Weighted Moments Estimators of the Parameters for the Extreme Value Distribution Based on the Multiply Type II Censored Sample

We propose the weighted moments estimators (WMEs) of the location and scale parameters for the extreme value distribution based on the multiply type II censored sample. Simulated mean squared errors (MSEs) of best linear unbiased estimator (BLUE) and exact MSEs of WMEs are compared to study the behavior of different estimation methods. The results show the best estimator among the WMEs and BLUE under different combinations of censoring schemes.


Introduction
The lifetime of a special component has an extreme value distribution with the location parameter and scale parameter if its cumulative distribution function (c.d.f.) is given by where −∞ < < ∞, > 0. If ( − )/ = , then the c.d.f. of can be written as ( ) = exp[− − ], −∞ < < ∞. The distribution has been applied in the fields of life testing and reliability [1,2]. Many researches in estimators of location and scale parameters for the extreme value distribution have been discussed. Balakrishnan and Chan [3] presented tables of the means, variances, and covariances of order statistics from extreme value distribution based on the complete sample. In life testing experiments, the experimenter might not always be in a position to observe the lifetimes of all the items put on test due to time limitations and/or other restrictions (such as dearth of bankroll or material resources). Therefore, multiplying type II censored sample may arise in practice. Let us suppose, for instance, that out of items put on life test, the first lifetimes (1) < (2) < ⋅ ⋅ ⋅ < ( ) have only been observed and the lifetimes for the rest ( − ) components remain unobserved or missing. This type of censoring is known as right type II censoring. Another way to get censored data is to observe the largest lifetimes ( − +1) < ( − +2) < ⋅ ⋅ ⋅ < ( ) . The lifetimes of first ( − ) components are missing, called left type II censoring scheme. Moreover, if the left and right censoring situations arise together, this is known as doubly type II censoring scheme [4][5][6][7][8][9]. In addition, a reverse situation to doubly type II censoring is midcensoring where the data on two extremes are available, but some middle observations are censored [10]. However, these censoring schemes are special cases of multiply type II censoring scheme. Furthermore, if the left and right censoring cases arise together, this is known as doubly type II censoring scheme. In addition, a reverse situation to doubly type II censoring is middle censoring where the data on two extremes are available, but some middle observations are censored. However, these censoring schemes are special cases of multiply type II censoring scheme [11,12].
In this paper, we propose the WMEs of the location parameter and the scale parameter from extreme value distribution based on the multiply type II censored sample. The WMEs are derived in Section 2. We also extend the methods of Balakrishnan and Cohen [13] and Balakrishnan and Chan [4] to find BLUEs of the location and scale parameters 2 The Scientific World Journal under multiply type II censored sample. To study the behavior of different estimation methods, simulated MSEs of BLUE and exact MSEs of WMEs are compared and discussed in Section 3. We obtain the best estimator of and among the WMEs and BLUE, respectively. Finally, conclusions are given in Section 4.
For the case of = 0, the WMEs of the scale parameter are proposed as (18) since it is reasonable with consecutive data. Consider the following: In the same way, the minimum values of MSE of̂, = 5, 6, 7, 8 can be obtained as where = ( )/ and = Var( )/ 2 , = 5, 6, 7, 8. Thus, the WMEŝof the location parameter for the case of = 0 can also be provided as follows: Next, we extend the method of Balakrishnan and Cohen [13] and Balakrishnan and Chan [4] to derive the BLUE of the location and scale parameters in an explicit form under a multiply type II censored sample for the general case. BLUÊ 9 and̂9 are obtained as follows:  ] , = ( ( ) − ) = ( ( ) ) , ( ( ) ) and Cov( ( ) , ( ) ) are as above definition.
Further, the MSEs of BLUÊ9 and̂9 are given by where 1, and are as above definition. In addition, we find the best estimators based on the smallest MSEs among the WMEs and BLUE, respectively, as follows:

The Comparative Numerical Results
To study the behavior of our proposed WMEs and BLUE, the simulated MSEs of BLUE and exact MSEs of WMEs are compared. The numerical results were obtained to cover different censoring schemes assigning appropriate values to , , , and s for sample size = 15, 25, = 1, and = 1. The study results are summarized in Tables 1 to 4. (v) WME6 is the best estimator for = 15, 25 and left censoring scheme. The magnitude of MSE decreases as sample size increases. The value of the smallest MSE divided by BLUE is denoted as WME/BLUE, and its range is between 0.821384 and 0.996789 for = 15, 25 and ̸ = 0. It means that our smallest MSE of WMEs is smaller than that of BLUE. The range of WME/BLUE is between 0.794971 and 0.987798 for = 15, 25 and = 0. It means that our smallest MSE of WMEs is smaller than that of BLUE.

Conclusion
The object of this paper is to propose WMEs of location parameter and scale parameter of the extreme value distribution based on the multiply type II censored sample. The comparison results of scale estimators find that our proposed WMEs are, in general, superior to BLUE under different combinations of censoring schemes and sample size. On the other hand, the comparison results of location estimators present that for = 15, our proposed WMEs are, in general, superior to BLUE and different censoring schemes except ( , , , ) = (2, 5, 2, 2), (3, 5, 2, 1), (3, 5, 1, 2), (2, 5, 3, 1), (2, 5, 1, 3), (4, 5, 1, 1), left and doubly censoring 6 The Scientific World Journal  schemes; for = 25 our proposed WMEs are, in general, superior to BLUE and all different censoring schemes except right censoring scheme.
WMEs and BLUE can further be used to predict the th ( − < ≤ ) ordered observation ( ) in a sample of size from the extreme value distribution based on multiply type II censored sample. Moreover, if observations = exp(− ) are from a Weibull distribution with a scale parameter and a shape parameter , then the best estimators of Weibull distribution can be obtained through proposing the best estimators of the extreme value distribution, where = exp(− ) and = 1/ . Moreover, by replacing by − , the corresponding distribution of − is also called extreme value distribution with c.d.f. as given by where −∞ < = − < ∞, > 0. Our method and the best linear unbiased method can be used to find the best estimator of and and predict the th ( − < ≤ ) ordered observation ( ) for extreme value distribution with c.d.f. as (25) based on negative transformation and the multiply type II censored sample.
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