In this paper, we propose a new estimation method based on a weighted linear regression framework to obtain some estimators for unknown parameters in a two-parameter Rayleigh distribution under a progressive Type-II censoring scheme. We also provide unbiased estimators of the location parameter and scale parameter which have a nuisance parameter, and an estimator based on a pivotal quantity which does not depend on the other parameter. The proposed weighted least square estimator (WLSE) of the location parameter is not dependent on the scale parameter. In addition, the WLSE of the scale parameter is not dependent on the location parameter. The results are compared with the maximum likelihood method and pivot-based estimation method. The assessments and comparisons are done using Monte Carlo simulations and real data analysis. The simulation results show that the estimators
In statistics, engineering, economics, and medical research, censoring is a condition in which the value of a measurement or observation is only partially known. In life testing experiments or reliability, the observations are often censored. Type-II censoring is the most famous censoring schemes. Type-II censoring occurs if an experiment has a set number of subjects or items and stops the experiment when a predetermined number are observed to have failed; the remaining subjects are then right-censored. But it is not possible to withdraw live units during the experiment. We apply a generalization of the Type-II censoring scheme, where it is possible to withdraw live units during the experiment, and it is well known for progressive Type-II censoring schemes. Progressive Type-II censoring has been suggest in the field of lifetime experiments. Its allowance for the removal of live units from the experiments at various stages is an attractive feature due to its potential to save cost and time for the experimenter.
One of the most popular distributions is the Weibull distribution in lifetime experiments. The Rayleigh distribution can be obtained when the shape parameter of the Weibull distribution is 2. The Rayleigh distribution was introduced by Rayleigh (1880). The Rayleigh distribution can be used to model the lifetime of an object or a service time. The Rayleigh distribution is related to several other distributions such as the chi-squared distribution, extreme value distribution, exponential distribution, normal distribution, Rice distributions and Weibull distribution.
The following are the probability density function (pdf)
Rayleigh (1880) introduced the Rayleigh distribution, which many studies have since investigated. Johnson
Observations are often censored in experiments for testing lifetime or reliability. Type-I and Type-II censoring are well-known censoring schemes, but live units cannot be withdrawn during the experiment in these schemes. We apply a generalization of the Type-II censoring scheme where it is possible to withdraw live units during the experiment, which is well known for progressive Type-II censoring schemes. This scheme can be explained as follows. Suppose there are
We propose a new approach based on the weighted linear regression framework of unknown parameters in the two-parameter Rayleigh distribution with pdf
This section provides the MLEs and pivot-based estimators of the location parameter
Let
where
where
Based on the pivotal quantity, we apply the method proposed by Seo and Kang (2016, 2017) to the two-parameter Rayleigh distribution to obtain the following estimator.
Let
Then,
which are independent and identically distributed from the standard exponential distribution (see Viveros and Balakrishnan, 1994). Wang (2009) found the pivotal quantity which has a
The pivotal quantity
Note that
Since
where
Therefore, we have
This completes the proof.
Note that the estimator
The result is denoted as
We obtain an estimator of the location parameter
Because
which are order statistics from the uniform (0, 1) distribution with sample size
Then, from
which has a
The MLE
Lu and Tao (2007) proposed estimation methods based on a linear regression-type framework. We extend the idea to the progressive Type-II censoring scheme and propose new estimators based on a weighted least-square method under the progressive Type-II censoring scheme in this section.
From
This leads to the following linear regression:
where
where
The following theorem provides new estimators of
Consider the weight
In terms of
The proof is completed by solving these equations simultaneously.
Let
Then, we have
This indicates that the numerator of the estimator
where
This completes the proof.
The estimator
This section examines the validity of the proposed method through Monte Carlo simulations and real data analysis.
To assess and compare the discussed estimators in the Sections 2 and 3, we report their mean squared errors (MSEs) and biases in Tables 1 and 2. The progressive Type-II censored samples are generated from a two-parameter Rayleigh distribution with
As mentioned, the estimators
Tables 1 and 2 show that, the estimator
We consider a real strength dataset that was originally reported by Badar and Priest (1982). The dataset includes the strength measured in GPA for single carbon fibers and impregnated 1000-carbon fiber tows. Dey
Table 4 shows that
This paper proposed a new estimation method based on a weighted linear regression framework to estimate the unknown parameters of a two-parameter Rayleigh distribution under the progressive Type-II censoring scheme. The proposed method was compared with the maximum likelihood method and an estimation method based on the pivotal quantity. In addition, we provided an unbiased estimator of the location parameter, which has a nuisance parameter as a scale parameter but is admissible, unlike the other estimators considered. The simulation results showed that the unbiased estimator-based
The weighted least square estimator (WLSE) of the location parameter is not dependent on the scale parameter. In addition, the WLSE of the scale parameter is also not dependent on the location parameter (Theorem 4). The WLSE of the scale parameter is always greater than 0. Therefore the range of the scale parameter is always satisfied (Theorem 5).
The mean squared errors (biases) of the estimators of
Scheme | ||||||
---|---|---|---|---|---|---|
20 | 18 | I | 0.02188 (0.11612) | 0.01550 (−0.00551) | 0.02277 (0.12684) | 0.03788 (0.16587) |
II | 0.02173 (0.11571) | 0.01515 (−0.00532) | 0.02282 (0.12669) | 0.03843 (0.16726) | ||
III | 0.02204 (0.11626) | 0.01533 (−0.00541) | 0.02284 (0.12685) | 0.03818 (0.16655) | ||
IV | 0.02173 (0.11578) | 0.01537 (−0.00550) | 0.02263 (0.12630) | 0.03820 (0.16662) | ||
14 | I | 0.02471 (0.12162) | 0.01706 (−0.00779) | 0.02352 (0.12936) | 0.03913 (0.16915) | |
II | 0.02373 (0.11928) | 0.01581 (−0.00697) | 0.02341 (0.12861) | 0.03954 (0.17015) | ||
III | 0.02478 (0.12148) | 0.01649 (−0.00739) | 0.02355 (0.12918) | 0.03916 (0.16925) | ||
IV | 0.02429 (0.12040) | 0.01658 (−0.00774) | 0.02298 (0.12780) | 0.03889 (0.16812) | ||
30 | 26 | I | 0.01318 (0.09163) | 0.00993 (−0.00327) | 0.01513 (0.10294) | 0.02493 (0.13394) |
II | 0.01305 (0.09126) | 0.00968 (−0.00314) | 0.01517 (0.10290) | 0.02541 (0.13527) | ||
III | 0.01336 (0.09189) | 0.00981 (−0.00321) | 0.01516 (0.10297) | 0.02512 (0.13454) | ||
IV | 0.01309 (0.09125) | 0.00984 (−0.00328) | 0.01496 (0.10236) | 0.02513 (0.13421) | ||
18 | I | 0.01466 (0.09518) | 0.01053 (−0.00456) | 0.01536 (0.10424) | 0.02544 (0.13620) | |
II | 0.01408 (0.09346) | 0.00969 (−0.00393) | 0.01539 (0.10391) | 0.02620 (0.13869) | ||
III | 0.01480 (0.09518) | 0.01012 (−0.00425) | 0.01545 (0.10427) | 0.02563 (0.13698) | ||
IV | 0.01412 (0.09366) | 0.01020 (−0.00454) | 0.01499 (0.10256) | 0.02530 (0.13526) | ||
40 | 36 | I | 0.00958 (0.07857) | 0.00720 (−0.00055) | 0.01152 (0.08997) | 0.01850 (0.11451) |
II | 0.00942 (0.07815) | 0.00708 (−0.00050) | 0.01150 (0.08991) | 0.01876 (0.11540) | ||
III | 0.00936 (0.07800) | 0.00714 (−0.00053) | 0.01151 (0.08994) | 0.01861 (0.11498) | ||
IV | 0.00939 (0.07809) | 0.00716 (−0.00056) | 0.01144 (0.08959) | 0.01867 (0.11502) | ||
28 | I | 0.00951 (0.07808) | 0.00732 (−0.00257) | 0.01162 (0.08943) | 0.01876 (0.11604) | |
II | 0.00939 (0.07744) | 0.00708 (−0.00233) | 0.01152 (0.08961) | 0.01925 (0.11794) | ||
III | 0.00965 (0.07805) | 0.00714 (−0.00245) | 0.01167 (0.08954) | 0.01893 (0.11675) | ||
IV | 0.00932 (0.07739) | 0.00719 (−0.00257) | 0.01161 (0.08846) | 0.01880 (0.11608) | ||
16 | I | 0.01157 (0.08391) | 0.00837 (−0.00489) | 0.01174 (0.09027) | 0.01963 (0.11965) | |
II | 0.01069 (0.08131) | 0.00718 (−0.00388) | 0.01160 (0.08994) | 0.02033 (0.12252) | ||
III | 0.01168 (0.08383) | 0.00786 (−0.00440) | 0.01172 (0.09046) | 0.01983 (0.12053) | ||
IV | 0.01091 (0.08189) | 0.00793 (−0.00483) | 0.01175 (0.08814) | 0.01912 (0.11739) |
The mean squared errors (biases) of the estimators of
Scheme | ||||||
---|---|---|---|---|---|---|
20 | 18 | I | 0.46591 (0.39721) | 0.16682 (0.04204) | 0.28832 (0.32407) | 0.35769 (−0.05326) |
II | 0.40350 (0.36992) | 0.15326 (0.03805) | 0.26142 (0.29992) | 0.30091 (−0.07389) | ||
III | 0.44619 (0.38687) | 0.16022 (0.04013) | 0.27668 (0.31295) | 0.33033 (−0.05876) | ||
IV | 0.44897 (0.38742) | 0.16065 (0.03965) | 0.27532 (0.31256) | 0.30707 (−0.07153) | ||
14 | I | 1.11865 (0.56512) | 0.30318 (0.06844) | 0.53712 (0.42310) | 0.76992 (−0.07398) | |
II | 0.67353 (0.44613) | 0.22639 (0.05125) | 0.37798 (0.33128) | 0.45654 (−0.08379) | ||
III | 0.91483 (0.51607) | 0.26586 (0.06048) | 0.48177 (0.38259) | 0.62273 (−0.06572) | ||
IV | 0.90011 (0.51375) | 0.26546 (0.05788) | 0.45264 (0.37803) | 0.49526 (−0.06645) | ||
30 | 26 | I | 0.21751 (0.28219) | 0.10272 (0.02802) | 0.17030 (0.25442) | 0.20453 (−0.03156) |
II | 0.18717 (0.25844) | 0.09252 (0.02451) | 0.14841 (0.22962) | 0.17011 (−0.05319) | ||
III | 0.20705 (0.27286) | 0.09777 (0.02635) | 0.16004 (0.24284) | 0.18812 (−0.03635) | ||
IV | 0.20501 (0.27276) | 0.09841 (0.02598) | 0.15904 (0.24317) | 0.17455 (−0.05241) | ||
18 | I | 0.52673 (0.42172) | 0.18040 (0.04666) | 0.32749 (0.34999) | 0.46649 (−0.05621) | |
II | 0.30468 (0.31619) | 0.12977 (0.03214) | 0.21060 (0.25289) | 0.26352 (−0.07190) | ||
III | 0.42688 (0.37807) | 0.15587 (0.03995) | 0.27173 (0.30692) | 0.37518 (−0.04926) | ||
IV | 0.42156 (0.37678) | 0.15728 (0.03786) | 0.26358 (0.30484) | 0.28694 (−0.05830) | ||
40 | 36 | I | 0.12106 (0.21567) | 0.06645 (0.02061) | 0.10970 (0.20719) | 0.12364 (−0.02572) |
II | 0.10917 (0.20138) | 0.06162 (0.01864) | 0.09862 (0.19164) | 0.11005 (−0.04145) | ||
III | 0.11407 (0.20761) | 0.06410 (0.01967) | 0.10425 (0.19973) | 0.11697 (−0.02993) | ||
IV | 0.11516 (0.20943) | 0.06452 (0.01952) | 0.10472 (0.20058) | 0.11211 (−0.04146) | ||
28 | I | 0.19497 (0.26893) | 0.09593 (0.02673) | 0.16612 (0.25335) | 0.20679 (−0.02887) | |
II | 0.14259 (0.22120) | 0.07663 (0.02011) | 0.12286 (0.20122) | 0.14451 (−0.05070) | ||
III | 0.17279 (0.24851) | 0.08657 (0.02363) | 0.14572 (0.22955) | 0.17911 (−0.02910) | ||
IV | 0.17107 (0.25031) | 0.08804 (0.02289) | 0.14529 (0.23122) | 0.15269 (−0.04846) | ||
6 | I | 0.76254 (0.49684) | 0.22886 (0.05820) | 0.42622 (0.39723) | 0.68288 (−0.08647) | |
II | 0.29968 (0.30785) | 0.13433 (0.03255) | 0.20786 (0.23311) | 0.28973 (−0.07801) | ||
III | 0.54592 (0.42113) | 0.18293 (0.04656) | 0.32679 (0.32876) | 0.50467 (−0.06786) | ||
IV | 0.50642 (0.41166) | 0.17992 (0.04134) | 0.29847 (0.31805) | 0.32605 (−0.04793) |
Observed progressive Type-II censored sample
0.562 | 0.564 | 0.729 | 0.802 | 0.950 | 1.053 | 1.111 | 1.115 | 1.194 | 1.208 |
1.247 | 1.256 | 1.271 | 1.277 | 1.348 | 1.390 | 1.429 | 1.474 | 1.503 | 1.520 |
1.524 | 1.551 | 1.551 | 1.609 | 1.632 |
Estimates of
0.455 | 0.378 | 0.471 | 0.562 | 0.407 | 0.337 | 0.403 | 0.495 |