where is as derived in Equation (9).

3.3. Bayesian Estimation Based on Non-Informative Prior

The non-informative prior distribution of the parameter is obtained from Equation

(3) and it is given by. Then, we obtain the Bayesian estimates of

in this case as follows:

1) Simple Random Sample:

(11)

and

(12)

2) Maximum ranked set sampling with unequal samples:

(13)

and

(14)

4. Simulation Results

To illustrate the performance of the derived Bayesian estimates of scale parameter of the Weibull distribution with informative and non-informative prior based on SRS and MRSSU, we carry out the Monte Carlo simulations using R-Software version 3.1.1. We compute bias, mean squared error and relative efficiency of the estimators by assuming the shape parameter is known. The numerical results obtained for fixed values of, [and 1] and sample size m [3, 4 and 5] for 1000 runs. The bias of the Bayesian estimates based on SRS and MRSSU are presented in Table 1 and Table 2, and MSE of the Bayesian estimates based on SRS and MRSSU is presented in Table 3 and Table 4.

Table 1. Bias of the Bayesian estimates based on SRS and MRSSU. For (when, ,).

Table 2. Bias of the Bayesian estimates based on SRS and MRSSU. For (when, ,).

The relative efficiency of the Bayesian estimates based on maximum ranked set sampling with unequal samples with respect to simple random sampling can be defined as follows

And are presented in Table 5.

Table 3. MSE of the Bayesian estimates based on SRS and MRSSU. For (when, ,).

Table 4. MSE of the Bayesian estimates based on SRS and MRSSU. For (when, ,).

Table 5. Relative efficiency when and.

5. Conclusions

We present Bayesian estimation based on SRS and MRSSU. The Weibull distribution is used as an application example to illustrate our results. We compute bias, MSE and relative efficiency of the derived Bayesian estimates and then make a comparison between SRS and MRSSU. Our observations of the results are stated in the following points:

1) From Table 1 and Table 2, first, we found that the Bayesian estimates of are all biased. Next, we found that the Bayesian estimates based on Jeffreys prior are less biased than gamma prior. Also, we observed that the Bayesian estimates based on MRSSU are considerably less biased than SRS.

2) From Table 3 and Table 4, it is observed that the mean squared error of all estimates decreases when sample size m increases. Next, we observed that the Bayesian estimates based on MRSSU have a much smaller mean squared error than the corresponding Bayesian estimates based on SRS in all cases considered.

3) From Table 5, we observe that the relative efficiency of the Bayesian estimator based on MRSSU w.r.t. SRS Bayesian estimators are greater than 1 and increases with m. Also, decreases in Linex function as m increases for.

Therefore, we conclude that the Bayesian estimates based on maximum ranked set sampling with unequal samples are more efficient than the corresponding Bayesian estimates of simple random sampling.

Finally, we conclude that the results of the simulation experiment showed that the Bayesian estimates based on maximum ranked set sampling with unequal samples are more efficient, when compared with the Bayesian estimates of simple random sampling.

Acknowledgements

The authors would like to thank the referees for their helpful comments that have led to an improved paper.

Cite this paper

Biradar, B. and Shivanna, B. (2016) Weibull-Bayesian Estimation Based on Maximum Ranked Set Sampling with Unequal Samples.*Open Journal of Statistics*, **6**, 1028-1036. doi: 10.4236/ojs.2016.66083.

Biradar, B. and Shivanna, B. (2016) Weibull-Bayesian Estimation Based on Maximum Ranked Set Sampling with Unequal Samples.

References

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https://doi.org/10.1071/AR9520385

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https://doi.org/10.1007/BF02911622

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https://doi.org/10.2307/2556166

[4] Shaibu, A.B. and Muttlak, H.A. (2004) Estimating the Parameters of the Normal, Exponential and Gamma Distributions Using Median and Extreme Ranked Set Samples. Statistics, 1, 75-98.

[5] Al-Omari, A.I., Jaber, K.H. and Al-Omari, A. (2008) Modified Ratio-Type Estimators of the Mean Using Extreme Ranked Set Sampling. Journal of Mathematics and Statistics, 4, 150-155.

https://doi.org/10.3844/jmssp.2008.150.155

[6] Islam, T., Shaibur, M.R. and Hossain, S.S. (2009) Effectivity of Modified Maximum Likelihood Estimators Using Selected Ranked Set Sampling Data. Austrian Journal of Statistics, 38, 109-120.

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https://doi.org/10.1080/01621459.1986.10478289

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[12] Sadek, A., Sultan, K.S. and Balakrishnan, N. (2009) Bayesian Estimation Based on Ranked Set Sampling Using Asymmetric Loss Function. Bulletin of the Malaysian Mathematical Sciences Society, 38, 707-718.

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http://dx.doi.org/10.14419/ijasp.v2i2.3373

[14] Al-Hadhrami, S.A. and Al-Omari, A.I. (2009) Bayesian Inference on the Variance of Normal Distribution Using Moving Extremes Ranked Set Sampling. Journal of Modern Applied Statistical Methods, 8, 273-281.

[15] Hassan, A.S. (2013) Maximum Likelihood and Bayes Estimators of the Unknown Parameters for Exponentiated Exponential Distribution Using Ranked Set Sampling. International Journal of Engineering Research and Applications, 3, 720-725.

[16] Mohammadi, M.Y. and Pazira, H. (2010) Classical and Bayesian Estimations on the Generalized Exponential Distribution Using Censored Data. International Journal of Mathematical Analysis, 4, 1417-1431.

[17] Ghafoori, S., Habibi Rad, A. and Doostparast, M. (2011) Bayesian Two-Sample Prediction with Type-II Censored Data for Some Lifetime Models. JIRSS, 10, 63-86.

[18] Al-Hadhrami, S.A. and Al-Omari, A.I. (2012) Bayes Estimation of the Mean of Normal Distribution Using Moving Extreme Ranked Set Sampling. Pakistan Journal of Statistics and Operation Research, VIII, 21-30.

[19] Mohie El-Din, M.M., Kotb, M.S. and Newer, H.A. (2015) Bayesian Estimation and Prediction for Pareto Distribution Based on Ranked Set Sampling. Journal of Statistics Applcations and Probability, 4, 211-221.

[20] Biradar, B.S. and Sanotsha, C.D. (2014) Estimation of the Mean of the Exponential Distribution Using Maximum Ranked Set Sampling with Unequal Samples. Open journal of Statistics, Scientific Research, 4, 641-649.

[21] Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York.

https://doi.org/10.1007/978-1-4757-4286-2

[1] McIntyre, G.A. (1952) A Method for Unbiased Selective Sampling Using Ranked Sets. Australian Journal of Agricultural Research, 3, 385-390.

https://doi.org/10.1071/AR9520385

[2] 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.

https://doi.org/10.1007/BF02911622

[3] Dell, T.R. and Clutter, J.L. (1972) Ranked Set Sampling Theory with Order Statistics Background. Biometrics, 28, 545-555.

https://doi.org/10.2307/2556166

[4] Shaibu, A.B. and Muttlak, H.A. (2004) Estimating the Parameters of the Normal, Exponential and Gamma Distributions Using Median and Extreme Ranked Set Samples. Statistics, 1, 75-98.

[5] Al-Omari, A.I., Jaber, K.H. and Al-Omari, A. (2008) Modified Ratio-Type Estimators of the Mean Using Extreme Ranked Set Sampling. Journal of Mathematics and Statistics, 4, 150-155.

https://doi.org/10.3844/jmssp.2008.150.155

[6] Islam, T., Shaibur, M.R. and Hossain, S.S. (2009) Effectivity of Modified Maximum Likelihood Estimators Using Selected Ranked Set Sampling Data. Austrian Journal of Statistics, 38, 109-120.

[7] Ibrahim, K. and Syam, M. (2010) Estimating the Population Mean Using Stratified Median Ranked Set Sampling. Applied Mathematical Sciences, 4, 2341-2354.

[8] Varian, H.R. (1975) A Bayesian Approach to Real Estate Assessment. North Holland, Amsterdam, 195-208.

[9] Zellner, A. (1986) Bayesian Estimation and Prediction Using Asymmetric Loss Functions. Journal of the American Statistical Association, 81, 446-451.

https://doi.org/10.1080/01621459.1986.10478289

[10] Al-Saleh, M.F. and Muttlak, H.A. (1998) A Note in Bayesian Estimation Using Ranked Set Sampling. Pakistan Journal of Statistics, 14, 49-56.

[11] Ahmed (2007) Bayesian Estimation of the Logormal Distrbution Mean Using Ranked SET Sampling. Basrah Journal of Science, 25, 101-112.

[12] Sadek, A., Sultan, K.S. and Balakrishnan, N. (2009) Bayesian Estimation Based on Ranked Set Sampling Using Asymmetric Loss Function. Bulletin of the Malaysian Mathematical Sciences Society, 38, 707-718.

[13] Sadek, A. and Alharbi, F. (2014) Weibull-Bayesian Analysis Based on Ranked Set Sampling. International Journal of Advanced Statistics and Probability, 2, 114-123.

http://dx.doi.org/10.14419/ijasp.v2i2.3373

[14] Al-Hadhrami, S.A. and Al-Omari, A.I. (2009) Bayesian Inference on the Variance of Normal Distribution Using Moving Extremes Ranked Set Sampling. Journal of Modern Applied Statistical Methods, 8, 273-281.

[15] Hassan, A.S. (2013) Maximum Likelihood and Bayes Estimators of the Unknown Parameters for Exponentiated Exponential Distribution Using Ranked Set Sampling. International Journal of Engineering Research and Applications, 3, 720-725.

[16] Mohammadi, M.Y. and Pazira, H. (2010) Classical and Bayesian Estimations on the Generalized Exponential Distribution Using Censored Data. International Journal of Mathematical Analysis, 4, 1417-1431.

[17] Ghafoori, S., Habibi Rad, A. and Doostparast, M. (2011) Bayesian Two-Sample Prediction with Type-II Censored Data for Some Lifetime Models. JIRSS, 10, 63-86.

[18] Al-Hadhrami, S.A. and Al-Omari, A.I. (2012) Bayes Estimation of the Mean of Normal Distribution Using Moving Extreme Ranked Set Sampling. Pakistan Journal of Statistics and Operation Research, VIII, 21-30.

[19] Mohie El-Din, M.M., Kotb, M.S. and Newer, H.A. (2015) Bayesian Estimation and Prediction for Pareto Distribution Based on Ranked Set Sampling. Journal of Statistics Applcations and Probability, 4, 211-221.

[20] Biradar, B.S. and Sanotsha, C.D. (2014) Estimation of the Mean of the Exponential Distribution Using Maximum Ranked Set Sampling with Unequal Samples. Open journal of Statistics, Scientific Research, 4, 641-649.

[21] Berger, J.O. (1985) Statistical Decision Theory and Bayesian Analysis. Springer-Verlag, New York.

https://doi.org/10.1007/978-1-4757-4286-2