e/5-1240792x67.png" /> (10)

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.
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