Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss

ABSTRACT

By exponentiating each of the components of a finite mixture of two exponential components model by a positive parameter, several shapes of hazard rate functions are obtained. Maximum likelihood and Bayes methods, based on square error loss function and objective prior, are used to obtain estimators based on balanced square error loss function for the parameters, survival and hazard rate functions of a mixture of two exponentiated exponential components model. Approximate interval estimators of the parameters of the model are obtained.

By exponentiating each of the components of a finite mixture of two exponential components model by a positive parameter, several shapes of hazard rate functions are obtained. Maximum likelihood and Bayes methods, based on square error loss function and objective prior, are used to obtain estimators based on balanced square error loss function for the parameters, survival and hazard rate functions of a mixture of two exponentiated exponential components model. Approximate interval estimators of the parameters of the model are obtained.

KEYWORDS

Finite Mixtures; Exponentiated Exponential Distribution; Maximum Likelihood Estimation; Bayes Estimation, Square Error and Balanced Square Error Loss Functions; Objective Prior

Finite Mixtures; Exponentiated Exponential Distribution; Maximum Likelihood Estimation; Bayes Estimation, Square Error and Balanced Square Error Loss Functions; Objective Prior

Cite this paper

E. 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.

E. AL-Hussaini and M. Hussein, "Estimation under a Finite Mixture of Exponentiated Exponential Components Model and Balanced Square Error Loss,"

References

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[13] E. K. AL-Hussaini, “Bayesian Prediction under a Mixture of Two Exponential Components Model Based on Type 1 Censoring,” Journal of Applied Statistical Science, Vol. 8, 1999, pp. 173-185.

[14] E. K. AL-Hussaini, “Bayesian Predictive Density of Order Statistics Based on Finite Mixture Models,” Journal of Statistical Planning and Inference, Vol. 113, No. 1, 2003, pp. 15-24. doi:10.1016/S0378-3758(01)00297-X

[15] H. Varian, “A Bayesian Approach to Real Estate Assessment,” In: S. E. Fienberg and A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics, North Holland, Amsterdam, 1975.

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[17] J. Ahmadi, M. J. Jozani, E. Marchand and A. Parsian, “Bayes Estimation Based on k-Record Data from a General Class of Distributions under Balanced Type Loss Functions,” Journal of Statistical Planning and Inference, Vol. 139, No. 3, 2009, pp. 1180-1189. doi:10.1016/j.jspi.2008.07.008

[18] A. Zellner, “Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions,” In: S. S. Gupta and J. O. Burger, Eds., Statistical Decision Theory and Related Topics V, Springer-Verlag, New York, pp. 377-390.

[19] W. Mendenhall and R. J. Hader, “Estimation of Parameters of Mixed Exponentially Distributed Failure Time Distributions from Censored Life Test Data,” Biometrika, Vol. 45, 1958, pp. 504-520.

[20] T. M. Apostol, “Mathematical Analysis,” Addison Wesley, Reading, 1957.

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

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

[23] W. Nelson, “Lifetime Data Analysis,” Wiley, New York, 1990.

[1] S. Newcomb, “A Generalized Theory of the Combination of Observations So as to Obtain the Best Result,” American Journal of Mathematics, Vol. 8, No. 4, 1886, pp. 343- 366. doi:10.2307/2369392

[2] K. Pearson, “Contributions to the Mathematical Theory of Evolution,” Philosophical Transactions A, Vol. 185, 1894, pp. 71-110.

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

[4] B. S. Everitt and D. J. Hand, “Finite Mixture Distributions,” Chapman & Hall, London, 1981. doi:10.1007/978-94-009-5897-5

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

[6] B. G. Lindsay, “Mixture Models: Theory, Geometry and Applications,” Institute of Mathematical Statistics, Hayward, 1995.

[7] G. J. McLachlan and D. Peel, “Finite Mixture Models,” Wiley, New York, 2000. doi:10.1002/0471721182

[8] E. K. AL-Hussaini and K. S. Sultan, “Reliability and Hazard Based on Finite Mixture Models,” In: N. Balak- rishnan and C. R. Rao, Eds., Advances in Reliability, Vol. 20, Elsevier, Amsterdam, 2001.

[9] E. K. AL-Hussaini and K. E. Ahmad, “Information Matrix for a Mixture of Two Inverse Gaussian Distributions,” Communications in Statistics—Simulation and Computation, Vol. 13, No. 6, 1984, pp. 785-800. doi:10.1080/03610918408812415

[10] K. E. AL-Hussaini and N. S. Abd-El-Hakim, “Failure Rate of the Inverse Gaussian-Weibull Model,” Annals of the Institute of Statistical Mathematics, Vol. 41, No. 3, 1989, pp. 617-622. doi:10.1007/BF00050672

[11] E. K. AL-Hussaini and N. S. Abd-El-Hakim, “Estimation of Parameters of the Inverse Gaussian-Weibull,” Communications in Statistics—Theory and Methods, Vol. 19, No. 5, 1990, pp. 1607-1622. doi:10.1080/03610929008830280

[12] E. K. AL-Hussaini and N. S. Abd-El-Hakim, “Efficiency of Schemes of Sampling from the Inverse Gaussian-Wei- bull Mixture Model,” Communications in Statistics— Theory and Methods, Vol. 21, No. 11, 1992, pp. 3143- 3169. doi:10.1080/03610929208830967

[13] E. K. AL-Hussaini, “Bayesian Prediction under a Mixture of Two Exponential Components Model Based on Type 1 Censoring,” Journal of Applied Statistical Science, Vol. 8, 1999, pp. 173-185.

[14] E. K. AL-Hussaini, “Bayesian Predictive Density of Order Statistics Based on Finite Mixture Models,” Journal of Statistical Planning and Inference, Vol. 113, No. 1, 2003, pp. 15-24. doi:10.1016/S0378-3758(01)00297-X

[15] H. Varian, “A Bayesian Approach to Real Estate Assessment,” In: S. E. Fienberg and A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics, North Holland, Amsterdam, 1975.

[16] R. D. Thompson and A. P. Basu, “Asymptotic Loss Function for Estimating System Reliability,” In: D. A. Berry, K. M. Chaloner and J. K. Geweke, Bayesian Analysis in Statistics and Econometrics, Eds., Wiley, New York, 1996.

[17] J. Ahmadi, M. J. Jozani, E. Marchand and A. Parsian, “Bayes Estimation Based on k-Record Data from a General Class of Distributions under Balanced Type Loss Functions,” Journal of Statistical Planning and Inference, Vol. 139, No. 3, 2009, pp. 1180-1189. doi:10.1016/j.jspi.2008.07.008

[18] A. Zellner, “Bayesian and Non-Bayesian Estimation Using Balanced Loss Functions,” In: S. S. Gupta and J. O. Burger, Eds., Statistical Decision Theory and Related Topics V, Springer-Verlag, New York, pp. 377-390.

[19] W. Mendenhall and R. J. Hader, “Estimation of Parameters of Mixed Exponentially Distributed Failure Time Distributions from Censored Life Test Data,” Biometrika, Vol. 45, 1958, pp. 504-520.

[20] T. M. Apostol, “Mathematical Analysis,” Addison Wesley, Reading, 1957.

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

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

[23] W. Nelson, “Lifetime Data Analysis,” Wiley, New York, 1990.