^{1}

^{2}

A modification of ranked set sampling (RSS) called maximum ranked set sampling with unequal sample (MRSSU) is considered for the Bayesian estimation of scale parameter α of the Weibull distribution. Under this method, we use Linex loss function, conjugate and Jeffreys prior distributions to derive the Bayesian estimate of α. In order to measure the efficiency of the obtained Bayesian estimates with respect to the Bayesian estimates of simple random sampling (SRS), we compute the bias, mean squared error (MSE) and asymptotic relative efficiency of the obtained Bayesian estimates using simulation. It is shown that the proposed estimates are found to be more efficient than the corresponding one based on SRS.

In certain practical problems, actual measurements of a variable interest are costly or time-consuming, but the ranking items according to the variable is relatively easy with- out actual measurement. Under such circumstances McIntyre [

Some research works have investigated ranked set sampling from a Bayesian point of view. Varian [

In this paper, we derive the Bayesian estimates of the Weibull scale parameter α based on gamma and Jeffreys prior distributions by MRSSU method proposed by Biradar and Santosha [

Let

And cumulative distribution function (cdf)

where

In order to derive, and to measure the performance of an estimator we use squared error, loss function (SEL) (see, Berger [

The Linex loss function for the parameter

where

where

In this section, we derive the Bayes estimates of the Weibull parameter

Let

Hence, the Bayesian estimation of

While the Bayesian estimate of

where,

Then,

Assume that the variable of interest X has density function

^{th} order statistic (maximum) of an SRS of size i from

Let MRSSU be drawn from Weibull distribution, then the density function of

Then the joint density of MRSSU in this case due to independence of

where

Then the posterior density of α is

The Bayes estimate of

Next, in order to derive the Bayesian estimation of

Now the Bayesian estimation of

where

The non-informative prior distribution of the parameter

(3) and it is given by

in this case as follows:

1) Simple Random Sample:

and

2) Maximum ranked set sampling with unequal samples:

and

To illustrate the performance of the derived Bayesian estimates of scale parameter

Bias( | Bias( | Bias( | Bias( | ||||||
---|---|---|---|---|---|---|---|---|---|

Jeffrey prior | Gamma prior | Jeffrey prior | Gamma prior | ||||||

m | SRS | MRSSU | SRS | MRSSU | c | SRS | MRSSU | SRS | MRSSU |

3 | 0.2487 | 0.1151 | 0.3420 | 0.1811 | 1 | 0.1379 | 0.0750 | 0.2433 | 0.1401 |

−1 | 0.4071 | 0.1689 | 0.5318 | 0.2335 | |||||

4 | 0.1585 | 0.0636 | 0.2355 | 0.1068 | 1 | 0.0914 | 0.0429 | 0.1733 | 0.0853 |

−1 | 0.2454 | 0.0874 | 0.3207 | 0.1314 | |||||

5 | 0.1255 | 0.0424 | 0.1936 | 0.0731 | 1 | 0.0794 | −0.0538 | 0.1481 | 0.0231 |

−1 | 0.1831 | −0.0064 | 0.2536 | 0.0664 |

Bias( | Bias( | Bias( | Bias( | ||||||
---|---|---|---|---|---|---|---|---|---|

Jeffrey prior | Gamma prior | Jeffrey prior | Gamma prior | ||||||

m | SRS | MRSSU | SRS | MRSSU | c | SRS | MRSSU | SRS | MRSSU |

3 | 0.4975 | 0.2302 | 0.4778 | 0.2781 | 1 | 0.1369 | 0.0853 | 0.2209 | 0.1484 |

−1 | 0.8876 | 0.4671 | 1.0272 | 0.4803 | |||||

4 | 0.3171 | 0.1271 | 0.3422 | 0.1700 | 1 | 0.0883 | 0.0493 | 0.1626 | 0.0954 |

−1 | 0.7232 | 0.2321 | 0.6499 | 0.2657 | |||||

5 | 0.2510 | 0.0848 | 0.2939 | 0.1190 | 1 | 0.0864 | −0.1250 | 0.1523 | −0.0088 |

−1 | 0.5139 | 0.0508 | 0.5115 | 0.1397 |

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

MSE( | MSE( | MSE( | MSE( | ||||||
---|---|---|---|---|---|---|---|---|---|

Jeffrey prior | Gamma prior | Jeffrey prior | Gamma prior | ||||||

m | SRS | MRSSU | SRS | MRSSU | c | SRS | MRSSU | SRS | MRSSU |

3 | 0.4193 | 0.1147 | 0.3899 | 0.1357 | 1 | 0.1849 | 0.0800 | 0.2179 | 0.0980 |

−1 | 1.0296 | 0.1870 | 1.2535 | 0.2015 | |||||

4 | 0.2850 | 0.0555 | 0.2470 | 0.0651 | 1 | 0.1389 | 0.0450 | 0.1524 | 0.0528 |

−1 | 0.4709 | 0.0712 | 0.4577 | 0.0823 | |||||

5 | 0.1584 | 0.0304 | 0.1696 | 0.0355 | 1 | 0.1004 | 0.0387 | 0.1170 | 0.0505 |

−1 | 0.2593 | 0.0615 | 0.2823 | 0.0542 |

MSE( | MSE( | MSE( | MSE( | ||||||
---|---|---|---|---|---|---|---|---|---|

Jeffrey prior | Gamma prior | Jeffrey prior | Gamma prior | ||||||

m | SRS | MRSSU | SRS | MRSSU | c | SRS | MRSSU | SRS | MRSSU |

3 | 1.6772 | 0.4586 | 0.8381 | 0.3874 | 1 | 0.4288 | 0.2397 | 0.3299 | 0.2208 |

−1 | 3.9104 | 1.2337 | 3.8436 | 0.8687 | |||||

4 | 1.1400 | 0.2220 | 0.5929 | 0.2103 | 1 | 0.3550 | 0.1507 | 0.2771 | 0.1453 |

−1 | 3.0419 | 0.3914 | 1.7406 | 0.3360 | |||||

5 | 0.6337 | 0.1215 | 0.4605 | 0.1228 | 1 | 0.2852 | 0.1327 | 0.2464 | 0.1168 |

−1 | 1.6544 | 0.3654 | 1.0991 | 0.2718 |

m | c | ||||||||

3 | 3.6570 | 3.6570 | 2.8736 | 2.1630 | 1 | 2.3096 | 1.7889 | 2.2232 | 1.4942 |

−1 | 5.5066 | 3.1696 | 6.2195 | 4.4246 | |||||

4 | 5.1355 | 5.1355 | 3.7963 | 2.8197 | 1 | 3.0854 | 2.3558 | 2.8862 | 1.9076 |

−1 | 6.6168 | 7.7715 | 5.5641 | 5.1806 | |||||

5 | 5.2143 | 5.2143 | 4.7793 | 3.7483 | 1 | 1.7104 | 2.1492 | 2.3176 | 2.1088 |

−1 | 2.3269 | 4.5273 | 5.2070 | 4.0432 |

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

2) From

3) From

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.

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

Biradar, B.S. and Shivanna, B.K. (2016) Weibull-Bayesian Es- timation Based on Maximum Ranked Set Sampling with Unequal Samples. Open Journal of Statistics, 6, 1028-1036. http://dx.doi.org/10.4236/ojs.2016.66083