The Scientific World Journal

Volume 2013 (2013), Article ID 281624, 8 pages

http://dx.doi.org/10.1155/2013/281624

Research Article

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

Jong-Wuu Wu,¹ Sheau-Chiann Chen,¹ Wen-Chuan Lee,² and Heng-Yi Lai¹

¹Department of Applied Mathematics, National Chiayi University, Chiayi, Taiwan

²Department of International Business, Chang Jung Christian University, Tainan 71101, Taiwan

Received 26 April 2013; Accepted 6 June 2013

Academic Editors: C. P. Bhunu, S. Hristova, and L. Jódar

Copyright © 2013 Jong-Wuu Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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.

1. 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 , . If , then the c.d.f. of can be written as . 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 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 . 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 [4–9]. 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 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.

2. The Weighted Moments Estimators and the Best Linear Unbiased Estimator

Suppose that the experimenter fails to observe the first , the last , and the middle observations. Observations are the available multiply type II censored sample from (1). To eliminate the effect of the location parameter, we subtract from each statistic, and a new available multiply type II censored sample can be written a . First, by using this censored data, we propose the following WMEs to estimate the scale parameter :

We apply (2) to (5) to compute their weights by minimizing their MSEs , , or equivalently

where , , and , . By using the results of Lieblein [14] and , taking as example, , , , , and are obtained as follows:

where

is Euler’s constant and [14].

Finally, by differentiating (6) with respect to and , we obtain

where , and are as above definitions, . Hence, by applying the result in (9) to (6), a minimum value of can be obtained as follows:

where , and are as above definitions, [12].

Based on (2) to (5), the estimators of location parameter for WMEs are computed by using moments method, and a weight is assigned in each equation from (11) to (14) as well. Thus, the estimators of location for WMEs are shown as follows:

where , , , and as (2) to (5), respectively,

Similarly, the weights are computed by minimizing MSEs of and the optimal weights in MSEs of are obtained as

The minimum value of MSEs of can then be written as

where , , , , and can be obtained based on (8) for .

For the case of , 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 of , can be obtained as

where and , . Thus, the WMEs of the location parameter for the case of can also be provided as follows:

where, , , , and , , , as (18). The MSEs of are obtained by using for .

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. BLUE and are obtained as follows:

where

and are as above definition.

Further, the MSEs of BLUE and 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:

3. 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 for sample size , , and . The study results are summarized in Tables 1 to 4.

Table 1: MSEs of BLUE and WMEs of the scale parameter at , , and .

Censoring scheme type II

The simulated MSEs of BLUE and exact MSEs of WMEs

BLUE
WME1
WME2
WME3
WME4
WME/BLUE

Multiply
2
5
2
2
0.070396
0.067245
0.066049*
0.066182
0.067211
0.938249

3
5
2
1
0.074345
0.069957
0.069488*
0.069732
0.069821
0.934669

3
5
1
2
0.081776
0.077573
0.075709*
0.075784
0.077564
0.925810

2
5
3
1
0.064955
0.061733
0.061551
0.061859
0.061543*
0.947471

2
5
1
3
0.077774
0.074938
0.072323*
0.072350
0.074945
0.929912

1
5
3
2
0.060664
0.058404
0.057795*
0.057900
0.058413
0.952707

1
5
2
3
0.066040
0.064033
0.062353
0.062348*
0.064107
0.944094

4
5
1
1
0.085554
0.079741
0.078920*
0.079045
0.079696
0.922458

1
3
7
1
0.059028
0.056334
0.057378
0.057511
0.056253*
0.952984

1
5
1
4
0.073128
0.071480
0.068433
0.068391*
0.071521
0.935223

Middle
0
3
6
0
0.045595
0.044507*
0.044635
0.044507*
0.044635
0.976138

BLUE
WME5
WME6
WME7
WME8
WME/BLUE

Right
0
0
0
8
0.101814
0.132434
0.102793
0.132434
0.084350*
0.828472

Left
8
3
0
0
0.145212
0.127135*
0.127135*
0.349345
0.349345
0.875513

Doubly
7
3
0
2
0.169211
0.155749
0.145217*
0.477876
0.300771
0.858200

The smallest MSE among the proposed WMEs and BLUE.

For comparison results of scale estimators for , Tables 1 and 2 indicate that (i) WME2 is the best estimator for , , and all different censoring schemes except , , , ; (ii) WME3 is the best estimator for , , and all different censoring schemes except ,  , ; (iii) WME8 is the best estimator for and right censoring scheme; (iv) WME5(=WME6) is the best estimator for and left censoring scheme; (v) WME6 is the best estimator for 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 , and its range is between 0.821384 and 0.996789 for and . It means that our smallest MSE of WMEs is smaller than that of BLUE. The range of is between 0.794971 and 0.987798 for and . It means that our smallest MSE of WMEs is smaller than that of BLUE.

Table 2: MSEs of BLUE and WMEs of the scale parameter at , , and .

Censoring scheme type II

The simulated MSEs of BLUE and exact MSEs of WMEs

BLUE
WME1
WME2
WME3
WME4
WME/BLUE

Multiply
2
5
2
2
0.033049
0.033973
0.032609
0.032605*
0.033950
0.986562

3
5
2
1
0.034131
0.034709
0.034021*
0.034042
0.034667
0.996789

3
5
1
2
0.036422
0.037066
0.035446*
0.035476
0.036991
0.973205

2
5
3
1
0.038106
0.031840
0.031332
0.031300*
0.031862
0.821384

2
5
1
3
0.034793
0.036526
0.034272
0.034250*
0.036474
0.984403

1
5
3
2
0.030754
0.031121
0.030042
0.029959*
0.031178
0.974163

1
5
2
3
0.031932
0.033203
0.031209
0.031211*
0.033194
0.977406

4
5
1
1
0.041570
0.037365
0.036608
0.036605*
0.037358
0.880563

1
3
7
1
0.030205
0.029124
0.028682*
0.028727
0.029052
0.949578

1
5
1
4
0.033436
0.035586
0.032923
0.032814*
0.035537
0.981404

Middle
0
3
6
0
0.026008
0.025694*
0.025757
0.025694*
0.025757
0.987927

BLUE
WME5
WME6
WME7
WME8
WME/BLUE

Right
0
0
0
8
0.038026
0.058844
0.046303
0.058844
0.037562*
0.987798

Left
8
3
0
0
0.060523
0.048114*
0.048114*
0.069671
0.069671
0.794971

Doubly
7
3
0
2
0.056175
0.053227
0.049471*
0.09685
0.064317
0.880659

The smallest MSE among the proposed WMEs and BLUE.

For comparison results of location estimators for , Tables 3 and 4 show that (i) WME1 is the best estimator for , , , , , and middle censoring scheme; (ii) WME1 is the best estimator for and all different censoring schemes except right censoring scheme; (iii) WME8 is the best estimator for and right censoring scheme; (iv) WME5 is the best estimator for and left and doubly censoring schemes. The magnitude of MSE decreases as sample size increases. The range of is between 0.861855 and 1.048036 for and . The range of is between 0.843873 and 1.02405 for and .

Table 3: MSEs of BLUE and WMEs of the location parameter at , , and .

Censoring scheme type II

The simulated MSEs of BLUE and exact MSEs of WMEs

BLUE
WME1
WME2
WME3
WME4
WME/BLUE

Multiply
2
5
2
2
0.081471*
0.084486
0.088129
0.087748
0.089356
1.037007

3
5
2
1
0.087399*
0.093767
0.097813
0.091590
0.098849
1.047952

3
5
1
2
0.087310*
0.093792
0.094834
0.091504
0.096280
1.048036

2
5
3
1
0.081571*
0.084650
0.090575
0.085397
0.091483
1.037746

2
5
1
3
0.081668*
0.084633
0.086168
0.090419
0.087180
1.036306

1
5
3
2
0.078327
0.076445*
0.083378
0.085178
0.083145
0.975973

1
5
2
3
0.078619
0.076260*
0.081964
0.089104
0.081176
0.969995

4
5
1
1
0.097020*
0.105156
0.107617
0.100723
0.107746
1.038167

1
3
7
1
0.083615
0.076417*
0.089073
0.089757
0.086154
0.913925

1
5
1
4
0.079463
0.076634*
0.081216
0.093448
0.079339
0.964399

Middle
0
3
6
0
0.075444
0.071020*
0.079400
0.084294
0.079400
0.941360

BLUE
WME5
WME6
WME7
WME8
WME/BLUE

Right
0
0
0
8
0.095452
0.105177
0.094909
0.105177
0.094913*
0.99431

Left
8
3
0
0
0.228809*
0.234311
0.234311
0.463530
0.463530
1.02405

Doubly
7
3
0
2
0.182548*
0.184806
0.186032
0.310332
0.268930
1.01237

The smallest MSE among the proposed WMEs and BLUE.

Table 4: MSEs of BLUE and WMEs of the location parameter at , , and .

Censoring scheme type II

The simulated MSEs of BLUE and exact MSEs of WMEs

BLUE
WME1
WME2
WME3
WME4
WME/BLUE

Multiply
2
5
2
2
0.047455
0.043392*
0.047048
0.046523
0.045351
0.914382

3
5
2
1
0.047910
0.046756*
0.049381
0.047447
0.048703
0.975913

3
5
1
2
0.048516
0.046415*
0.048644
0.047332
0.047459
0.956695

2
5
3
1
0.048501
0.043917*
0.047599
0.046681
0.046598
0.905486

2
5
1
3
0.046475
0.043009*
0.046365
0.046338
0.044072
0.925422

1
5
3
2
0.046091
0.040401*
0.045429
0.046284
0.043109
0.876549

1
5
2
3
0.045630
0.039876*
0.045084
0.045986
0.041846
0.873899

4
5
1
1
0.051705
0.050038
0.051508
0.048829*
0.051088
0.944386

1
3
7
1
0.047423
0.041238*
0.047731
0.050815
0.045522
0.861855

1
5
1
4
0.045836
0.039504*
0.044303
0.045540
0.040560
0.861855

Middle
0
3
6
0
0.043759
0.039707*
0.043286
0.052436
0.043286
0.907402

BLUE
WME5
WME6
WME7
WME8
WME/BLUE

Right
0
0
0
8
0.047960*
0.048812
0.049585
0.048812
0.04907
1.017765

Left
8
3
0
0
0.078955
0.066628*
0.066628*
0.076515
0.076515
0.843873

Doubly
7
3
0
2
0.062959
0.060392*
0.061361
0.068265
0.064084
0.959227

The smallest MSE among the proposed WMEs and BLUE.

4. 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 , our proposed WMEs are, in general, superior to BLUE and different censoring schemes except , , , , , , left and doubly censoring schemes; for 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 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 and . Moreover, by replacing by , the corresponding distribution of is also called extreme value distribution with c.d.f. as given by

where , . 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.

Acknowledgments

The authors thank the referees for their suggestions and helpful comments in revising the paper. This research was partially supported by the National Science Council, Taiwan (Plans nos. NSC 100-2221-E-309-002, NSC 101-2118-M-415-001, and NSC 101-2221-E-309-004).

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