Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm

Affiliation(s)

Department of Mathematics, Umm Al-Qura University, Makkah Al-Mukarramah, KSA.

Department of Mathematics, Taif University, Taif, KSA.

Department of Mathematics, Umm Al-Qura University, Makkah Al-Mukarramah, KSA.

Department of Mathematics, Taif University, Taif, KSA.

ABSTRACT

This article considers the problem in obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two Rayleigh (MTR) distributions based on generalized order statistics (GOS). We consider one-sample and two-sample prediction schemes using the Markov chain Monte Carlo (MCMC) algorithm. The conjugate prior is used to carry out the Bayesian analysis. The results are specialized to upper record values. Numerical example is presented in the methods proposed in this paper.

This article considers the problem in obtaining the maximum likelihood prediction (point and interval) and Bayesian prediction (point and interval) for a future observation from mixture of two Rayleigh (MTR) distributions based on generalized order statistics (GOS). We consider one-sample and two-sample prediction schemes using the Markov chain Monte Carlo (MCMC) algorithm. The conjugate prior is used to carry out the Bayesian analysis. The results are specialized to upper record values. Numerical example is presented in the methods proposed in this paper.

KEYWORDS

Mixture Distributions; Rayleigh Distribution; Generalized Order Statistics; Record Values; MCMC

Mixture Distributions; Rayleigh Distribution; Generalized Order Statistics; Record Values; MCMC

Cite this paper

T. Abushal and A. Al-Zaydi, "Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm,"*Open Journal of Statistics*, Vol. 2 No. 3, 2012, pp. 356-367. doi: 10.4236/ojs.2012.23044.

T. Abushal and A. Al-Zaydi, "Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm,"

References

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[2] M. Ahsanullah, “Generalized Order Statistics from Two Parameter Uniform Distribution,” Communications in Statistics—Theory and Methods, Vol. 25, No. 10, 1996, pp. 2311-2318. doi:10.1080/03610929608831840

[3] M. Ahsanullah, “Generalized Order Statistics from Exponential Distributiuon,” Journal of Statistical Planning and Inference, Vol. 85, No. 1-2, 2000, pp. 85-91. doi:10.1016/S0378-3758(99)00068-3

[4] U. Kamps and U. Gather, “Characteristic Property of Generalized Order Statistics for Exponential Distributions,” Applicationes Mathematicae (Warsaw), Vol. 24, No. 4, 1997, pp. 383-391.

[5] E. Cramer and U. Kamps, “Relations for Expectations of Functions of Generalized Order Statistics,” Journal of Statistical Planning and Inference, Vol. 89, No. 1-2, 2000, pp. 79-89. doi:10.1016/S0378-3758(00)00074-4

[6] M. Habibullah and M. Ahsanullah, “Estimation of Parameters of a Pareto Distribution by Generalized Order statistics,” Communications in Statistics—Theory and Methods, Vol. 29, No. 7, 2000, pp. 1597-1609. doi:10.1080/03610920008832567

[7] Z. F. Jaheen, “On Bayesian Prediction of Generalized Order Statistics,” Journal of Statistical Theory and Applications, Vol. 1, No. 3, 2002, pp.191-204.

[8] Z. F. Jaheen, “Estimation Based on Generalized Order Statistics from the Burr Model,” Communications in Statistics—Theory and Methods, Vol. 34, No. 4, 2005, pp. 785-794. doi:10.1081/STA-200054408

[9] E. K. Al-Hussaini and A. A. Ahmad, “On Bayesian Predictive Distributions of Generalized Order Statistics,” Metrika, Vol. 57, No. 2, 2003, pp. 165-176. doi:10.1007/s001840200207

[10] E. K. Al-Hussaini, “Generalized Order Statistics: Prospective and Applications,” Journal of Applied Statistical Science, Vol. 13, No. 1, 2004, pp. 59-85.

[11] A. A. Ahmad and T. A. Abu-Shal, “Recurrence Relations for Moment Generating Functions of Nonadjacent Generalized Order Statistics Based on a Class of Doubly Truncated Distributions,” Journal of Statistical Theory and Applications, Vol. 6, No. 2, 2007, pp. 174-189.

[12] A. A. Ahmad and T. A. Abu-Shal, “Recurrence Relations for Moment Generating Functions of Generalized Order Statistics from Doubly Truncated Continuous Distributions,” Journal of Statistical Theory and Applications, Vol. 6, No. 2, 2008, pp. 243-257.

[13] A. A. Ahmad, “Relations for Single and Product Moments of Generalized Order Statistics from Doubly Truncated Burr type XII Distribution,” Journal of the Egyptian Mathematical Society, Vol. 15, No. 1, 2007, pp. 117- 128.

[14] A. A. Ahmad, “Single and Product Moments of Generalized Order Statistics from Linear Exponential Distribution,” Communications in Statistics—Theory and Methods, Vol. 37, No. 8, 2008, pp. 1162-1172. doi:10.1080/03610920701713344

[15] Z. A. Aboeleneen, “Inference for Weibull Distribution under Generalized Order Statistics,” Mathematics and Computers in Simulation, Vol. 81, No. 1, 2010, pp. 26-36. doi:10.1016/j.matcom.2010.06.013

[16] S. Abu El Fotouh, “Estimation for the Parameters of the Weibull Extension Model Based on Generalized Order Statistics,” International Journal of Contemporary Mathematical Sciencess, Vol. 6, No. 36, 2011, pp. 1749-1760.

[17] S. F. Ateya, “Prediction under Generalized Exponential Distribution Using MCMC Algorithm,” International Mathematical Forum, Vol. 6, No. 63, 2011, pp. 3111-3119.

[18] S. F. Ateya and A. A. Ahmad, “Inferences Based on Generalized Order Statistics under Truncated Type I Generalized Logistic Distribution,” Statistics, Vol. 45, No. 4, 2011, pp. 389-402. doi:10.1080/02331881003650149

[19] B. S. Everitt and D. J. Hand, “Finite Mixture Distributions,” Cambridge University Press, Cambridge, 1981.

[20] D. M. Titterington, A. F. M. Smith and U. E. Makov, “Statistical Analysis of Finite Mixture Distributions,” John Wiley and Sons, New York, 1985.

[21] K. E. Ahmad, “Identifiability of Finite Mixtures Using a New Transform,” Annals of the Institute of Statistical Mathematics, Vol. 40, No. 2, 1988, pp. 261-265. doi:10.1007/BF00052342

[22] G. J. McLachlan and K. E. Basford, “Mixture Models: Inferences and Applications to Clustering,” Marcel Dekker, New York, 1988.

[23] Z. F. Jaheen, “On Record Statistics from a Mixture of Two Exponential Distributions,” Journal of Statistical Computation and Simulation, Vol. 75, No. 1, 2005, pp. 1- 11. doi:10.1080/00949650410001646924

[24] K. E. Ahmad, Z. F. Jaheen and Heba S. Mohammed, “Bayesian Prediction Based on Type-I Censored Data from a Mixture of Burr Type XII Distribution and Its Reciprocal,” Statistics, Vol. 1, No. 1, 2011, pp. 1-11. doi:10.1080/02331888.2011.555550

[25] E. K. AL-Hussaini and M. Hussein, “Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss,” Open Journal of Statistics, Vol. 2, No. 1, 2012, pp. 28-38. doi:10.4236/ojs.2012.21004

[26] M. A. M. Ali-Mousa, “Bayesian Prediction Based on Pareto Doubly Censored Data,” Statistics, Vol. 37, No. 1, 2003, pp. 65-72. doi:10.1080/0233188021000004639

[27] A. S. Papadapoulos and W. J. Padgett, “On Bayes Estimation for Mixtures of Two Exponential-Life-Distributions from Right-Censored Samples,” IEEE Transactions on Reliability, Vol. 35, No. 1, 1986, pp. 102-105. doi:10.1109/TR.1986.4335364

[28] A. F. Attia, “On estimation for mixtures of 2 Rayleigh Distribution with Censoring,” Microelectronics Reliability, Vol. 33, No. 6, 1993, pp. 859-867. doi:10.1016/0026-2714(93)90259-2

[29] K. E. Ahmad, H. M. Moustafa and A. M. Abd-El-Rahman, “Approximate Bayes Estimation for Mixtures of Two Weibull Distributions under Type II Censoring,” Journal of Statistical Computation and Simulation, Vol. 58, No. 3, 1997, pp. 269-285. doi:10.1080/00949659708811835

[30] A. A. Soliman, “Estimators for the Finite Mixture of Rayleigh Model Based on Progressively Censored Data,” Communications in Statistics—Theory and Methods, Vol. 35, No. 5, 2006, pp. 803-820. doi:10.1080/03610920500501379

[31] M. Saleem and M. Aslam, “Bayesian Analysis of the Two Component Mixture of the Rayleigh Distribution Assuming the Uniform and the Jeffreys Priors,” Journal of Applied Statistical Science, Vol. 16, No. 4, 2008, pp. 105-113.

[32] M. Saleem and M. Aslam, “On Prior Selection for the Mixture of Rayleigh Distribution Using Predictive Intervals,” Pakistan Journal of Statistics, Vol. 24, No. 1, 2007, pp. 21-35.

[33] M. Saleem and M. Irfan, “On Properties of the Bayes estimates of the Rayleigh Mixture Parameters: A Simulation Study,” Pakistan Journal of Statistics., Vol. 26, No. 3, 2010, pp. 547-555.

[34] A. Zellner, “Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions,” In: J. O. Berger and S. S. Gupta, Eds., Statistical Decision Theory and Methods. V, Springer, New York, 1994, pp. 339-390.

[35] M. J. Jozani, E. Marchand and A. Parsian, “Bayes Estimation under a General Class of Balanced Loss Functions,” Universite de Sherbrooke, Sherbrooke, 2006.

[36] E. K. Al-Hussaini and K. E. Ahmad, “On the Identifiability of Finite Mixtures of Distributions,” IEEE Transactions on Information Theory, Vol. 27, No. 5, 1981, pp. 664-668. doi:10.1109/TIT.1981.1056389

[37] K. E. Ahmad and E. K. Al-Hussaini, “Remarks on the Non-Identifiability of Mixtures of Distributions,” Annals of the Institute of Statistical Mathematics, Vol. 34, No. 1, 1982, pp. 543-544. doi:10.1007/BF02481052

[38] S. J. Press, “Subjective and Objective Bayesian Statistics: Principles, Models and Applications,” Wiley, New York, 2003.

[39] J. Aitchison and I.R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975.

[1] U. Kamps, “A Concept of Generalized Order Statistics,” Journal of Statistical Planning and Inference, Vol. 48, No. 1, 1995, pp. 1-23. doi:10.1016/0378-3758(94)00147-N

[2] M. Ahsanullah, “Generalized Order Statistics from Two Parameter Uniform Distribution,” Communications in Statistics—Theory and Methods, Vol. 25, No. 10, 1996, pp. 2311-2318. doi:10.1080/03610929608831840

[3] M. Ahsanullah, “Generalized Order Statistics from Exponential Distributiuon,” Journal of Statistical Planning and Inference, Vol. 85, No. 1-2, 2000, pp. 85-91. doi:10.1016/S0378-3758(99)00068-3

[4] U. Kamps and U. Gather, “Characteristic Property of Generalized Order Statistics for Exponential Distributions,” Applicationes Mathematicae (Warsaw), Vol. 24, No. 4, 1997, pp. 383-391.

[5] E. Cramer and U. Kamps, “Relations for Expectations of Functions of Generalized Order Statistics,” Journal of Statistical Planning and Inference, Vol. 89, No. 1-2, 2000, pp. 79-89. doi:10.1016/S0378-3758(00)00074-4

[6] M. Habibullah and M. Ahsanullah, “Estimation of Parameters of a Pareto Distribution by Generalized Order statistics,” Communications in Statistics—Theory and Methods, Vol. 29, No. 7, 2000, pp. 1597-1609. doi:10.1080/03610920008832567

[7] Z. F. Jaheen, “On Bayesian Prediction of Generalized Order Statistics,” Journal of Statistical Theory and Applications, Vol. 1, No. 3, 2002, pp.191-204.

[8] Z. F. Jaheen, “Estimation Based on Generalized Order Statistics from the Burr Model,” Communications in Statistics—Theory and Methods, Vol. 34, No. 4, 2005, pp. 785-794. doi:10.1081/STA-200054408

[9] E. K. Al-Hussaini and A. A. Ahmad, “On Bayesian Predictive Distributions of Generalized Order Statistics,” Metrika, Vol. 57, No. 2, 2003, pp. 165-176. doi:10.1007/s001840200207

[10] E. K. Al-Hussaini, “Generalized Order Statistics: Prospective and Applications,” Journal of Applied Statistical Science, Vol. 13, No. 1, 2004, pp. 59-85.

[11] A. A. Ahmad and T. A. Abu-Shal, “Recurrence Relations for Moment Generating Functions of Nonadjacent Generalized Order Statistics Based on a Class of Doubly Truncated Distributions,” Journal of Statistical Theory and Applications, Vol. 6, No. 2, 2007, pp. 174-189.

[12] A. A. Ahmad and T. A. Abu-Shal, “Recurrence Relations for Moment Generating Functions of Generalized Order Statistics from Doubly Truncated Continuous Distributions,” Journal of Statistical Theory and Applications, Vol. 6, No. 2, 2008, pp. 243-257.

[13] A. A. Ahmad, “Relations for Single and Product Moments of Generalized Order Statistics from Doubly Truncated Burr type XII Distribution,” Journal of the Egyptian Mathematical Society, Vol. 15, No. 1, 2007, pp. 117- 128.

[14] A. A. Ahmad, “Single and Product Moments of Generalized Order Statistics from Linear Exponential Distribution,” Communications in Statistics—Theory and Methods, Vol. 37, No. 8, 2008, pp. 1162-1172. doi:10.1080/03610920701713344

[15] Z. A. Aboeleneen, “Inference for Weibull Distribution under Generalized Order Statistics,” Mathematics and Computers in Simulation, Vol. 81, No. 1, 2010, pp. 26-36. doi:10.1016/j.matcom.2010.06.013

[16] S. Abu El Fotouh, “Estimation for the Parameters of the Weibull Extension Model Based on Generalized Order Statistics,” International Journal of Contemporary Mathematical Sciencess, Vol. 6, No. 36, 2011, pp. 1749-1760.

[17] S. F. Ateya, “Prediction under Generalized Exponential Distribution Using MCMC Algorithm,” International Mathematical Forum, Vol. 6, No. 63, 2011, pp. 3111-3119.

[18] S. F. Ateya and A. A. Ahmad, “Inferences Based on Generalized Order Statistics under Truncated Type I Generalized Logistic Distribution,” Statistics, Vol. 45, No. 4, 2011, pp. 389-402. doi:10.1080/02331881003650149

[19] B. S. Everitt and D. J. Hand, “Finite Mixture Distributions,” Cambridge University Press, Cambridge, 1981.

[20] D. M. Titterington, A. F. M. Smith and U. E. Makov, “Statistical Analysis of Finite Mixture Distributions,” John Wiley and Sons, New York, 1985.

[21] K. E. Ahmad, “Identifiability of Finite Mixtures Using a New Transform,” Annals of the Institute of Statistical Mathematics, Vol. 40, No. 2, 1988, pp. 261-265. doi:10.1007/BF00052342

[22] G. J. McLachlan and K. E. Basford, “Mixture Models: Inferences and Applications to Clustering,” Marcel Dekker, New York, 1988.

[23] Z. F. Jaheen, “On Record Statistics from a Mixture of Two Exponential Distributions,” Journal of Statistical Computation and Simulation, Vol. 75, No. 1, 2005, pp. 1- 11. doi:10.1080/00949650410001646924

[24] K. E. Ahmad, Z. F. Jaheen and Heba S. Mohammed, “Bayesian Prediction Based on Type-I Censored Data from a Mixture of Burr Type XII Distribution and Its Reciprocal,” Statistics, Vol. 1, No. 1, 2011, pp. 1-11. doi:10.1080/02331888.2011.555550

[25] E. K. AL-Hussaini and M. Hussein, “Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss,” Open Journal of Statistics, Vol. 2, No. 1, 2012, pp. 28-38. doi:10.4236/ojs.2012.21004

[26] M. A. M. Ali-Mousa, “Bayesian Prediction Based on Pareto Doubly Censored Data,” Statistics, Vol. 37, No. 1, 2003, pp. 65-72. doi:10.1080/0233188021000004639

[27] A. S. Papadapoulos and W. J. Padgett, “On Bayes Estimation for Mixtures of Two Exponential-Life-Distributions from Right-Censored Samples,” IEEE Transactions on Reliability, Vol. 35, No. 1, 1986, pp. 102-105. doi:10.1109/TR.1986.4335364

[28] A. F. Attia, “On estimation for mixtures of 2 Rayleigh Distribution with Censoring,” Microelectronics Reliability, Vol. 33, No. 6, 1993, pp. 859-867. doi:10.1016/0026-2714(93)90259-2

[29] K. E. Ahmad, H. M. Moustafa and A. M. Abd-El-Rahman, “Approximate Bayes Estimation for Mixtures of Two Weibull Distributions under Type II Censoring,” Journal of Statistical Computation and Simulation, Vol. 58, No. 3, 1997, pp. 269-285. doi:10.1080/00949659708811835

[30] A. A. Soliman, “Estimators for the Finite Mixture of Rayleigh Model Based on Progressively Censored Data,” Communications in Statistics—Theory and Methods, Vol. 35, No. 5, 2006, pp. 803-820. doi:10.1080/03610920500501379

[31] M. Saleem and M. Aslam, “Bayesian Analysis of the Two Component Mixture of the Rayleigh Distribution Assuming the Uniform and the Jeffreys Priors,” Journal of Applied Statistical Science, Vol. 16, No. 4, 2008, pp. 105-113.

[32] M. Saleem and M. Aslam, “On Prior Selection for the Mixture of Rayleigh Distribution Using Predictive Intervals,” Pakistan Journal of Statistics, Vol. 24, No. 1, 2007, pp. 21-35.

[33] M. Saleem and M. Irfan, “On Properties of the Bayes estimates of the Rayleigh Mixture Parameters: A Simulation Study,” Pakistan Journal of Statistics., Vol. 26, No. 3, 2010, pp. 547-555.

[34] A. Zellner, “Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions,” In: J. O. Berger and S. S. Gupta, Eds., Statistical Decision Theory and Methods. V, Springer, New York, 1994, pp. 339-390.

[35] M. J. Jozani, E. Marchand and A. Parsian, “Bayes Estimation under a General Class of Balanced Loss Functions,” Universite de Sherbrooke, Sherbrooke, 2006.

[36] E. K. Al-Hussaini and K. E. Ahmad, “On the Identifiability of Finite Mixtures of Distributions,” IEEE Transactions on Information Theory, Vol. 27, No. 5, 1981, pp. 664-668. doi:10.1109/TIT.1981.1056389

[37] K. E. Ahmad and E. K. Al-Hussaini, “Remarks on the Non-Identifiability of Mixtures of Distributions,” Annals of the Institute of Statistical Mathematics, Vol. 34, No. 1, 1982, pp. 543-544. doi:10.1007/BF02481052

[38] S. J. Press, “Subjective and Objective Bayesian Statistics: Principles, Models and Applications,” Wiley, New York, 2003.

[39] J. Aitchison and I.R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975.