory and Methods, Vol. 22, 1993, 615-641. |

[25] | E. K. AL-Hussaini and Z. F. Jaheen, “Bayes Estimation of the Parameters, Reliability and Failure Rate Functions of the Burr Type XII Failure Model,” Journal of Statistical Computation and Simulation, Vol. 41, No. 1-2, 1992, pp. 31-40. doi:10.1080/00949659208811389 |

[26] | E. K. AL-Hussaini and Z. F. Jaheen, “Approximate Bayes Estimators Applied to the Burr Model,” Communications in Statistics-Theory and Methods, Vol. 23, No. 1, 1994, pp. 99-121. |

[27] | D. Moore and A. S. Papadopoulos, “The Burr Type XII Distribution as a Failure Model under Various Loss Functions,” Microelectronics Reliability, Vol. 40, No. 12, 2000, pp. 2117-2122. doi:10.1016/S0026-2714(00)00031-7 |

[28] | A. H. Khan and A. I. Khan, “Moments of Order Statistics from Burr’s Distribution and Its Characterization,” Metron-International Journal of Statistics, Vol. 45, 1987, pp. 21-29. |

[29] | E. K. AL-Hussaini, “A Characterization of the Burr Type XII Distribution,” Applied Mathematics Letters, Vol. 4, No. 1, 1991, pp. 59-61. doi:10.1016/0893-9659(91)90123-D |

[30] | A. H. Abdel-Hamid, “Constant-Partially Accelerated Life Tests for Burr XII Distribution with Progressive Type II Censoring,” Computational Statistics & Data Analysis, Vol. 53, No. 7, 2009, pp. 2511-2523. doi:10.1016/j.csda.2009.01.018 |

[31] | A. M. Nigm, “Prediction Bounds for the Burr Model,” Communications in Statistics-Theory and Methods, Vol. 17, No. 1, 1988, 287-296. doi:10.1080/03610928808829622 |

[32] | E. K. AL-Hussaini and Z. F. Jaheen, “Bayes Prediction Bounds for the Burr Type XII Failure Model,” Communications in Statistics-Theory and Methods, Vol. 24, No. 7, 1995, 1829-1842. doi:10.1080/03610929508831589 |

[33] | E. K. AL-Hussaini and Z. F. Jaheen, “Bayesian Prediction Bounds for the Burr Type XII Distribution in the Presence of Outliers,” Journal of Statistical Planning and Inference, Vol. 55, No. 1, 1996, pp. 23-37. doi:10.1016/0378-3758(95)00184-0 |

[34] | E. K. AL-Hussaini, “Bayesian Predictive Density of Order Statistics Based on Finite Mixture Models,” Journal of Statistical Planning and Inference, Vol. 113, No. 1, 2003, pp. 15-24. doi:10.1016/S0378-3758(01)00297-X |

[35] | E. K. AL-Hussaini and A. A. Ahmad, “On Bayesian Interval Prediction of Future Records,” Test, Vol. 12, No. 1, 2003, pp. 79-99. doi:10.1007/BF02595812 |

[36] | Q. X. Shao, H. Wong, J. Xia, and W.-C Ip, “Models for Extremes Using the Extended Three-Parameter Burr XII System with Application to Flood Frequency Analysis,” Hydrological Sciences, Vol. 49, No. 4, 2004, pp. 685-702. |

[37] | A. W. Marshall and I. Olkin, “A New Method for Adding a Parameter to a Family of Distributions with Applications to the Exponential and Weibull Families,” Biometrika, Vol. 84, No. 3, 1997, pp. 641-652. doi:10.1093/biomet/84.3.641 |

[38] | E. K. AL-Hussaini and M. Ghitany, “On Certain Countable Mixtures of Absolutely Continuous Distributions,” Metron-International Journal of Statistics, Vol. 18, No. 1, 2005, pp. 39-53. |

[39] | E. K. AL-Hussaini and M. A. Gharib, “A New Family of Distributions as a Countable Mixture with Poisson Added Parameter,” Journal of Statistics Theory and Applications, Vol. 8, pp. 169-190. |

[40] | P. F. Verhulst, “Recheches Mathématiques sur la loid’accroissement de la Population, Nouveaux mémoire de l’Academie Royale de Sciences et Belle-Lettres de Bruxelles, Vol. 18, 1845, pp. 1-42. |

[41] | P. F. Verhulst, “Notice sur la Loique la Population Poursuitdans son Accroissement,” Correspondanc ema- thématiqueet physique, publiée par L.A.L. Quetelet, Vol. 10, 1838, pp. 113-121. |

[42] | J. C. Ahuja and S. W. Nash, “The Generalized Gompertz- Verhulst Family of Distributions,” Sankhyā, Vol. 29, 1967, No. 2, pp. 141-161. |

[43] | P. F. Verhulst, “Deuxiémemémoire sur la loid’accroissment de la Population,” Mémoire de l’Academie Royale de Sciences, des Lettreset de Beaux-Arts de Belgique, Series 2, Vol. 20, 1847, pp. 1-32. |

[44] | E. K. AL-Hussaini, “On Exponentiated Class of Distributions,” Journal of Statistics Theory and Applications, Vol. 9, 2010, pp. 41-63. |

[45] | E. K. AL-Hussaini, “Inference Based on Censored Samples from Exponentiated Populations,” Test, Vol. 19, No. 3, 2010, pp. 487-513. doi:10.1007/s11749-010-0183-5 |

[46] | R. C. Gupta and R. D. Gupta, “Proportional Reversed Hazard Rate Model and Its Applications,” Journal of Statistical Planning and Inference, Vol. 137, No. 11, 2007, pp. 3525-3536. doi:10.1016/j.jspi.2007.03.029 |

[47] | E. L. Lehmann, “The Power of Rank Tests,” The Annals of Mathematical Statistics, Vol. 24, No. 1, 1953, 28-43. doi:10.1214/aoms/1177729080 |

[48] | H. Varian, “A Bayesian Approach to Real Estate Assessment,” In: S. E. Fienberg, A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics, North-Holland, Amsterdam, 1975, pp. 195-208. |

[49] | R. D. Thompson and A. P. Basu, “Asymmetric loss Function for Estimating System Reliability,” In: D.A. Berry, K. M. Chaloner and J. K. Geweke, Eds., Bayesian Analysis in Statistics and Econometrics, Wiley series in Probability and Statistics, New York,1996. |

[50] | T. M. Apostol, “Mathematical Analysis,” 3rd Edition, Addison-Wesley, Boston, 1957. |

[51] | J. F. Lawless, “Statistical Models and Methods for Lifetime Data,” 2nd Edition, Wiley, New York, 2003. |

[52] | M. K. Cowles and B. P. Carlin, “Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review,” Journal of the American Statistical Association, Vol. 91, No. 434, 1996, pp. 883-904. doi:10.2307/2291683 |

[53] | A. Gelman and D.B.Rubin, “Inference from Iterative Simulation Using Multiple Sequences,” Statistical Science, Vol. 7, No. 4, 1992, pp. 457-472. doi:10.1214/ss/1177011136 |

[54] | G. O. Roberts, A. Gelman and W. R. Gilks, “Weak Convergence and optimal Scaling of random Walk Metropolis Algorithms,” The Annals of Applied Probability, Vol. 7, No. 1, 1997, pp. 110-120. doi:10.1214/aoap/1034625254 |

[55] | L. Tierney, “Markov Chains for Exploring Posterior Distributions,” The Annals of Statistics, Vol. 22, No. 4, 1994, pp. 1701-1762. doi:10.1214/aos/1176325750 |

[56] | D. Gamerman and H. P. Lopes, “Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference,” 2nd Edition, Chapman and Hall, London, 2006. |

[57] | D. Sinha, T. Maiti, J. Ibrahim and B. Ouyang, “Current Methods for Recurrent Events Data with Dependent Termination: A Bayesian Perspective,” Journal of the American Statistical Association, Vol. 103, 2008, pp. 866-878. doi:10.1198/016214508000000201 |

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