Inferences under a Class of Finite Mixture Distributions Based on Generalized Order Statistics

Affiliation(s)

Department of Mathematics, Assiut University, Assiut, Egypt.

Department of Mathematics, Taif University, Taif, KSA.

Department of Mathematics, Assiut University, Assiut, Egypt.

Department of Mathematics, Taif University, Taif, KSA.

ABSTRACT

The main purpose of this paper is to obtain estimates of parameters, reliability and hazard rate functions of a heterogeneous population represented by finite mixture of two general components. The doubly Type II censoring of generalized order statistics scheme is used. Maximum likelihood and Bayes methods of estimation are used for this purpose. The two methods of estimation are compared via a Monte Carlo Simulation study.

The main purpose of this paper is to obtain estimates of parameters, reliability and hazard rate functions of a heterogeneous population represented by finite mixture of two general components. The doubly Type II censoring of generalized order statistics scheme is used. Maximum likelihood and Bayes methods of estimation are used for this purpose. The two methods of estimation are compared via a Monte Carlo Simulation study.

Cite this paper

A. Ahmad and A. AL-Zaydi, "Inferences under a Class of Finite Mixture Distributions Based on Generalized Order Statistics,"*Open Journal of Statistics*, Vol. 3 No. 4, 2013, pp. 231-244. doi: 10.4236/ojs.2013.34027.

A. Ahmad and A. AL-Zaydi, "Inferences under a Class of Finite Mixture Distributions Based on Generalized Order Statistics,"

References

[1] 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

[2] E. K. AL-Hussaini, G. R. Al-Dayian and S. A. Adham, “On Finite Mixture of Two-Component Gompertz Lifetime Model,” Journal of Statistical Computation and Simulation, Vol. 67, No. 1, 2000, pp. 1-20.

[3] E. K. AL-Hussaini and S. F. Ateya, “Bayes Estimations under a Mixture of Truncated Type I Generalized Logistic Components Model,” Journal of Applied Statistical Science, Vol. 4 No. 2, 2005, pp. 183-208.

[4] 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. 111. doi:10.1080/00949650410001646924

[5] E. K. AL-Hussaini and R. A. AL-Jarallah, “Bayes Inference under a Finite Mixture of Two Compound Gompertz Components Model,” Cairo University, 2006.

[6] K. S. Sultan, M. A. Ismail and A. S. Al-Moisheer, “Mixture of Two Inverse Weibull Distributions: Properties and Estimation,” Computational Statistics & Data Analysis, Vol. 51, No. 11, 2007, pp. 5377-5387. doi:10.1016/j.csda.2006.09.016

[7] 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

[8] 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

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

[10] 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

[11] 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

[12] 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

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

[14] Z. F. Jaheen, “On Bayesian Prediction of Generalized Order Statistics,” Journal of Applied Statistical Science, Vol. 1, No. 3, 2002, pp.191-204.

[15] 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

[16] 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. , 2007, pp. 117128.

[17] 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. 11621172. doi:10.1080/03610920701713344

[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] T. A. Abu-Shal and A. M. AL-Zaydi, “Estimation Based on Generalized Order Statistics from a Mixture of Two Rayleigh Distributions,” International Journal of Statistics and Probability, Vol. 1, No. 2, 2012, pp. 79-90.

[20] T. A. Abu-Shal and A. M. 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

[21] B. S. Everitt and D. J. Hand, “Finite Mixture Distributions,” University Press, Cambridge, 1981. doi:10.1007/978-94-009-5897-5

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

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

[24] H. Teicher, “Identifiability of Finite Mixtures,” The Annals of Mathematical Statistics, Vol. 34, No. 4, 1963, pp. 1265-1269. doi:10.1214/aoms/1177703862

[25] E. K. AL-Hussaini and K. E. Ahmad, “On the Identifiability of Finite Mixtures of Distributions,” Transactions on Information Theory, Vol. 27, No. 5, 1981, pp. 664668.

[26] K. E. Ahmad, “Identifiability of Finite Mixtures Using a New Transform,” The Annals of Mathematical Statistics, Vol. 40, No. 2, 1988, pp. 261-265. doi:10.1007/BF00052342

[27] A. A. Ahmad, “On Bayesian Interval Prediction of Future Generalized Order Statistics Using Doubly Censoring,” Statistics, Vol. 45, No. 5, 2011, pp. 413425. doi:10.1080/02331881003650123

[28] 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

[29] E. K. AL-Hussaini, “Prediction, Observables from a General Class of Distributions,” Journal of Statistical Planning and Inference, Vol. 79, No. 1, 1999, pp. 79-91. doi:10.1016/S0378-3758(98)00228-6

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

[1] 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

[2] E. K. AL-Hussaini, G. R. Al-Dayian and S. A. Adham, “On Finite Mixture of Two-Component Gompertz Lifetime Model,” Journal of Statistical Computation and Simulation, Vol. 67, No. 1, 2000, pp. 1-20.

[3] E. K. AL-Hussaini and S. F. Ateya, “Bayes Estimations under a Mixture of Truncated Type I Generalized Logistic Components Model,” Journal of Applied Statistical Science, Vol. 4 No. 2, 2005, pp. 183-208.

[4] 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. 111. doi:10.1080/00949650410001646924

[5] E. K. AL-Hussaini and R. A. AL-Jarallah, “Bayes Inference under a Finite Mixture of Two Compound Gompertz Components Model,” Cairo University, 2006.

[6] K. S. Sultan, M. A. Ismail and A. S. Al-Moisheer, “Mixture of Two Inverse Weibull Distributions: Properties and Estimation,” Computational Statistics & Data Analysis, Vol. 51, No. 11, 2007, pp. 5377-5387. doi:10.1016/j.csda.2006.09.016

[7] 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

[8] 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

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

[10] 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

[11] 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

[12] 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

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

[14] Z. F. Jaheen, “On Bayesian Prediction of Generalized Order Statistics,” Journal of Applied Statistical Science, Vol. 1, No. 3, 2002, pp.191-204.

[15] 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

[16] 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. , 2007, pp. 117128.

[17] 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. 11621172. doi:10.1080/03610920701713344

[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] T. A. Abu-Shal and A. M. AL-Zaydi, “Estimation Based on Generalized Order Statistics from a Mixture of Two Rayleigh Distributions,” International Journal of Statistics and Probability, Vol. 1, No. 2, 2012, pp. 79-90.

[20] T. A. Abu-Shal and A. M. 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

[21] B. S. Everitt and D. J. Hand, “Finite Mixture Distributions,” University Press, Cambridge, 1981. doi:10.1007/978-94-009-5897-5

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

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

[24] H. Teicher, “Identifiability of Finite Mixtures,” The Annals of Mathematical Statistics, Vol. 34, No. 4, 1963, pp. 1265-1269. doi:10.1214/aoms/1177703862

[25] E. K. AL-Hussaini and K. E. Ahmad, “On the Identifiability of Finite Mixtures of Distributions,” Transactions on Information Theory, Vol. 27, No. 5, 1981, pp. 664668.

[26] K. E. Ahmad, “Identifiability of Finite Mixtures Using a New Transform,” The Annals of Mathematical Statistics, Vol. 40, No. 2, 1988, pp. 261-265. doi:10.1007/BF00052342

[27] A. A. Ahmad, “On Bayesian Interval Prediction of Future Generalized Order Statistics Using Doubly Censoring,” Statistics, Vol. 45, No. 5, 2011, pp. 413425. doi:10.1080/02331881003650123

[28] 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

[29] E. K. AL-Hussaini, “Prediction, Observables from a General Class of Distributions,” Journal of Statistical Planning and Inference, Vol. 79, No. 1, 1999, pp. 79-91. doi:10.1016/S0378-3758(98)00228-6

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