Shrinkage Testimator in Gamma Type-II Censored Data under LINEX Loss Function
ABSTRACT
Prakash and Singh presented the shrinkage testimators under the invariant version of LINEX loss function for the scale parameter of an exponential distribution in presence Type-II censored data. In this paper, we extend this approach to gamma distribution, as Prakash and Singh’s paper is a special case of this paper. In fact, some shrinkage testimators for the scale parameter of a gamma distribution, when Type-II censored data are available, have been suggested under the LINEX loss function assuming the shape parameter is to be known. The comparisons of the proposed testimators have been made with improved estimator. All these estimators are compared empirically using Monte Carlo simulation.
Prakash and Singh presented the shrinkage testimators under the invariant version of LINEX loss function for the scale parameter of an exponential distribution in presence Type-II censored data. In this paper, we extend this approach to gamma distribution, as Prakash and Singh’s paper is a special case of this paper. In fact, some shrinkage testimators for the scale parameter of a gamma distribution, when Type-II censored data are available, have been suggested under the LINEX loss function assuming the shape parameter is to be known. The comparisons of the proposed testimators have been made with improved estimator. All these estimators are compared empirically using Monte Carlo simulation.
Cite this paper
A. Shadrokh and H. Pazira, "Shrinkage Testimator in Gamma Type-II Censored Data under LINEX Loss Function," Open Journal of Statistics, Vol. 3 No. 4, 2013, pp. 245-257. doi: 10.4236/ojs.2013.34028.
A. Shadrokh and H. Pazira, "Shrinkage Testimator in Gamma Type-II Censored Data under LINEX Loss Function," Open Journal of Statistics, Vol. 3 No. 4, 2013, pp. 245-257. doi: 10.4236/ojs.2013.34028.
References
[1] A. Parsian and S. N. U. A. Kirmani, “Estimation under LINEX loss function,” In: A. Ulla, Ed., Handbook of Applied Econometrics and Statistical Inference, 165 Dekker, New York, 2002, pp. 53-76. [2] H. Pazira and A. Shadrokh, “Comparison of LINEX and Precautionary Bayes Estimators on the Gamma Distribution Using Censored Data,” Journal of Statistics and Management Systems, Vol. 14, No. 3, 2011, pp. 617-638. doi:10.1080/09720510.2011.10701575 [3] T. S. Ferguson, “Mathematical Statistics: A Decision Theoretic Approach,” Academic Press, New York, 1967. [4] A. Zellner and M.S. Geisel, “Sensitivity of Control to Uncertainty and Form of the Criterion Function,” In the future of statistics, Ed Donald G. Watts, Academic Press, New York, 1968, pp. 269-289. [5] J. Aitchison and I. R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975. doi:10.1017/CBO9780511569647 [6] H. R. Varian, “A Bayesian Approach to Real Estate Assessment,” In: L. J. Savage, S. E. Feinberg and A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics: In Honor of L. J. Savage, North-Holland Pub. Co., Amsterdam, 1975, pp. 195-208. [7] J. O. Berger, “Statistical Decision Theory-Foundation Concepts and Methods,” Springer-Verlag, New York, 1980. doi:10.1007/978-1-4757-1727-3 [8] A. P. Basu and N. Ebrahimi, “Bayesian Approach to Life Testing and Reliability Estimation Using Asymmetric Loss Function,” Journal of Statistical Planning and Inferences, Vol. 29, No. 1-2, 1991, pp. 21-31. doi:10.1016/0378-3758(92)90118-C [9] B. N. Pandey, “Testimator of the Scale Parameter of the Exponential Distribution Using LINEX Loss Function,” Communication in Statistics-Theory and Methods, Vol. 26, No. 6, 1997, pp. 2191-2202. doi:10.1016/0026-2714(88)90294-6 [10] A. A. Soliman, “Comparison of Linex and QuadraticBayes Estimators Foe the Rayleigh Distribution,” Communication in Statistics-Theory and Methods, Vol. 29, No. 1, 2000, pp. 95-107. doi:10.1080/03610920008832471 [11] G. Prakash and D. C. Singh, “Shrinkage Estimation in Exponential Type-II Censored Data under LINEX Loss,” Journal of the Korean Statistical Society, Vol. 37, No. 1, 2008, pp. 53-61. doi:10.1016/j.jkss.2007.07.002 [12] A. Parsian and N. S. Farsipour, “Estimation of the Mean of the Selected Population under Asymmetric Loss Function,” Metrika, Vol. 50, No. 2, 1999, pp. 89-107. [13] U. Singh, P. K. Gupta and S. K. Upadhyay, “Estimation of Exponentiated Weibull Shape Parameters under LINEX Loss Function,” Communication in Statistics-Simulation, Vol. 31, No. 4, 2002, pp. 523-537. doi:10.1081/SAC-120004310 [14] N. Misra and E. V. D. Meulen, “On Estimating the Mean of the Selected Normal Population under the LINEX Loss Function,” Metrika, Vol. 26, No. 9, 2003, pp. 173-184. doi:10.1080/03610929708832041 [15] J. Ahmadi, M. Doostparast and A. Parsian, “Estimation and Prediction in a Two-Parameter Exponential Distribution Based on K-Record Values under LINEX Loss Function,” Communications in Statistics-Theory and Methods, Vol. 34, No. 4, 2005, pp. 795-805. doi:10.1081/STA-200054393 [16] Y. Xiao, Y. Takada and N. Shi, “Minimax Confidence Bound of the Normal Mean under an Asymmetric Loss Function,” Annals of Statistical Mathematics, Vol. 57, No. 1, 2005, pp. 167-182. doi:10.1007/BF02506886 [17] D. C. Singh, G. Prakash and P. Singh, “Shrinkage Testimator for the Shape Parameter of Pareto Distribution Using LINEX Loss Function,” Communication in Statistics-Theory and Methods, Vol. 36, No. 4, 2007, pp. 741753. doi:10.1080/03610920601033694 [18] A. K. Rao and R. S. Srivastava, “Bayesian Estimation of the Scale Parameter of Gamma Distribution under Linex Loss Function with Censoring,” In: B. N. Pandey, Ed., Statistical Techniques in Life-Testing, Reliability, Sampling Theory and Quality Control, Narosa Pub House, Daryaganj, 2002, pp. 287-295. [19] J. R. Thompson, “Some Shrunken Techniques for Estimateing the Mean,” Journal of the American Statistical Association, Vol. 63, No. 321, 1968, pp. 113-122. doi:10.2307/2283832 [20] V. B. Waikar, F. J. Schuurmann and T. E. Raghunathan, “On a Two Stage Shrunken Testimator of the Mean of a Normal Distribution,” Communications in Statistics-Theory and Methods, Vol. 13, No. 15, 1984, pp. 1901-1913. doi:10.1080/03610928408828802 [21] S. R. Adke, V. B. Waikar and F. J. Schuurmann, “A Two Stage Shrinkage Testimator for the Mean of an Exponential Distribution,” Communication in Statistics-Theory and Methods, Vol. 16, No. 6, 1987, pp. 1821-1834. doi:10.1080/03610928708829474 [22] B. N., Pandey, H. J. Malik and R. Srivastava, “Shrinkage Testimator for the Variance of a Normal Distribution at Single and Double Stages,” Microelectron Reliability, Vol. 28, No. 6, 1988, pp. 929-944. doi:10.1016/0026-2714(88)90294-6
[1] A. Parsian and S. N. U. A. Kirmani, “Estimation under LINEX loss function,” In: A. Ulla, Ed., Handbook of Applied Econometrics and Statistical Inference, 165 Dekker, New York, 2002, pp. 53-76. [2] H. Pazira and A. Shadrokh, “Comparison of LINEX and Precautionary Bayes Estimators on the Gamma Distribution Using Censored Data,” Journal of Statistics and Management Systems, Vol. 14, No. 3, 2011, pp. 617-638. doi:10.1080/09720510.2011.10701575 [3] T. S. Ferguson, “Mathematical Statistics: A Decision Theoretic Approach,” Academic Press, New York, 1967. [4] A. Zellner and M.S. Geisel, “Sensitivity of Control to Uncertainty and Form of the Criterion Function,” In the future of statistics, Ed Donald G. Watts, Academic Press, New York, 1968, pp. 269-289. [5] J. Aitchison and I. R. Dunsmore, “Statistical Prediction Analysis,” Cambridge University Press, Cambridge, 1975. doi:10.1017/CBO9780511569647 [6] H. R. Varian, “A Bayesian Approach to Real Estate Assessment,” In: L. J. Savage, S. E. Feinberg and A. Zellner, Eds., Studies in Bayesian Econometrics and Statistics: In Honor of L. J. Savage, North-Holland Pub. Co., Amsterdam, 1975, pp. 195-208. [7] J. O. Berger, “Statistical Decision Theory-Foundation Concepts and Methods,” Springer-Verlag, New York, 1980. doi:10.1007/978-1-4757-1727-3 [8] A. P. Basu and N. Ebrahimi, “Bayesian Approach to Life Testing and Reliability Estimation Using Asymmetric Loss Function,” Journal of Statistical Planning and Inferences, Vol. 29, No. 1-2, 1991, pp. 21-31. doi:10.1016/0378-3758(92)90118-C [9] B. N. Pandey, “Testimator of the Scale Parameter of the Exponential Distribution Using LINEX Loss Function,” Communication in Statistics-Theory and Methods, Vol. 26, No. 6, 1997, pp. 2191-2202. doi:10.1016/0026-2714(88)90294-6 [10] A. A. Soliman, “Comparison of Linex and QuadraticBayes Estimators Foe the Rayleigh Distribution,” Communication in Statistics-Theory and Methods, Vol. 29, No. 1, 2000, pp. 95-107. doi:10.1080/03610920008832471 [11] G. Prakash and D. C. Singh, “Shrinkage Estimation in Exponential Type-II Censored Data under LINEX Loss,” Journal of the Korean Statistical Society, Vol. 37, No. 1, 2008, pp. 53-61. doi:10.1016/j.jkss.2007.07.002 [12] A. Parsian and N. S. Farsipour, “Estimation of the Mean of the Selected Population under Asymmetric Loss Function,” Metrika, Vol. 50, No. 2, 1999, pp. 89-107. [13] U. Singh, P. K. Gupta and S. K. Upadhyay, “Estimation of Exponentiated Weibull Shape Parameters under LINEX Loss Function,” Communication in Statistics-Simulation, Vol. 31, No. 4, 2002, pp. 523-537. doi:10.1081/SAC-120004310 [14] N. Misra and E. V. D. Meulen, “On Estimating the Mean of the Selected Normal Population under the LINEX Loss Function,” Metrika, Vol. 26, No. 9, 2003, pp. 173-184. doi:10.1080/03610929708832041 [15] J. Ahmadi, M. Doostparast and A. Parsian, “Estimation and Prediction in a Two-Parameter Exponential Distribution Based on K-Record Values under LINEX Loss Function,” Communications in Statistics-Theory and Methods, Vol. 34, No. 4, 2005, pp. 795-805. doi:10.1081/STA-200054393 [16] Y. Xiao, Y. Takada and N. Shi, “Minimax Confidence Bound of the Normal Mean under an Asymmetric Loss Function,” Annals of Statistical Mathematics, Vol. 57, No. 1, 2005, pp. 167-182. doi:10.1007/BF02506886 [17] D. C. Singh, G. Prakash and P. Singh, “Shrinkage Testimator for the Shape Parameter of Pareto Distribution Using LINEX Loss Function,” Communication in Statistics-Theory and Methods, Vol. 36, No. 4, 2007, pp. 741753. doi:10.1080/03610920601033694 [18] A. K. Rao and R. S. Srivastava, “Bayesian Estimation of the Scale Parameter of Gamma Distribution under Linex Loss Function with Censoring,” In: B. N. Pandey, Ed., Statistical Techniques in Life-Testing, Reliability, Sampling Theory and Quality Control, Narosa Pub House, Daryaganj, 2002, pp. 287-295. [19] J. R. Thompson, “Some Shrunken Techniques for Estimateing the Mean,” Journal of the American Statistical Association, Vol. 63, No. 321, 1968, pp. 113-122. doi:10.2307/2283832 [20] V. B. Waikar, F. J. Schuurmann and T. E. Raghunathan, “On a Two Stage Shrunken Testimator of the Mean of a Normal Distribution,” Communications in Statistics-Theory and Methods, Vol. 13, No. 15, 1984, pp. 1901-1913. doi:10.1080/03610928408828802 [21] S. R. Adke, V. B. Waikar and F. J. Schuurmann, “A Two Stage Shrinkage Testimator for the Mean of an Exponential Distribution,” Communication in Statistics-Theory and Methods, Vol. 16, No. 6, 1987, pp. 1821-1834. doi:10.1080/03610928708829474 [22] B. N., Pandey, H. J. Malik and R. Srivastava, “Shrinkage Testimator for the Variance of a Normal Distribution at Single and Double Stages,” Microelectron Reliability, Vol. 28, No. 6, 1988, pp. 929-944. doi:10.1016/0026-2714(88)90294-6