Bayesian Prediction of Future Generalized Order Statistics from a Class of Finite Mixture Distributions

Affiliation(s)

^{1}
Department of Mathematics, Assiut University, Assiut, Egypt.

^{2}
Department of Mathematics, Taif University, Taif, Saudi Arabia.

ABSTRACT

This article is concerned with the problem of prediction for the future generalized order statistics from a mixture of two general components based on doubly type II censored sample. We consider the one sample prediction and two sample prediction techniques. Bayesian prediction intervals for the median of future sample of generalized order statistics having odd and even sizes are obtained. Our results are specialized to ordinary order statistics and ordinary upper record values. A mixture of two Gompertz components model is given as an application. Numerical computations are given to illustrate the procedures.

KEYWORDS

Generalized Order Statistics, Bayesian Prediction, Heterogeneous Population, Doubly Type II Censored Samples, One- and Two-Sample Schemes

Generalized Order Statistics, Bayesian Prediction, Heterogeneous Population, Doubly Type II Censored Samples, One- and Two-Sample Schemes

Cite this paper

A. Ahmad, A. and Al-Zaydi, A. (2015) Bayesian Prediction of Future Generalized Order Statistics from a Class of Finite Mixture Distributions.*Open Journal of Statistics*, **5**, 585-599. doi: 10.4236/ojs.2015.56060.

A. Ahmad, A. and Al-Zaydi, A. (2015) Bayesian Prediction of Future Generalized Order Statistics from a Class of Finite Mixture Distributions.

References

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http://dx.doi.org/10.1007/978-1-4899-4467-2

[2] Aitchison, J. and Dunsmore, I.R. (1975) Statistical Prediction Analysis. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511569647

[3] AL-Hussaini, E.K. (1999) Prediction, Observables from a General Class of Distributions. Journal of Statistical Planning and Inference, 79, 79-91.

http://dx.doi.org/10.1016/S0378-3758(98)00228-6

[4] AL-Hussaini, E.K. (1999) Bayesian Prediction under a Mixture of Two Exponential Components Model Based on Type I Censoring. Journal of Applied Statistical Science, 8, 173-185.

[5] AL-Hussaini, E.K. (2001) On Bayesian Prediction of Future Median. Communications in Statistics—Theory and Methods, 30, 1395-1410.

http://dx.doi.org/10.1081/STA-100104752

[6] AL-Hussaini, E.K., Nigm, A.M. and Jaheen, Z.F. (2001) Bayesian Prediction Based on Finite Mixtures of Lomax Components Model and Type I Censoring. Statistics, 35, 259-268.

http://dx.doi.org/10.1080/02331880108802735

[7] AL-Hussaini, E.K. (2003) Bayesian Predictive Density of Order Statistics Based on Finite Mixture Models. Journal of Statistical Planning and Inference, 113, 15-24.

http://dx.doi.org/10.1016/S0378-3758(01)00297-X

[8] AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Interval Prediction of Future Records. Test, 12, 79-99.

http://dx.doi.org/10.1007/BF02595812

[9] Jaheen, Z.F. (2003) Bayesian Prediction under a Mixture of Two-Component Gompertz Lifetime Model. Test, 12, 413-426. http://dx.doi.org/10.1007/BF02595722

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

[11] AL-Hussaini, E.K. and Osman, M.I. (1997) On the Median of a Finite Mixture. Journal of Statistical Computation and Simulation, 58, 121-144.

http://dx.doi.org/10.1080/00949659708811826

[12] AL-Hussaini, E.K. and Jaheen, Z.F. (1999) Parametric Prediction Bounds for the Future Median of the Exponential Distribution. Statistics: A Journal of Theoretical and Applied Statistics, 32, 267-275.

http://dx.doi.org/10.1080/02331889908802667

[13] Kamps, U. (1995) A Concept of Generalized Order Statistics. Journal of Statistical Planning and Inference, 48, 1-23.

http://dx.doi.org/10.1016/0378-3758(94)00147-N

[14] Ahsanullah, M. (1996) Generalized Order Statistics from Two Parameter Uniform Distribution. Communications in Statistics: Theory and Methods, 25, 2311-2318.

http://dx.doi.org/10.1080/03610929608831840

[15] Ahsanullah, M. (2000) Generalized Order Statistics from Exponential Distribution. Journal of Statistical Planning and Inference, 85, 85-91.

http://dx.doi.org/10.1016/S0378-3758(99)00068-3

[16] Kamps, U. and Gather, U. (1997) Characteristic Property of Generalized Order Statistics for Exponential Distributions. Applicationes Mathematicae, 24, 383-391.

[17] Cramer, E. and Kamps, U. (2000) Relations for Expectations of Functions of Generalized Order Statistics. Journal of Statistical Planning and Inference, 89, 79-89.

http://dx.doi.org/10.1016/S0378-3758(00)00074-4

[18] Cramer, E. and Kamps, U. (2001) On Distributions of Generalized Order Statistics. Statistics: A Journal of Theoretical and Applied Statistics, 35, 269-280.

http://dx.doi.org/10.1080/02331880108802736

[19] Habibullah, M. and Ahsanullah, M. (2000) Estimation of Parameters of a Pareto Distribution by Generalized Order Statistics. Communications in Statistics: Theory and Methods, 29, 1597-1609.

http://dx.doi.org/10.1080/03610920008832567

[20] AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Interval Prediction of Future Records. Test, 12, 79-99.

http://dx.doi.org/10.1007/BF02595812

[21] AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Predictive Distributions of Generalized Order Statistics. Metrika, 57, 165-176.

http://dx.doi.org/10.1007/s001840200207

[22] AL-Hussaini, E.K. (2004) Generalized Order Statistics: Prospective and Applications. Journal of Applied Statistical Science, 13, 59-85.

[23] Jaheen, Z.F. (2002) On Bayesian Prediction of Generalized Order Statistics. Journal of Statistical Theory and Applications, 1, 191-204.

[24] Jaheen, Z.F. (2005) Estimation Based on Generalized Order Statistics from the Burr Model. Communications in Statistics: Theory and Methods, 34, 785-794.

http://dx.doi.org/10.1081/STA-200054408

[25] Ahmad, A.A. (2007) Relations for Single and Product Moments of Generalized Order Statistics from Doubly Truncated Burr Type XII Distribution. Journal of the Egyptian Mathematical Society, 15, 117-128.

[26] Ahmad, A.A. (2008) Single and Product Moments of Generalized Order Statistics from Linear Exponential Distribution. Communications in Statistics: Theory and Methods, 37, 1162-1172.

http://dx.doi.org/10.1080/03610920701713344

[27] Ahmad, A.A. (2011) On Bayesian Interval Prediction of Future Generalized Order Statistics Using Doubly Censoring. Statistics: A Journal of Theoretical and Applied Statistics, 45, 413-425.

http://dx.doi.org/10.1080/02331881003650123

[28] Ateya, S.F. and Ahmad, A.A. (2011) Inferences Based on Generalized Order Statistics under Truncated Type I Generalized Logistic Distribution. Statistics: A Journal of Theoretical and Applied Statistics, 45, 389-402.

http://dx.doi.org/10.1080/02331881003650149

[29] Abu-Shal, T.A. and AL-Zaydi, A.M. (2012) Estimation Based on Generalized Order Statistics from a Mixture of Two Rayleigh Distributions. International Journal of Statistics and Probability, 1, 79-90.

[30] Abu-Shal, T.A. and AL-Zaydi, A.M. (2012) Bayesian Prediction Based on Generalized Order Statistics from a Mixture of Two Exponentiated Weibull Distribution via MCMC Simulation. International Journal of Statistics and Probability, 1, 20-34.

[31] Abu-Shal, T.A. and AL-Zaydi, A.M. (2012) Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm. Open Journal of Statistics, 2, 356-367.

http://dx.doi.org/10.4236/ojs.2012.23044

[32] Abu-Shal, T.A. (2013) On Bayesian Prediction of Future Median Generalized Order Statistics Using Doubly Censored Data from Type-I Generalized Logistic Model. Journal of Statistical and Econometric Methods, 2, 61-79.

[33] Ahmad, A.A. and AL-Zaydi, A.M. (2013) Inferences under a Class of Finite Mixture Distributions Based on Generalized Order Statistics. Open Journal of Statistics, 3, 231-244.

http://dx.doi.org/10.4236/ojs.2013.34027

[34] Everitt, B.S. and Hand, D.J. (1981) Finite Mixture Distributions. Springer Netherlands, Dordrecht.

http://dx.doi.org/10.1007/978-94-009-5897-5

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

[36] McLachlan, G.J. and Basford, K.E. (1988) Mixture Models: Inferences and Applications to Clustering. Marcel Dekker, New York.

[37] Teicher, H. (1963) Identifiability of Finite Mixtures. The Annals of Mathematical Statistics, 34, 1265-1269.

http://dx.doi.org/10.1214/aoms/1177703862

[38] AL-Hussaini, E.K. and Ahmad, K.E. (1981) On the Identifiability of Finite Mixtures of Distributions. IEEE Transactions on Information Theory, 27, 664-668.

http://dx.doi.org/10.1109/TIT.1981.1056389

[39] Ahmad, K.E. (1988) Identifiability of Finite Mixtures Using a New Transform. Annals of the Institute of Statistical Mathematics, 40, 261-265.

http://dx.doi.org/10.1007/BF00052342

[40] AL-Hussaini, E.K., Al-Dayian, G.R. and Adham, S.A. (2000) On Finite Mixture of Two-Component Gompertz Lifetime Model. Journal of Statistical Computation and Simulation, 67, 1-20.

[1] Geisser, S. (1993) Predictive Inference: An Introduction. Chapman and Hall, London.

http://dx.doi.org/10.1007/978-1-4899-4467-2

[2] Aitchison, J. and Dunsmore, I.R. (1975) Statistical Prediction Analysis. Cambridge University Press, Cambridge.

http://dx.doi.org/10.1017/CBO9780511569647

[3] AL-Hussaini, E.K. (1999) Prediction, Observables from a General Class of Distributions. Journal of Statistical Planning and Inference, 79, 79-91.

http://dx.doi.org/10.1016/S0378-3758(98)00228-6

[4] AL-Hussaini, E.K. (1999) Bayesian Prediction under a Mixture of Two Exponential Components Model Based on Type I Censoring. Journal of Applied Statistical Science, 8, 173-185.

[5] AL-Hussaini, E.K. (2001) On Bayesian Prediction of Future Median. Communications in Statistics—Theory and Methods, 30, 1395-1410.

http://dx.doi.org/10.1081/STA-100104752

[6] AL-Hussaini, E.K., Nigm, A.M. and Jaheen, Z.F. (2001) Bayesian Prediction Based on Finite Mixtures of Lomax Components Model and Type I Censoring. Statistics, 35, 259-268.

http://dx.doi.org/10.1080/02331880108802735

[7] AL-Hussaini, E.K. (2003) Bayesian Predictive Density of Order Statistics Based on Finite Mixture Models. Journal of Statistical Planning and Inference, 113, 15-24.

http://dx.doi.org/10.1016/S0378-3758(01)00297-X

[8] AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Interval Prediction of Future Records. Test, 12, 79-99.

http://dx.doi.org/10.1007/BF02595812

[9] Jaheen, Z.F. (2003) Bayesian Prediction under a Mixture of Two-Component Gompertz Lifetime Model. Test, 12, 413-426. http://dx.doi.org/10.1007/BF02595722

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

[11] AL-Hussaini, E.K. and Osman, M.I. (1997) On the Median of a Finite Mixture. Journal of Statistical Computation and Simulation, 58, 121-144.

http://dx.doi.org/10.1080/00949659708811826

[12] AL-Hussaini, E.K. and Jaheen, Z.F. (1999) Parametric Prediction Bounds for the Future Median of the Exponential Distribution. Statistics: A Journal of Theoretical and Applied Statistics, 32, 267-275.

http://dx.doi.org/10.1080/02331889908802667

[13] Kamps, U. (1995) A Concept of Generalized Order Statistics. Journal of Statistical Planning and Inference, 48, 1-23.

http://dx.doi.org/10.1016/0378-3758(94)00147-N

[14] Ahsanullah, M. (1996) Generalized Order Statistics from Two Parameter Uniform Distribution. Communications in Statistics: Theory and Methods, 25, 2311-2318.

http://dx.doi.org/10.1080/03610929608831840

[15] Ahsanullah, M. (2000) Generalized Order Statistics from Exponential Distribution. Journal of Statistical Planning and Inference, 85, 85-91.

http://dx.doi.org/10.1016/S0378-3758(99)00068-3

[16] Kamps, U. and Gather, U. (1997) Characteristic Property of Generalized Order Statistics for Exponential Distributions. Applicationes Mathematicae, 24, 383-391.

[17] Cramer, E. and Kamps, U. (2000) Relations for Expectations of Functions of Generalized Order Statistics. Journal of Statistical Planning and Inference, 89, 79-89.

http://dx.doi.org/10.1016/S0378-3758(00)00074-4

[18] Cramer, E. and Kamps, U. (2001) On Distributions of Generalized Order Statistics. Statistics: A Journal of Theoretical and Applied Statistics, 35, 269-280.

http://dx.doi.org/10.1080/02331880108802736

[19] Habibullah, M. and Ahsanullah, M. (2000) Estimation of Parameters of a Pareto Distribution by Generalized Order Statistics. Communications in Statistics: Theory and Methods, 29, 1597-1609.

http://dx.doi.org/10.1080/03610920008832567

[20] AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Interval Prediction of Future Records. Test, 12, 79-99.

http://dx.doi.org/10.1007/BF02595812

[21] AL-Hussaini, E.K. and Ahmad, A.A. (2003) On Bayesian Predictive Distributions of Generalized Order Statistics. Metrika, 57, 165-176.

http://dx.doi.org/10.1007/s001840200207

[22] AL-Hussaini, E.K. (2004) Generalized Order Statistics: Prospective and Applications. Journal of Applied Statistical Science, 13, 59-85.

[23] Jaheen, Z.F. (2002) On Bayesian Prediction of Generalized Order Statistics. Journal of Statistical Theory and Applications, 1, 191-204.

[24] Jaheen, Z.F. (2005) Estimation Based on Generalized Order Statistics from the Burr Model. Communications in Statistics: Theory and Methods, 34, 785-794.

http://dx.doi.org/10.1081/STA-200054408

[25] Ahmad, A.A. (2007) Relations for Single and Product Moments of Generalized Order Statistics from Doubly Truncated Burr Type XII Distribution. Journal of the Egyptian Mathematical Society, 15, 117-128.

[26] Ahmad, A.A. (2008) Single and Product Moments of Generalized Order Statistics from Linear Exponential Distribution. Communications in Statistics: Theory and Methods, 37, 1162-1172.

http://dx.doi.org/10.1080/03610920701713344

[27] Ahmad, A.A. (2011) On Bayesian Interval Prediction of Future Generalized Order Statistics Using Doubly Censoring. Statistics: A Journal of Theoretical and Applied Statistics, 45, 413-425.

http://dx.doi.org/10.1080/02331881003650123

[28] Ateya, S.F. and Ahmad, A.A. (2011) Inferences Based on Generalized Order Statistics under Truncated Type I Generalized Logistic Distribution. Statistics: A Journal of Theoretical and Applied Statistics, 45, 389-402.

http://dx.doi.org/10.1080/02331881003650149

[29] Abu-Shal, T.A. and AL-Zaydi, A.M. (2012) Estimation Based on Generalized Order Statistics from a Mixture of Two Rayleigh Distributions. International Journal of Statistics and Probability, 1, 79-90.

[30] Abu-Shal, T.A. and AL-Zaydi, A.M. (2012) Bayesian Prediction Based on Generalized Order Statistics from a Mixture of Two Exponentiated Weibull Distribution via MCMC Simulation. International Journal of Statistics and Probability, 1, 20-34.

[31] Abu-Shal, T.A. and AL-Zaydi, A.M. (2012) Prediction Based on Generalized Order Statistics from a Mixture of Rayleigh Distributions Using MCMC Algorithm. Open Journal of Statistics, 2, 356-367.

http://dx.doi.org/10.4236/ojs.2012.23044

[32] Abu-Shal, T.A. (2013) On Bayesian Prediction of Future Median Generalized Order Statistics Using Doubly Censored Data from Type-I Generalized Logistic Model. Journal of Statistical and Econometric Methods, 2, 61-79.

[33] Ahmad, A.A. and AL-Zaydi, A.M. (2013) Inferences under a Class of Finite Mixture Distributions Based on Generalized Order Statistics. Open Journal of Statistics, 3, 231-244.

http://dx.doi.org/10.4236/ojs.2013.34027

[34] Everitt, B.S. and Hand, D.J. (1981) Finite Mixture Distributions. Springer Netherlands, Dordrecht.

http://dx.doi.org/10.1007/978-94-009-5897-5

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

[36] McLachlan, G.J. and Basford, K.E. (1988) Mixture Models: Inferences and Applications to Clustering. Marcel Dekker, New York.

[37] Teicher, H. (1963) Identifiability of Finite Mixtures. The Annals of Mathematical Statistics, 34, 1265-1269.

http://dx.doi.org/10.1214/aoms/1177703862

[38] AL-Hussaini, E.K. and Ahmad, K.E. (1981) On the Identifiability of Finite Mixtures of Distributions. IEEE Transactions on Information Theory, 27, 664-668.

http://dx.doi.org/10.1109/TIT.1981.1056389

[39] Ahmad, K.E. (1988) Identifiability of Finite Mixtures Using a New Transform. Annals of the Institute of Statistical Mathematics, 40, 261-265.

http://dx.doi.org/10.1007/BF00052342

[40] AL-Hussaini, E.K., Al-Dayian, G.R. and Adham, S.A. (2000) On Finite Mixture of Two-Component Gompertz Lifetime Model. Journal of Statistical Computation and Simulation, 67, 1-20.