Bayes Prediction of Future Observables from Exponentiated Populations with Fixed and Random Sample Size

ABSTRACT

Bayesian predictive probability density function is obtained when the underlying pop-ulation distribution is exponentiated and subjective prior is used. The corresponding predictive survival function is then obtained and used in constructing 100(1 – ?)% predictive interval, using one- and two- sample schemes when the size of the future sample is fixed and random. In the random case, the size of the future sample is assumed to follow the truncated Poisson distribution with parameter λ. Special attention is paid to the exponentiated Burr type XII population, from which the data are drawn. Two illustrative examples are given, one of which uses simulated data and the other uses data that represent the breaking strength of 64 single carbon fibers of length 10, found in Lawless [40].

Bayesian predictive probability density function is obtained when the underlying pop-ulation distribution is exponentiated and subjective prior is used. The corresponding predictive survival function is then obtained and used in constructing 100(1 – ?)% predictive interval, using one- and two- sample schemes when the size of the future sample is fixed and random. In the random case, the size of the future sample is assumed to follow the truncated Poisson distribution with parameter λ. Special attention is paid to the exponentiated Burr type XII population, from which the data are drawn. Two illustrative examples are given, one of which uses simulated data and the other uses data that represent the breaking strength of 64 single carbon fibers of length 10, found in Lawless [40].

KEYWORDS

Predictive Density And Survival Functions, One- And Two-Sample Schemes, Bayes Prediction, Exponentiated Population. Exponentiated Burr Type XII Distribution, Data Of Carbon Fibers

Predictive Density And Survival Functions, One- And Two-Sample Schemes, Bayes Prediction, Exponentiated Population. Exponentiated Burr Type XII Distribution, Data Of Carbon Fibers

Cite this paper

nullE. AL-Hussaini and M. Hussein, "Bayes Prediction of Future Observables from Exponentiated Populations with Fixed and Random Sample Size,"*Open Journal of Statistics*, Vol. 1 No. 1, 2011, pp. 24-32. doi: 10.4236/ojs.2011.11004.

nullE. AL-Hussaini and M. Hussein, "Bayes Prediction of Future Observables from Exponentiated Populations with Fixed and Random Sample Size,"

References

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[2] E.K. AL-Hussaini,"A Characterization of the Burr type XII distribution," Appl. Math. Lett., Vol. 1, 1991, pp. 59-61.

[3] E.K. AL-Hussaini,"Bayesian prediction under a mixture of two exponential components model based on type I censoring," J. Appl. Statist. Sc. Vol. 8, 1999, pp. 173-185.

[4] E.K. AL-Hussaini, "Prediction: advances and new research," Presented as an invited topical paper in the International Mathematics Conference [Mathematics and the 21st century ], Cairo, Egypt, January 2000.

[5] E.K. AL-Hussaini, On Bayes prediction of future median. Commun. Statist.- Theory Meth., Vol. 30, 2001, pp. 1395-1410.

[6] E.K. AL-Hussaini, "Bayesian predictive density of order statistics based on finite mixture models,". J. Statist. Plann. Inf. Vol. 113, 2003, pp. 15-24.

[7] E.K. AL-Hussaini, "On the exponentiated class of distributions," J. Statist. Theory Appl ., Vol. 9, 2010a, pp. 41-64.

[8] E.K. AL-Hussaini,"Inference based on censored samples from exponentiated populations," Test., Vol. 19, 2010b, pp. 487-513.

[9] E.K. AL-Hussaini and M.A. Gharib,"A new family of distributions as a countable mixture with Poisson added parameter," J. Statist. Theory Appl., Vol. 8, 2009, pp. 169-190.

[10] E.K. AL-Hussaini and M. Ghitany, "On certain countable mixtures of absolutely continuous distributions," Metron,Vol. LXIII, pp. 39-53.

[11] E.K. AL-Hussaini and M. Hussein, "Estimation using censored data from exponentiated Burr type XII distribution," (Submitted).

[12] 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," J. Statist. Comput. Simul.,Vol. 41, 1992, pp. 31- 40.

[13] E.K. AL-Hussaini and Z.F. Jaheen, "Approximate Bayes estimators applied to the Burr model," Commun. Statist.-Theory Meth., Vol. 23, 1994, pp. 99-121.

[14] E.K. AL-Hussaini and Z.F. Jaheen," (1995), Bayes prediction bounds for the Burr type XII failure model," Commun. Statist.-Theory Meth., Vol. 24, 1995. pp. 1829-1842.

[15] E.K. AL-Hussaini and Z.F. Jaheen,"Bayesian prediction bounds for the Burr type XII distribution in the presence of outliers," J. Statist. Plann. Inf., Vol. 55, 1996, pp. 23-37.

[16] E.K. AL-Hussaini and Z.F. Jaheen, "Parametric prediction bounds for the future median of the exponential distribution," Statistics, Vol. 32, 1999, pp. 267 \-275.

[17] 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, 1992. pp. 33-42.

[18] E.K. AL-Hussaini A.M. Nigm and Z.F. Jaheen," Bayesian prediction based on finite mixtures of Lomax components model and type I censoring," Statistics, Vol. 35, 2001, 259-268.

[19] M.A. Ali Mousa and Z.F. Jaheen,"Bayesian prediction for the Burr type XII model based on doubly censored data," Statistics, Vol. 48, 1997, pp. 337-344.

[20] B.C. Arnold and S.J. Press,"Bayesian inference for Pareto populations," J. Economt-rics, Vol. 21, 1983, pp. 287-306.

[21] L.A. Baharith,"On the four-parameter Burr type XII life time model," M.Sc. thesis.King Abdulaziz University, Jeddah, Saudi Arabia, 1997.

[22] J. Bernardo and A. Smith, "Bayesian Theory," Wiley, New York, 1994.

[23] I.W. Burr, "Cumulative frequency functions," Ann. Math. Statist., Vol. 1, 1942, pp. 215-232.

[24] I.W. Burr and P.J. Cislak, (1968)," On a general system of distributions:I. Its curve shaped characteristics; II. The sample median," J. Amer. Statist. Assoc. Vol. 63, 1968, pp. 627-635.

[25] S.D. Dubey,"Statistical contributions to reliability engineerings," ARL TR 72-0155, AD 1774537, 1972.

[26] S.D. Dubey, "Statistical treatment of certain life testing and reliability problems,"ARL TR 73-0155, AD 1774537, 1973.

[27] I. Dunsmore, "The Bayesian predictive distribution in life testing models. Technom-trics Vol. 16, 1974, pp. 455-460.

[28] I.G. Evans and A.M. Nigm, "Bayesian prediction for two-parameter Weibull lifetime model," Commun. Statist.-Theory Meth., Vol. 9, 1980a, pp. 649-658.

[29] I.G. Evans and A.M. Nigm, "Bayesian 1-sample prediction for the 2-parameter Wei-bull distribution," IEEE Trans. Rel., Vol. R-29, 1980b, pp. 410-413.

[30] I.G. Evans and A.M. Nigm,"Bayesian prediction for the left truncated exponential distribution," Technometrics 22, 1980c, pp. 201-204.

[31] I.G. Evans and A.S. Ragab, "Bayesian inferences given a type-2 censored sample from Burr distribution," Commun. Statist.-Theory Meth.,Vol. 12, 1983, pp. 1569-1580.

[32] S. Geisser, "Predicting Pareto and exponential observables," Canad. J. Statist. Vol. 12, 1984, pp. 143-152.

[33] S. Geisser,"Prediction Inference: An Introduction," Chapman and Hall, London, 1993.

[34] R.C. Gupta and R.D. Gupta,"On the distribution of order statistics for a random sample size," Statist. Neelandica. Vol. 38 , 1984, pp. 13-19.

[35] R.C. Gupta and R.D. Gupta, "Proportional reversed hazard rate model and its applications," J. Statist. Plann. Inf., Vol. 137 , 2007, pp. 3525-3536.

[36] H.A. Howlader, HPD prediction intervals for Rayleigh distribution," IEEE Trans. Rel., Vol. 34, 1985, pp. 121-123.

[37] Z.F. Jaheen,"Bayesian estimations and predictions based on single Burr type XII models and their finite mixture," Ph.D. dissertation, University of Assiut, Egypt, 1993.

[38] K. Kaminsky and P. Nelson, "Prediction of order statistics," In Balakrishnan, N. and Rao, C.R. Eds. Handbook of Statistics, vol.17, Elsevier Science, Amesterdam, The Netherland, 1998, pp. 431-450.

[39] A.H. Khan and A.I. Khan, "Moments of order statistics from Burr's distribution and its characterization," Metron, Vol. 45, 1987, pp. 21-29.

[40] J.F. Lawless,"Statistical Models and Methods for Lifetime Data," 2nd ed, Wiley, 2003.

[41] E.L. Lehmann,"The power of rank tests," Ann. Math. Statist. 24,1953, pp. 28-43.

[42] A.W. Lewis, The Burr distribution as a general parametric family in survivorship and reliability theory applications. Ph.D. thesis, University of North Carolina, 1981.

[43] T. Lwin,"Estimating the tale of the Paretian law," Scand. Actuar. J. Vol. 55, 1972, pp. 170-178.

[44] G.S. Lingappaiah,"Bayesian approach to prediction and the spacings in the exponential distribution," Ann. Instit. Statist. Math., Vol. 31, 1979, pp. 391-401.

[45] G.S. Lingappaiah,"Bayesian approach to the estimation of parameters in the Burr's XII distribution with outliers," J. Orissa Math. Soc., Vol. 1, 1983. pp. 55-59.

[46] H.J. Malik,"Bayesian estimation of the Paretian index. Scand. Actuar. J., Vol. 53, 1970, pp. 6-9.

[47] 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 1997, pp. 641 -652.

[48] H.N. Nagaraja,"Prediction problems," In: Balackrishnan, N. and Basu, A.P., Eds., The exponential distribution: theory and applications, Gordon and Breach, New York, 1995, pp.139-163.

[49] A.M Nigm, "Prediction bounds for the Burr model," Commun. Statist.-Theory Meth., Vol. 17, 1988, pp. 287- 296.

[50] A.M. Nigm, and H.I. Hamdy, "Bayesian prediction bounds for the Pareto lifetime," Commun. Statist.-Theory Meth., Vol. 16, 1987, pp. 1761-1772.

[51] A.M. Nigm, E.K. AL-Hussaini, and Z.F. Jaheen,"Bayesian two-sample prediction under the Lomax model with fixed and random sample size. J. Appl. Statist. Sc., Vol. 15, 2007, pp. 381-390.

[52] J.K. Patel,"Prediction intervals-a review," Commun. Statist.-Theory Meth., Vol. 18, 1989, pp. 2393-2465.

[53] A.S. Papadopoulos,"The Burr distribution as a life time model from a Bayesian approach," IEEE Trans. Rel., Vol. R-27, 1978, pp. 369-371.

[54] M.Z. Raqab,"Predictors of future order statistics from type II censoring samples," Ph.D. dissertation, Ohio State University, Colombus, 1992.

[55] A. Shah and D.V. Gokhale, "On maximum product of spacings (MPS) estimation for Burr XII distribution,"Commun. Statist.-Theory Meth. Vol. 22, 1993, pp. 615-641.

[56] S.K. Sinha, "On the prediction limits for Rayleigh life distribution," Culcutta. Statist. Assoc. Bulletin Vol. 39, 1990, pp. 104-109.

[57] S.K. Sinha, and H.A. Howlader,"On the sampling of the Bayes estimators of the Pareto parameter with proper and improper priors and associated goodness of fit. Tech. Report No. 103. Dept. Statist., university of Mintoba, Winnipeg, Canada, 1980

[58] S.K. Sinha, and H.A. Howlader, Credible and HPD intervals of the parameter and reliability of Rayleigh distribution. IEEE Trans. Rel., Vol. 32, 1983, pp. 217-220.

[59] Tadikamalla, P.R. (1980). A look at the Burr and related distributions. Inter. Statist. Review, Vol. 48, 1980, pp. 337-344.

[60] K. Takahasi,"Note on the multivariate Burr's distribution," Ann. Instit. Statist. Math.1965, Vol. 17, 1965, pp. 257 -260.

[61] P.F. Verhulst, "Notice sur la loi population dans son accroissement," Correspondan-ce mathématique et physique, publiée par L.A.L. Quetelet, Vol. 10, 1838, pp. 113-121.

[62] P.F. Verhulst, "Recheches mathématiques sur la loi d'accroissement de la populati-on," Nouvelle mémoire de l'Academie Royale de Sciences et Belle-Lettres de Bruxelles [i.e. Mémoire Series 2], Vol. 18, 1845, pp. 1-42.

[63] P.F. Verhulst, "Deuxiéme mémoire sur la loi d'accroissment de la population," Mé-moire de l'Academie Royale des Sciences, des Lettres et de Beaux-Arts de Belgique, Seri-es 2, Vol. 20, 1847, pp. 1-32.

[64] A. Zellner, "An Introduction to Bayesian Inference in Econometrics," Wiley, 1971.

[1] J.C. Ahuja and S.W. Nash,"The generalized Gompertz-Verhulst family of distributio-ns," Sankhyā, Vol. 29, 1967, pp. 141-161.

[2] E.K. AL-Hussaini,"A Characterization of the Burr type XII distribution," Appl. Math. Lett., Vol. 1, 1991, pp. 59-61.

[3] E.K. AL-Hussaini,"Bayesian prediction under a mixture of two exponential components model based on type I censoring," J. Appl. Statist. Sc. Vol. 8, 1999, pp. 173-185.

[4] E.K. AL-Hussaini, "Prediction: advances and new research," Presented as an invited topical paper in the International Mathematics Conference [Mathematics and the 21st century ], Cairo, Egypt, January 2000.

[5] E.K. AL-Hussaini, On Bayes prediction of future median. Commun. Statist.- Theory Meth., Vol. 30, 2001, pp. 1395-1410.

[6] E.K. AL-Hussaini, "Bayesian predictive density of order statistics based on finite mixture models,". J. Statist. Plann. Inf. Vol. 113, 2003, pp. 15-24.

[7] E.K. AL-Hussaini, "On the exponentiated class of distributions," J. Statist. Theory Appl ., Vol. 9, 2010a, pp. 41-64.

[8] E.K. AL-Hussaini,"Inference based on censored samples from exponentiated populations," Test., Vol. 19, 2010b, pp. 487-513.

[9] E.K. AL-Hussaini and M.A. Gharib,"A new family of distributions as a countable mixture with Poisson added parameter," J. Statist. Theory Appl., Vol. 8, 2009, pp. 169-190.

[10] E.K. AL-Hussaini and M. Ghitany, "On certain countable mixtures of absolutely continuous distributions," Metron,Vol. LXIII, pp. 39-53.

[11] E.K. AL-Hussaini and M. Hussein, "Estimation using censored data from exponentiated Burr type XII distribution," (Submitted).

[12] 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," J. Statist. Comput. Simul.,Vol. 41, 1992, pp. 31- 40.

[13] E.K. AL-Hussaini and Z.F. Jaheen, "Approximate Bayes estimators applied to the Burr model," Commun. Statist.-Theory Meth., Vol. 23, 1994, pp. 99-121.

[14] E.K. AL-Hussaini and Z.F. Jaheen," (1995), Bayes prediction bounds for the Burr type XII failure model," Commun. Statist.-Theory Meth., Vol. 24, 1995. pp. 1829-1842.

[15] E.K. AL-Hussaini and Z.F. Jaheen,"Bayesian prediction bounds for the Burr type XII distribution in the presence of outliers," J. Statist. Plann. Inf., Vol. 55, 1996, pp. 23-37.

[16] E.K. AL-Hussaini and Z.F. Jaheen, "Parametric prediction bounds for the future median of the exponential distribution," Statistics, Vol. 32, 1999, pp. 267 \-275.

[17] 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, 1992. pp. 33-42.

[18] E.K. AL-Hussaini A.M. Nigm and Z.F. Jaheen," Bayesian prediction based on finite mixtures of Lomax components model and type I censoring," Statistics, Vol. 35, 2001, 259-268.

[19] M.A. Ali Mousa and Z.F. Jaheen,"Bayesian prediction for the Burr type XII model based on doubly censored data," Statistics, Vol. 48, 1997, pp. 337-344.

[20] B.C. Arnold and S.J. Press,"Bayesian inference for Pareto populations," J. Economt-rics, Vol. 21, 1983, pp. 287-306.

[21] L.A. Baharith,"On the four-parameter Burr type XII life time model," M.Sc. thesis.King Abdulaziz University, Jeddah, Saudi Arabia, 1997.

[22] J. Bernardo and A. Smith, "Bayesian Theory," Wiley, New York, 1994.

[23] I.W. Burr, "Cumulative frequency functions," Ann. Math. Statist., Vol. 1, 1942, pp. 215-232.

[24] I.W. Burr and P.J. Cislak, (1968)," On a general system of distributions:I. Its curve shaped characteristics; II. The sample median," J. Amer. Statist. Assoc. Vol. 63, 1968, pp. 627-635.

[25] S.D. Dubey,"Statistical contributions to reliability engineerings," ARL TR 72-0155, AD 1774537, 1972.

[26] S.D. Dubey, "Statistical treatment of certain life testing and reliability problems,"ARL TR 73-0155, AD 1774537, 1973.

[27] I. Dunsmore, "The Bayesian predictive distribution in life testing models. Technom-trics Vol. 16, 1974, pp. 455-460.

[28] I.G. Evans and A.M. Nigm, "Bayesian prediction for two-parameter Weibull lifetime model," Commun. Statist.-Theory Meth., Vol. 9, 1980a, pp. 649-658.

[29] I.G. Evans and A.M. Nigm, "Bayesian 1-sample prediction for the 2-parameter Wei-bull distribution," IEEE Trans. Rel., Vol. R-29, 1980b, pp. 410-413.

[30] I.G. Evans and A.M. Nigm,"Bayesian prediction for the left truncated exponential distribution," Technometrics 22, 1980c, pp. 201-204.

[31] I.G. Evans and A.S. Ragab, "Bayesian inferences given a type-2 censored sample from Burr distribution," Commun. Statist.-Theory Meth.,Vol. 12, 1983, pp. 1569-1580.

[32] S. Geisser, "Predicting Pareto and exponential observables," Canad. J. Statist. Vol. 12, 1984, pp. 143-152.

[33] S. Geisser,"Prediction Inference: An Introduction," Chapman and Hall, London, 1993.

[34] R.C. Gupta and R.D. Gupta,"On the distribution of order statistics for a random sample size," Statist. Neelandica. Vol. 38 , 1984, pp. 13-19.

[35] R.C. Gupta and R.D. Gupta, "Proportional reversed hazard rate model and its applications," J. Statist. Plann. Inf., Vol. 137 , 2007, pp. 3525-3536.

[36] H.A. Howlader, HPD prediction intervals for Rayleigh distribution," IEEE Trans. Rel., Vol. 34, 1985, pp. 121-123.

[37] Z.F. Jaheen,"Bayesian estimations and predictions based on single Burr type XII models and their finite mixture," Ph.D. dissertation, University of Assiut, Egypt, 1993.

[38] K. Kaminsky and P. Nelson, "Prediction of order statistics," In Balakrishnan, N. and Rao, C.R. Eds. Handbook of Statistics, vol.17, Elsevier Science, Amesterdam, The Netherland, 1998, pp. 431-450.

[39] A.H. Khan and A.I. Khan, "Moments of order statistics from Burr's distribution and its characterization," Metron, Vol. 45, 1987, pp. 21-29.

[40] J.F. Lawless,"Statistical Models and Methods for Lifetime Data," 2nd ed, Wiley, 2003.

[41] E.L. Lehmann,"The power of rank tests," Ann. Math. Statist. 24,1953, pp. 28-43.

[42] A.W. Lewis, The Burr distribution as a general parametric family in survivorship and reliability theory applications. Ph.D. thesis, University of North Carolina, 1981.

[43] T. Lwin,"Estimating the tale of the Paretian law," Scand. Actuar. J. Vol. 55, 1972, pp. 170-178.

[44] G.S. Lingappaiah,"Bayesian approach to prediction and the spacings in the exponential distribution," Ann. Instit. Statist. Math., Vol. 31, 1979, pp. 391-401.

[45] G.S. Lingappaiah,"Bayesian approach to the estimation of parameters in the Burr's XII distribution with outliers," J. Orissa Math. Soc., Vol. 1, 1983. pp. 55-59.

[46] H.J. Malik,"Bayesian estimation of the Paretian index. Scand. Actuar. J., Vol. 53, 1970, pp. 6-9.

[47] 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 1997, pp. 641 -652.

[48] H.N. Nagaraja,"Prediction problems," In: Balackrishnan, N. and Basu, A.P., Eds., The exponential distribution: theory and applications, Gordon and Breach, New York, 1995, pp.139-163.

[49] A.M Nigm, "Prediction bounds for the Burr model," Commun. Statist.-Theory Meth., Vol. 17, 1988, pp. 287- 296.

[50] A.M. Nigm, and H.I. Hamdy, "Bayesian prediction bounds for the Pareto lifetime," Commun. Statist.-Theory Meth., Vol. 16, 1987, pp. 1761-1772.

[51] A.M. Nigm, E.K. AL-Hussaini, and Z.F. Jaheen,"Bayesian two-sample prediction under the Lomax model with fixed and random sample size. J. Appl. Statist. Sc., Vol. 15, 2007, pp. 381-390.

[52] J.K. Patel,"Prediction intervals-a review," Commun. Statist.-Theory Meth., Vol. 18, 1989, pp. 2393-2465.

[53] A.S. Papadopoulos,"The Burr distribution as a life time model from a Bayesian approach," IEEE Trans. Rel., Vol. R-27, 1978, pp. 369-371.

[54] M.Z. Raqab,"Predictors of future order statistics from type II censoring samples," Ph.D. dissertation, Ohio State University, Colombus, 1992.

[55] A. Shah and D.V. Gokhale, "On maximum product of spacings (MPS) estimation for Burr XII distribution,"Commun. Statist.-Theory Meth. Vol. 22, 1993, pp. 615-641.

[56] S.K. Sinha, "On the prediction limits for Rayleigh life distribution," Culcutta. Statist. Assoc. Bulletin Vol. 39, 1990, pp. 104-109.

[57] S.K. Sinha, and H.A. Howlader,"On the sampling of the Bayes estimators of the Pareto parameter with proper and improper priors and associated goodness of fit. Tech. Report No. 103. Dept. Statist., university of Mintoba, Winnipeg, Canada, 1980

[58] S.K. Sinha, and H.A. Howlader, Credible and HPD intervals of the parameter and reliability of Rayleigh distribution. IEEE Trans. Rel., Vol. 32, 1983, pp. 217-220.

[59] Tadikamalla, P.R. (1980). A look at the Burr and related distributions. Inter. Statist. Review, Vol. 48, 1980, pp. 337-344.

[60] K. Takahasi,"Note on the multivariate Burr's distribution," Ann. Instit. Statist. Math.1965, Vol. 17, 1965, pp. 257 -260.

[61] P.F. Verhulst, "Notice sur la loi population dans son accroissement," Correspondan-ce mathématique et physique, publiée par L.A.L. Quetelet, Vol. 10, 1838, pp. 113-121.

[62] P.F. Verhulst, "Recheches mathématiques sur la loi d'accroissement de la populati-on," Nouvelle mémoire de l'Academie Royale de Sciences et Belle-Lettres de Bruxelles [i.e. Mémoire Series 2], Vol. 18, 1845, pp. 1-42.

[63] P.F. Verhulst, "Deuxiéme mémoire sur la loi d'accroissment de la population," Mé-moire de l'Academie Royale des Sciences, des Lettres et de Beaux-Arts de Belgique, Seri-es 2, Vol. 20, 1847, pp. 1-32.

[64] A. Zellner, "An Introduction to Bayesian Inference in Econometrics," Wiley, 1971.