Estimation of the Mean of the Exponential Distribution Using Maximum Ranked Set Sampling with Unequal Samples
Affiliation(s)
1 Department of Studies in Statistics, University of Mysore, Mysore, India. 2 All India Institute of Speech and Hearing, Mysore, India.
1 Department of Studies in Statistics, University of Mysore, Mysore, India. 2 All India Institute of Speech and Hearing, Mysore, India.
ABSTRACT
In this paper maximum ranked set sampling procedure with unequal samples (MRSSU) is proposed. Maximum likelihood estimator and modified maximum likelihood estimator are obtained and their properties are studied under exponential distribution. These methods are studied under both perfect and imperfect ranking (with errors in ranking). These estimators are then compared with estimators based on simple random sampling (SRS) and ranked set sampling (RSS) procedures. It is shown that relative efficiencies of the estimators based on MRSSU are better than those of the estimator based on SRS. Simulation results show that efficiency of proposed estimator is better than estimator based on RSS under ranking error.
In this paper maximum ranked set sampling procedure with unequal samples (MRSSU) is proposed. Maximum likelihood estimator and modified maximum likelihood estimator are obtained and their properties are studied under exponential distribution. These methods are studied under both perfect and imperfect ranking (with errors in ranking). These estimators are then compared with estimators based on simple random sampling (SRS) and ranked set sampling (RSS) procedures. It is shown that relative efficiencies of the estimators based on MRSSU are better than those of the estimator based on SRS. Simulation results show that efficiency of proposed estimator is better than estimator based on RSS under ranking error.
KEYWORDS
Efficiency, Error in Ranking, Maximum Likelihood Estimator, Modified Maximum Likelihood Estimator, Ranked Set Sampling
Efficiency, Error in Ranking, Maximum Likelihood Estimator, Modified Maximum Likelihood Estimator, Ranked Set Sampling
Cite this paper
Biradar, B. and Santosha, C. (2014) Estimation of the Mean of the Exponential Distribution Using Maximum Ranked Set Sampling with Unequal Samples. Open Journal of Statistics, 4, 641-649. doi: 10.4236/ojs.2014.48060.
Biradar, B. and Santosha, C. (2014) Estimation of the Mean of the Exponential Distribution Using Maximum Ranked Set Sampling with Unequal Samples. Open Journal of Statistics, 4, 641-649. doi: 10.4236/ojs.2014.48060.
References
[1] McIntyre, G.A. (1952) A Method for Unbiased Selective Sampling Using Ranked Sets. Australian Journal of Agricultural Research, 3, 385-390.
http://dx.doi.org/10.1071/AR9520385 [2] Dell, T.R. and Clutter, J.L. (1972) Ranked Set Sampling Theory with Order Statistics Background. Biometrics, 28, 545-555.
http://dx.doi.org/10.2307/2556166 [3] Takahasi, K. and Wakimoto, K. (1968) On Unbiased Estimates of the Population Mean Based on the Sample Stratified by Means of Ordering. Annals of the Institute of Statistical Mathematics, 20, 1-31.
http://dx.doi.org/10.1007/BF02911622 [4] Lam, K., Sinha, B.K. and Wu, Z. (1994) Estimation of Parameters in Two-Parameter Exponential Distribution Using Ranked Set Sampling. Annals of the Institute of Statistical Mathematics, 46, 723-736.
http://dx.doi.org/10.1007/BF00773478 [5] Stokes, L. (1995) Parametric Ranked Set Sampling. Annals of the Institute of Statistical Mathematics, 47, 465-482. [6] Samwi, H., Ahmad, M. and Abu-Dayyeh, W. (1996) Estimating the Population Mean Using Extreme Ranked Set Sampling. Biometrical Journal, 38, 577-586.
http://dx.doi.org/10.1002/bimj.4710380506 [7] Al-Odat, M.T. and Al-Saleh, M.F. (2001) A Variation of Ranked Set Sampling. Journal of Applied Statistical Science, 10, 137-146. [8] Al-Saleh, M.F. and Al-Hadrami, S. (2003a) Parametric Estimation for the Location Parameter for Symmetric Distributions Using Moving Extremes Ranked Set Sampling with Application to Trees Data. Environmetrics, 14, 651-664.
http://dx.doi.org/10.1002/env.610 [9] Abu-Dayyeh, W. and Al Sawi, E. (2009) Modified Inference about the Mean of the Exponential Distribution Using Moving Extreme Ranked Set Sampling. Statistical Papers, 50, 249-259.
http://dx.doi.org/10.1007/s00362-007-0072-5 [10] Mehrotra, K.G. and Nanda, P. (1974) Unbiased Estimator of Parameter by Order Statistics in the Case of Censored Samples. Biometrika, 61, 601-606.
http://dx.doi.org/10.1093/biomet/61.3.601 [11] Zheng, G. and Al-Saleh, M.F. (2002) Modified Maximum Likelihood Estimators Based on RSS. Annals of the Institute of Statistical Mathematics, 54, 641-658.
http://dx.doi.org/10.1023/A:1022475413950 [12] Al-Saleh, M.F. and Al-Hadrami, S. (2003b) Estimation of the Mean of the Exponential Distribution Using Moving Extremes Ranked Set Sampling. Statistical Papers, 44, 367-382.
http://dx.doi.org/10.1007/s00362-003-0161-z [13] Chen, W.X., Xie, M.Y. and Wu, M. (2013) Parametric Estimation for the Scale Parameter for Scale Distributions Using Moving Extremes Ranked Set Sampling. Statistics and Probability Letters, 83, 2060-2066.
http://dx.doi.org/10.1016/j.spl.2013.05.015 [14] Lehmann, E.L. (1983) Theory of Point Estimation. John Willey and Sons Inc., New York.
http://dx.doi.org/10.1007/978-1-4757-2769-2 [15] David, H.A. and Levine, D.N. (1972) Ranked Set Sampling in the Presence of Judgment Ranking Error. Biometrics, 28, 553-555.
[1] McIntyre, G.A. (1952) A Method for Unbiased Selective Sampling Using Ranked Sets. Australian Journal of Agricultural Research, 3, 385-390.
http://dx.doi.org/10.1071/AR9520385 [2] Dell, T.R. and Clutter, J.L. (1972) Ranked Set Sampling Theory with Order Statistics Background. Biometrics, 28, 545-555.
http://dx.doi.org/10.2307/2556166 [3] Takahasi, K. and Wakimoto, K. (1968) On Unbiased Estimates of the Population Mean Based on the Sample Stratified by Means of Ordering. Annals of the Institute of Statistical Mathematics, 20, 1-31.
http://dx.doi.org/10.1007/BF02911622 [4] Lam, K., Sinha, B.K. and Wu, Z. (1994) Estimation of Parameters in Two-Parameter Exponential Distribution Using Ranked Set Sampling. Annals of the Institute of Statistical Mathematics, 46, 723-736.
http://dx.doi.org/10.1007/BF00773478 [5] Stokes, L. (1995) Parametric Ranked Set Sampling. Annals of the Institute of Statistical Mathematics, 47, 465-482. [6] Samwi, H., Ahmad, M. and Abu-Dayyeh, W. (1996) Estimating the Population Mean Using Extreme Ranked Set Sampling. Biometrical Journal, 38, 577-586.
http://dx.doi.org/10.1002/bimj.4710380506 [7] Al-Odat, M.T. and Al-Saleh, M.F. (2001) A Variation of Ranked Set Sampling. Journal of Applied Statistical Science, 10, 137-146. [8] Al-Saleh, M.F. and Al-Hadrami, S. (2003a) Parametric Estimation for the Location Parameter for Symmetric Distributions Using Moving Extremes Ranked Set Sampling with Application to Trees Data. Environmetrics, 14, 651-664.
http://dx.doi.org/10.1002/env.610 [9] Abu-Dayyeh, W. and Al Sawi, E. (2009) Modified Inference about the Mean of the Exponential Distribution Using Moving Extreme Ranked Set Sampling. Statistical Papers, 50, 249-259.
http://dx.doi.org/10.1007/s00362-007-0072-5 [10] Mehrotra, K.G. and Nanda, P. (1974) Unbiased Estimator of Parameter by Order Statistics in the Case of Censored Samples. Biometrika, 61, 601-606.
http://dx.doi.org/10.1093/biomet/61.3.601 [11] Zheng, G. and Al-Saleh, M.F. (2002) Modified Maximum Likelihood Estimators Based on RSS. Annals of the Institute of Statistical Mathematics, 54, 641-658.
http://dx.doi.org/10.1023/A:1022475413950 [12] Al-Saleh, M.F. and Al-Hadrami, S. (2003b) Estimation of the Mean of the Exponential Distribution Using Moving Extremes Ranked Set Sampling. Statistical Papers, 44, 367-382.
http://dx.doi.org/10.1007/s00362-003-0161-z [13] Chen, W.X., Xie, M.Y. and Wu, M. (2013) Parametric Estimation for the Scale Parameter for Scale Distributions Using Moving Extremes Ranked Set Sampling. Statistics and Probability Letters, 83, 2060-2066.
http://dx.doi.org/10.1016/j.spl.2013.05.015 [14] Lehmann, E.L. (1983) Theory of Point Estimation. John Willey and Sons Inc., New York.
http://dx.doi.org/10.1007/978-1-4757-2769-2 [15] David, H.A. and Levine, D.N. (1972) Ranked Set Sampling in the Presence of Judgment Ranking Error. Biometrics, 28, 553-555.