Parameter Estimations for Generalized RayleighDistribution under Progressively Type-I IntervalCensored Data

ABSTRACT

In this paper, inference on parameter estimation of the generalized Rayleigh distribution are investigated for progressively type-I interval censored samples. The estimators of distribution parameters via maximum likelihood, moment method and probability plot are derived, and their performance are compared based on simulation results in terms of the mean squared error and bias. A case application of plasma cell myeloma data is used for illustrating the proposed estimation methods.

In this paper, inference on parameter estimation of the generalized Rayleigh distribution are investigated for progressively type-I interval censored samples. The estimators of distribution parameters via maximum likelihood, moment method and probability plot are derived, and their performance are compared based on simulation results in terms of the mean squared error and bias. A case application of plasma cell myeloma data is used for illustrating the proposed estimation methods.

Cite this paper

nullY. Lio, D. Chen and T. Tsai, "Parameter Estimations for Generalized RayleighDistribution under Progressively Type-I IntervalCensored Data,"*Open Journal of Statistics*, Vol. 1 No. 2, 2011, pp. 46-57. doi: 10.4236/ojs.2011.12006.

nullY. Lio, D. Chen and T. Tsai, "Parameter Estimations for Generalized RayleighDistribution under Progressively Type-I IntervalCensored Data,"

References

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[12] N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distribution,” Vol. 1, 2nd Edition, John Wiley and Sons, New York, 1995.

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[14] R. Aggarwala, “Progressively Interval Censoring: Some Mathematical Results with Application to Inference,” Communications in Statistics-Theory and Methods, Vol. 30, No. 8, 2010, pp. 1921-1935.

[15] H. Ng and Z. Wang, “Statistical Estimation for the Parameters of Weibull Distribution Based on Progressively type-I Interval Censored Sample,” Journal of Statistical Computation and Simulation, Vol. 79, No. 2, 2009, 145-159. doi:10.1080/00949650701648822

[16] D. G. Chen and Y. L. Lio, “Parameter Estimations for Generalized Exponential Distribution under Progressive Type-I Interval Censoring,” Computational Statistics and Data Analysis, Vol. 54, No. 6, 2010, pp. 1581-1591. doi:10.1016/j.csda.2010.01.007

[17] A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society: Series B, Vol. 39, No. 1, 1977, pp. 1-38.

[18] R Development Core Team, “A Language and Environment for Statistical Computing,” R Foundation for Statistical Computing, Vienna, 2006.

[19] R. Ihaka and R. Gentleman, “R: A Language for Data Analysis and graphics,” Journal of Computational and Graphical Statistics, Vol. 5, No. 3, 1996, pp. 299-314. doi:10.2307/1390807

[20] C. D. Kemp and W. Kemp, “Repid Generation of Frequency Tables,” Applied Statistics, Vol. 36, No. 3, 1987, pp. 277-282. doi:10.2307/2347786

[21] P. P. Carbone, L. E. Kellerhouse and E. A. Gehan, “Plasmacytic Myeloma: A Study of the Relationship of Survival to Various Clinical Manifestations and Anomalous Protein Type in 112 Patients,” The American Journal of Medince, Vol. 42, No. 6, 1967, pp. 937-948. doi:10.1016/0002-9343(67)90074-5

[22] J. Lawless, “Statistical Models and Methods for Lifetime Data,” John Wiley and Sons, New York, 1982.

[1] I. W. Burr, “Cumulative Frequency Distribution,” Annual of Mathematical Statistics, Vol. 13, No. 2, 1942, pp. 215-232.

[2] K. E. Ahmad, M. E. Fakhry and Z. F. Jaheen, “Empirical Bayes Estimation of P (Y < X) and Characterization of Burr-Type X Model,” Journal of Statistical Planning and Inference, Vol. 64, No. 2, 1997, pp. 297-308. doi:10.1016/S0378-3758(97)00038-4

[3] Z. F. Jaheen, “Bayesian Approach to Prediction with Outliers from the Burr Type X Model,” Microeleclron Reliability, Vol. 35, No. 4, 1995, pp. 45-47. doi:10.1016/0026-2714(94)00056-T

[4] Z. F. Jaheen, “Empirical Bayes Estimation of the Reliability and Failure Rate Functions of the Burr Type X Failure Model,” Journal Applied Statistical Science, Vol. 3, No. 2, 1996, pp. 281-285.

[5] D. Kundu and M. Z. Raqab, “Generalized Rayleigh Distribution: Different Methods of Estimation,” Computational Statistics and Data Analysis, Vol. 49, No. 1, 2005, pp. 187-200. doi:10.1016/j.csda.2004.05.008

[6] M. Z. Raqab, “Order Statistics from the Burr Type X Model,” Computational Mathematical Applications, Vol. 36, No. 4, 1998, pp. 111-120. doi:10.1016/S0898-1221(98)00143-6

[7] M. Z. Raqab and D. Kundu, “Burr Type X Distribution: Revisited,” 2003. http://home.iitk.ac.in/ kundu/paper118.pdf

[8] H. A. Sartawi and M. S. Abu-Salih, “Bayes Prediction Bounds for the Burr Type X Model,” Communication in Statistics: Theory and Methods, Vol. 20, No. 7, 1991, pp. 2307-2330. doi:10.1080/03610929108830633

[9] J. Q. Surles and W. J. Padgett, “Inference for P(Y < X) in the Burr Type X Model,” Journal Applied Statistical Science, Vol. 7, No. 2, 1998, pp. 225-238.

[10] J. Q. Surles and W. J. Padgett, “Inference for Reliability and Stress-Strength for a Scaled Burr Type X Model,” Lifetime Data Analysis, Vol. 7, No. 2, 2001, pp. 187-200. doi:10.1023/A:1011352923990

[11] J. Q. Surles and W. J. Padgett, “Some Properties of a Scaled Burr Type X Model,” Joournal Statisticsal Planning and Inference, Vol. 7, No. 2, 2004, pp. 187-200.

[12] N. L. Johnson, S. Kotz and N. Balakrishnan, “Continuous Univariate Distribution,” Vol. 1, 2nd Edition, John Wiley and Sons, New York, 1995.

[13] N. Balakrishnan, and R. Aggarwala, “Progressive Censoring: Theory, Methods and Applications,” Birkha User, Boston, 2000.

[14] R. Aggarwala, “Progressively Interval Censoring: Some Mathematical Results with Application to Inference,” Communications in Statistics-Theory and Methods, Vol. 30, No. 8, 2010, pp. 1921-1935.

[15] H. Ng and Z. Wang, “Statistical Estimation for the Parameters of Weibull Distribution Based on Progressively type-I Interval Censored Sample,” Journal of Statistical Computation and Simulation, Vol. 79, No. 2, 2009, 145-159. doi:10.1080/00949650701648822

[16] D. G. Chen and Y. L. Lio, “Parameter Estimations for Generalized Exponential Distribution under Progressive Type-I Interval Censoring,” Computational Statistics and Data Analysis, Vol. 54, No. 6, 2010, pp. 1581-1591. doi:10.1016/j.csda.2010.01.007

[17] A. P. Dempster, N. M. Laird and D. B. Rubin, “Maximum Likelihood from Incomplete Data via the EM Algorithm,” Journal of the Royal Statistical Society: Series B, Vol. 39, No. 1, 1977, pp. 1-38.

[18] R Development Core Team, “A Language and Environment for Statistical Computing,” R Foundation for Statistical Computing, Vienna, 2006.

[19] R. Ihaka and R. Gentleman, “R: A Language for Data Analysis and graphics,” Journal of Computational and Graphical Statistics, Vol. 5, No. 3, 1996, pp. 299-314. doi:10.2307/1390807

[20] C. D. Kemp and W. Kemp, “Repid Generation of Frequency Tables,” Applied Statistics, Vol. 36, No. 3, 1987, pp. 277-282. doi:10.2307/2347786

[21] P. P. Carbone, L. E. Kellerhouse and E. A. Gehan, “Plasmacytic Myeloma: A Study of the Relationship of Survival to Various Clinical Manifestations and Anomalous Protein Type in 112 Patients,” The American Journal of Medince, Vol. 42, No. 6, 1967, pp. 937-948. doi:10.1016/0002-9343(67)90074-5

[22] J. Lawless, “Statistical Models and Methods for Lifetime Data,” John Wiley and Sons, New York, 1982.