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

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 𝑌 of a special component has an extreme value distribution with the location parameter 𝜃 and scale parameter 𝜎 if its cumulative distribution function (c.d.f.) is given by 𝐹_𝑌𝑦=exp−𝑒^{−(𝑦−𝜃)/𝜎},−∞<𝑦<∞, (1) where −∞<𝜃<∞, 𝜎>0. If (𝑌−𝜃)/𝜎=𝑋, then the c.d.f. of 𝑋 can be written as 𝐹_𝑋(𝑥)=exp[−𝑒^−𝑥],−∞<𝑥<∞. 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 𝑛 items put on life test, the first 𝑟 lifetimes 𝑌₍₁₎<𝑌₍₂₎<⋯<𝑌_(𝑟) have only been observed and the lifetimes for the rest (𝑁−𝑟) 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 𝑠 lifetimes 𝑌_{(𝑁−𝑠+1)}<𝑌_{(𝑁−𝑠+2)}<⋅⋅⋅<𝑌_(𝑁). The lifetimes of first (𝑁−𝑠) 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 𝜃 and the scale parameter 𝜎 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 𝜃 and 𝜎 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 𝑟, the last 𝑠, and the middle 𝑙 observations. Observations 𝑌_(𝑟+1)<𝑌_(𝑟+2)<⋯<𝑌_(𝑟+𝑘)<𝑌_{(𝑟+𝑘+𝑙+1)}<𝑌_{(𝑟+𝑘+𝑙+2)}<⋯<𝑌_{(𝑁−𝑠)} are the available multiply type II censored sample from (1). To eliminate the effect of the location parameter, we subtract 𝑌_(𝑟+1) from each statistic, and a new available multiply type II censored sample can be written a 0<𝑌_(𝑟+2)−𝑌_(𝑟+1)<⋯<𝑌_(𝑟+𝑘)−𝑌_(𝑟+1)<𝑌_{(𝑟+𝑘+𝑙+1)}−𝑌_(𝑟+1)<𝑌_{(𝑟+𝑘+𝑙+2)}−𝑌_(𝑟+1)<⋯<𝑌_{(𝑁−𝑠)}−𝑌_(𝑟+1). First, by using this censored data, we propose the following WMEs to estimate the scale parameter 𝜎: 𝜎₁=𝑤₁₁^𝑟+𝑘_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)+𝑤₁₂^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)−𝑌_(𝑟+1)=𝑤₁₁𝐶₁₁+𝑤₁₂𝐶₁₂,𝜎₂=𝑤₂₁^𝑟+𝑘_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)+𝑙𝑌_(𝑟+𝑘)−𝑌_(𝑟+1)+𝑤₂₂^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)−𝑌_(𝑟+1)+𝑠𝑌_{(𝑁−𝑠)}−𝑌_(𝑟+1)=𝑤₂₁𝐶₂₁+𝑤₂₂𝐶₂₂,𝜎₃=𝑤₃₁^𝑟+𝑘_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)+𝑤₃₂^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)−𝑌_(𝑟+1)+𝑠𝑌_{(𝑁−𝑠)}−𝑌_(𝑟+1)=𝑤₃₁𝐶₃₁+𝑤₃₂𝐶₃₂,𝜎₄=𝑤₄₁^𝑟+𝑘_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)+𝑙𝑌_(𝑟+𝑘)−𝑌_(𝑟+1)+𝑤₄₂^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)−𝑌_(𝑟+1)=𝑤₄₁𝐶₄₁+𝑤₄₂𝐶₄₂. (2)(3)(4)(5)

We apply (2) to (5) to compute their weights by minimizing their MSEs min_{𝑤𝑖1,𝑤𝑖2}MSE(𝜎_𝑖), 𝑖=1,2,3,4, or equivalently min_{𝑤𝑖1,𝑤𝑖2}𝑤²_𝑖1𝑏_𝑖1+𝑤²_𝑖2𝑏_𝑖2+2𝑤_𝑖1𝑤_𝑖2𝑏_𝑖3+𝑤_𝑖1𝑎_𝑖1+𝑤_𝑖2𝑎_𝑖2−1²,𝑖=1,2,3,4, (6) where 𝑎_𝑖𝑗=𝐸(𝐶_𝑖𝑗)/𝜎, 𝑏_𝑖𝑗=Var(𝐶_𝑖𝑗)/𝜎²,𝑖=1,2,3,4, 𝑗=1,2 and 𝑏_𝑖3=Cov(𝐶_𝑖1,𝐶_𝑖2)/𝜎², 𝑖=1,2,3,4. By using the results of Lieblein [14] and 𝑌_(𝑖)=𝜃+𝜎𝑋_(𝑖), taking 𝑖=1 as example, 𝑎₁₁, 𝑎₁₂, 𝑏₁₁, 𝑏₁₂, and 𝑏₁₃ are obtained as follows: 𝑎₁₁=^𝑟+𝑘_𝑖=𝑟+2𝐸𝑋_(𝑖)−(𝑘−1)𝐸𝑋_(𝑟+1),𝑎₁₂=^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝐸𝑋_(𝑖)−(𝑁−𝑠−𝑟−𝑘−𝑙)𝐸𝑋_(𝑟+1),𝑏₁₁=^𝑟+𝑘_𝑖=𝑟+2Var𝑋_(𝑖)+2^{𝑟+𝑘−1}_𝑖=𝑟+2^𝑟+𝑘_𝑗=𝑖+1Cov𝑋_(𝑖),𝑋_(𝑗)+(𝑘−1)²Var𝑋_(𝑟+1)−2(𝑘−1)^𝑟+𝑘_𝑖=𝑟+2Cov𝑋_(𝑖),𝑋_(𝑟+1),𝑏₁₂=^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}Var𝑋_(𝑖)+2^{𝑁−𝑠−1}_{𝑖=𝑟+𝑘+𝑙+1}^𝑁−𝑠_𝑗=𝑖+1Cov𝑋_(𝑖),𝑋_(𝑗)+(𝑁−𝑠−𝑟−𝑘−𝑙)²Var𝑋_(𝑟+1)−2(𝑁−𝑠−𝑟−𝑘−𝑙)^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}Cov𝑋_(𝑖),𝑋_(𝑟+1),𝑏₁₃=^𝑟+𝑘_𝑖=𝑟+2^𝑁−𝑠_{𝑗=𝑟+𝑘+𝑙+1}Cov𝑋_(𝑖),𝑋_(𝑗)−(𝑁−𝑠−𝑟−𝑘−𝑙)^𝑟+𝑘_𝑖=𝑟+2Cov𝑋_(𝑖),𝑋_(𝑟+1)−(𝑘−1)^𝑁−𝑠_{𝑗=𝑟+𝑘+𝑙+1}Cov𝑋_(𝑟+1),𝑋_(𝑗)+(𝑘−1)(𝑁−𝑠−𝑟−𝑘−𝑙)Var𝑋_(𝑟+1), (7) where 𝐸𝑋_(𝑖)=𝑁𝐶^𝑁−1_𝑖−1^{𝑁−𝑖+1}_𝑝=1(−1)^𝑝−1×𝐶^𝑁−𝑖_𝑝−11,𝑋𝑖+𝑝−1𝛾+log𝑖+𝑝−1Cov_(𝑖),𝑋_(𝑗)=𝑖𝑗𝐶^𝑁_𝑗𝐶_𝑖^{𝑗−1𝑗−𝑖}_𝑝=1^{𝑁−𝑗+1}_𝑞=1(−1)^{𝑝+𝑞−2}×𝐶^{𝑗−𝑖−1}_𝑝−1𝐶^𝑁−𝑗_𝑞−1𝜙−𝑖+𝑝−1,𝑗−𝑖−𝑝+𝑞𝑁𝐶^𝑁−1_𝑖−1^{𝑁−𝑖+1}_𝑝=1(−1)^𝑝−1𝐶^𝑁−𝑖_𝑝−1×1×𝑖+𝑝−1𝛾+log𝑖+𝑝−1𝑁𝐶^𝑁−1_𝑗−1^{𝑁−𝑗+1}_𝑞=1(−1)^𝑞−1𝐶^𝑁−𝑗_𝑞−1×1,𝑗+𝑞−1𝛾+log𝑗+𝑞−1 (8)

𝛾=−Γ^′(1) is Euler’s constant and 𝜙(𝑡,𝑢)=∫^∞_−∞∫^𝑦_−∞𝑥𝑦𝑒^{−𝑥−𝑡𝑒−𝑥}𝑒^{−𝑦−𝑢𝑒−𝑦}𝑑𝑥𝑑𝑦 [14].

Finally, by differentiating (6) with respect to 𝑤_𝑖1 and 𝑤_𝑖2, we obtain 𝑤_𝑖1=𝑏_𝑖2𝑎_𝑖1−𝑏_𝑖3𝑎_𝑖2𝑏_𝑖1+𝑎²_𝑖1𝑏_𝑖2+𝑎²_𝑖2−𝑏_𝑖3+𝑎_𝑖1𝑎_𝑖2²,𝑤_𝑖2=𝑏_𝑖1𝑎_𝑖2−𝑏_𝑖3𝑎_𝑖1𝑏_𝑖1+𝑎²_𝑖1𝑏_𝑖2+𝑎²_𝑖2−𝑏_𝑖3+𝑎_𝑖1𝑎_𝑖2², (9) where 𝑎_𝑖𝑗,𝑏_𝑖𝑗, 𝑗=1,2 and 𝑏_𝑖3 are as above definitions, 𝑖=1,2,3,4. Hence, by applying the result in (9) to (6), a minimum value of MSE(𝜎_𝑖) can be obtained as follows: MSE𝜎_𝑖=𝑏_𝑖1𝑏_𝑖2−𝑏²_𝑖3𝑏_𝑖1+𝑎²_𝑖1𝑏_𝑖2+𝑎²_𝑖2−𝑏_𝑖3+𝑎_𝑖1𝑎_𝑖2²,𝑖=1,2,3,4, (10) where 𝑎_𝑖𝑗,𝑏_𝑖𝑗, 𝑗=1,2 and 𝑏_𝑖3 are as above definitions, 𝑖=1,2,3,4 [12].

Based on (2) to (5), the estimators of location parameter 𝜃 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: 𝜃₁=𝑤₁∑^𝑟+𝑘_𝑖=𝑟+1𝑌_(𝑖)𝑘−𝐸₁₁𝑘𝜎₁+1−𝑤₁×∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)−𝐸𝑁−𝑠−𝑟−𝑘−𝑙¹²𝜎𝑁−𝑠−𝑟−𝑘−𝑙¹=𝜃+𝜎𝑤₁𝐹₁₁+1−𝑤₁𝐹₁₂,𝜃₂=𝑤₂∑^𝑟+𝑘_𝑖=𝑟+1𝑌_(𝑖)+𝑙𝑌_(𝑟+𝑘)−𝐸𝑘+𝑙²¹𝜎𝑘+𝑙²+1−𝑤₂×∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)+𝑠𝑌_{(𝑁−𝑠)}−𝐸𝑁−𝑟−𝑘−𝑙²²𝜎𝑁−𝑟−𝑘−𝑙²=𝜃+𝜎𝑤₂𝐹₂₁+1−𝑤₂𝐹₂₂,𝜃₃=𝑤₃∑^𝑟+𝑘_𝑖=𝑟+1𝑌_(𝑖)𝑘−𝐸₁₁𝑘𝜎₃+1−𝑤₃×∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)+𝑠𝑌_{(𝑁−𝑠)}−𝐸𝑁−𝑟−𝑘−𝑙²²𝜎𝑁−𝑟−𝑘−𝑙³=𝜃+𝜎𝑤₃₃𝐹₃₁+1−𝑤₃𝐹₃₂,𝜃₄=𝑤₄∑^𝑟+𝑘_𝑖=𝑟+1𝑌_(𝑖)+𝑙𝑌_(𝑟+𝑘)−𝐸𝑘+𝑙²¹𝜎𝑘+𝑙⁴+1−𝑤₄×∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑌_(𝑖)−𝐸𝑁−𝑠−𝑟−𝑘−𝑙¹²𝜎𝑁−𝑠−𝑟−𝑘−𝑙⁴=𝜃+𝜎𝑤₄𝐹₄₁+1−𝑤₄𝐹₄₂, (11)(12)(13)(14) where 𝜎₁, 𝜎₂, 𝜎₃, and 𝜎₄ as (2) to (5), respectively, 𝐸₁₁=𝐸^𝑟+𝑘_𝑖=𝑟+1𝑋_(𝑖),𝐸₁₂=𝐸^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑋_(𝑖),𝐸₂₁=𝐸^𝑟+𝑘_𝑖=𝑟+1𝑋_(𝑖)+𝑙𝑋_(𝑟+𝑘),𝐸₂₂=𝐸^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑋_(𝑖)+𝑠𝑋_{(𝑁−𝑠)},𝐹₁₁=∑^𝑟+𝑘_𝑖=𝑟+1𝑋_(𝑖)𝑘−𝐸₁₁𝑘𝜎₁𝜎,𝐹₁₂=∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑋_(𝑖)−𝐸𝑁−𝑠−𝑟−𝑘−𝑙¹²𝜎𝑁−𝑠−𝑟−𝑘−𝑙𝜎¹,𝐹₂₁=∑^𝑟+𝑘_𝑖=𝑟+1𝑋_(𝑖)+𝑙𝑋_(𝑟+𝑘)−𝐸𝑘+𝑙²¹𝜎𝑘+𝑙𝜎²,𝐹₂₂=∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑋_(𝑖)+𝑠𝑋_{(𝑁−𝑠)}−𝐸𝑁−𝑟−𝑘−𝑙²²𝜎𝑁−𝑟−𝑘−𝑙𝜎²,𝐹₃₁=∑^𝑟+𝑘_𝑖=𝑟+1𝑋_(𝑖)𝑘−𝐸₁₁𝑘𝜎₃𝜎,𝐹₃₂=∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑋_(𝑖)+𝑠𝑋_{(𝑁−𝑠)}−𝐸𝑁−𝑟−𝑘−𝑙²²𝜎𝑁−𝑟−𝑘−𝑙𝜎³,𝐹₄₁=∑^𝑟+𝑘_𝑖=𝑟+1𝑋_(𝑖)+𝑙𝑋_(𝑟+𝑘)−𝐸𝑘+𝑙²¹𝜎𝑘+𝑙𝜎⁴,𝐹₄₂=∑^𝑁−𝑠_{𝑖=𝑟+𝑘+𝑙+1}𝑋_(𝑖)−𝐸𝑁−𝑠−𝑟−𝑘−𝑙¹²𝜎𝑁−𝑠−𝑟−𝑘−𝑙𝜎⁴. (15) Similarly, the weights are computed by minimizing MSEs of 𝜃_𝑖 and the optimal weights in MSEs of 𝜃_𝑖 are obtained as 𝑤_𝑖=Var𝐹_𝑖2−Cov𝐹_𝑖1,𝐹_𝑖2−𝐸𝐹_𝑖1𝐸𝐹_𝑖2+𝐸𝐹_𝑖2²×Var𝐹_𝑖1+Var𝐹_𝑖2−2Cov𝐹_𝑖1,𝐹_𝑖2+𝐸𝐹_𝑖1²−2𝐸𝐹_𝑖1𝐸𝐹_𝑖2+𝐸𝐹_𝑖2²⁻¹. (16)

The minimum value of MSEs of 𝜃_𝑖 can then be written as MSE𝜃_𝑖=𝑤²_𝑖Var𝐹_𝑖1+Var𝐹_𝑖2−2Cov𝐹_𝑖1,𝐹_𝑖2+𝐸(𝐹_𝑖1)²−2𝐸𝐹_𝑖1𝐸𝐹_𝑖2+𝐸𝐹_𝑖2²−2𝑤_𝑖Var𝐹_𝑖2−Cov𝐹_𝑖1,𝐹_𝑖2−𝐸𝐹_𝑖1𝐸𝐹_𝑖2+𝐸𝐹_𝑖2²+Var𝐹_𝑖2+𝐸𝐹_𝑖2², (17) where 𝐸(𝐹_𝑖1), 𝐸(𝐹_𝑖2), Var(𝐹_𝑖1), Var(𝐹_𝑖2), and Cov(𝐹_𝑖1,𝐹_𝑖2) can be obtained based on (8) for 𝑖=1,2,3,4.

For the case of 𝑙=0, the WMEs of the scale parameter 𝜎 are proposed as (18) since it is reasonable with consecutive data. Consider the following: 𝜎₅=𝑤₅^𝑁−𝑠_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)=𝑤₅𝐶₅,𝜎₆=𝑤₆^𝑁−𝑠_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)+𝑠𝑌_{(𝑁−𝑠)}−𝑌_(𝑟+1)=𝑤₆𝐶₆,𝜎₇=𝑤₇𝑟𝑌_(𝑟+2)−𝑌_(𝑟+1)+^𝑁−𝑠_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)=𝑤₇𝐶₇,𝜎₈=𝑤₈𝑟𝑌_(𝑟+2)−𝑌_(𝑟+1)+^𝑁−𝑠_𝑖=𝑟+2𝑌_(𝑖)−𝑌_(𝑟+1)+𝑠𝑌_{(𝑁−𝑠)}−𝑌_(𝑟+1)=𝑤₈𝐶₈. (18)

In the same way, the minimum values of MSE of 𝜎_𝑖, 𝑖=5,6,7,8 can be obtained as MSE𝜎_𝑖=𝑎²_𝑖𝑏_𝑖+𝑎⁴_𝑖𝑏_𝑖+𝑎²_𝑖²−2𝑎²_𝑖𝑏_𝑖+𝑎²_𝑖+1, (19) where 𝑎_𝑖=𝐸(𝐶_𝑖)/𝜎 and 𝑏_𝑖=Var(𝐶_𝑖)/𝜎², 𝑖=5,6,7,8. Thus, the WMEs 𝜃_𝑖 of the location parameter 𝜃 for the case of 𝑙=0 can also be provided as follows: 𝜃₅=∑^𝑁−𝑠_𝑖=𝑟+1𝑌_(𝑖)−𝐸𝑁−𝑠−𝑟⁵𝜎𝑁−𝑠−𝑟⁵,𝜃₆=∑^𝑁−𝑠_𝑖=𝑟+1𝑌_(𝑖)+𝑠𝑌_{(𝑁−𝑠)}−𝐸𝑁−𝑟⁶𝜎𝑁−𝑟⁶,𝜃₇=𝑟𝑌_(𝑟+1)+∑^𝑁−𝑠_𝑖=𝑟+1𝑌_(𝑖)−𝐸𝑁−𝑠⁷𝜎𝑁−𝑠⁷,𝜃₈=𝑟𝑌_(𝑟+1)+∑^𝑁−𝑠_𝑖=𝑟+1𝑌_(𝑖)+𝑠𝑌_{(𝑁−𝑠)}𝑁−𝐸₈𝑁𝜎₈, (20) where𝐸₅=𝐸[∑^𝑁−𝑠_𝑖=𝑟+1𝑋_(𝑖)], 𝐸₆=𝐸[∑^𝑁−𝑠_𝑖=𝑟+1𝑋_(𝑖)+𝑠𝑋_{(𝑁−𝑠)}], 𝐸₇=𝐸[𝑟𝑋_(𝑟+1)+∑^𝑁−𝑠_𝑖=𝑟+1𝑋_(𝑖)], 𝐸₈=𝐸[𝑟𝑋_(𝑟+1)+∑^𝑁−𝑠_𝑖=𝑟+1𝑋_(𝑖)+𝑠𝑋_{(𝑁−𝑠)}], and 𝜎₅, 𝜎₆, 𝜎₇, 𝜎₈ as (18). The MSEs of 𝜃_𝑖 are obtained by using MSE(𝜃_𝑖)=Var(𝜃_𝑖)+[𝐸(𝜃_𝑖)−𝜃]² for 𝑖=5,6,7,8.

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 𝜃₉ and 𝜎₉ are obtained as follows: 𝜃₉=𝛼^𝑇𝛽⁻¹𝛼1^𝑇𝛽⁻¹−𝛼^𝑇𝛽⁻¹1𝛼^𝑇𝛽⁻¹𝛼^𝑇𝛽⁻¹𝛼1^𝑇𝛽⁻¹1−𝛼^𝑇𝛽⁻¹1²𝜎𝑌,₉=1^𝑇𝛽⁻¹1𝛼^𝑇𝛽⁻¹−1^𝑇𝛽⁻¹𝛼1^𝑇𝛽⁻¹𝛼^𝑇𝛽⁻¹𝛼1^𝑇𝛽⁻¹1−𝛼^𝑇𝛽⁻¹1²𝑌, (21) where1=(1,1,…,1,1,…,1)^𝑇_{(𝑁−𝑠−𝑟−𝑙)×1}𝑌=𝑌_(𝑟+1),𝑌_(𝑟+2),…,𝑌_(𝑟+𝑘),𝑌_{(𝑟+𝑘+𝑙+1)},𝑌_{(𝑟+𝑘+𝑙+2)},…,𝑌_{(𝑁−𝑠)}^𝑇𝛼=𝛼_(𝑟+1),𝛼_(𝑟+2),…,𝛼_(𝑟+𝑘),𝛼_{(𝑟+𝑘+𝑙+1)},𝛼_{(𝑟+𝑘+𝑙+2)},…,𝛼_{(𝑁−𝑠)},𝛽𝛽=^{(𝑟+1,𝑟+1)}𝛽_{(𝑟+1,𝑟+2)}⋯𝛽_{(𝑟+1,𝑟+𝑘)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑟+1,𝑁−𝑠)}𝛽_{(𝑟+1,𝑟+2)}𝛽_{(𝑟+2,𝑟+2)}⋯𝛽_{(𝑟+2,𝑟+𝑘)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑟+2,𝑁−𝑠)}𝛽⋮⋮⋱⋮⋮⋱⋮_{(𝑟+1,𝑟+𝑘)}𝛽_{(𝑟+2,𝑟+𝑘)}⋯𝛽_{(𝑟+𝑘,𝑟+𝑘)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑟+𝑘,𝑁−𝑠)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}𝛽_{(𝑟+2,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑟+𝑘,𝑟+𝑘+𝑙+1)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑟+𝑘+𝑙+1,𝑁−𝑠)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+2)}𝛽_{(𝑟+2,𝑟+𝑘+𝑙+2)}⋯𝛽_{(𝑟+𝑘,𝑟+𝑘+𝑙+2)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑟+𝑘+𝑙+2,𝑁−𝑠)}𝛽⋮⋮⋱⋮⋮⋱⋮_{(𝑟+1,𝑁−𝑠)}𝛽_{(𝑟+2,𝑁−𝑠)}⋯𝛽_{(𝑟+𝑘,𝑁−𝑠)}𝛽_{(𝑟+1,𝑟+𝑘+𝑙+1)}⋯𝛽_{(𝑁−𝑠,𝑁−𝑠)},𝛼_𝑖𝑌=𝐸^(𝑖)−𝜃𝜎=𝐸𝑋_(𝑖),𝛽_(𝑖,𝑗)𝑌=Cov^(𝑖)−𝜃𝜎,𝑌_(𝑗)−𝜃𝜎=Cov𝑋_(𝑖),𝑋_(𝑗). (22)

𝐸(𝑋_(𝑖)) and Cov(𝑋_(𝑖),𝑋_(𝑗)) are as above definition.

Further, the MSEs of BLUE 𝜃₉ and 𝜎₉ are given by MSE𝜃₉=Var𝜃₉=𝜎²𝛼^𝑇𝛽⁻¹𝛼𝛼^𝑇𝛽⁻¹𝛼1^𝑇𝛽⁻¹1−𝛼^𝑇𝛽⁻¹1²,MSE𝜎₉=Var𝜎₉=𝜎²1^𝑇𝛽⁻¹1𝛼^𝑇𝛽⁻¹𝛼1^𝑇𝛽⁻¹1−𝛼^𝑇𝛽⁻¹1², (23) where 1, 𝛼 and 𝛽 are as above definition. In addition, we find the best estimators based on the smallest MSEs among the WMEs and BLUE, respectively, as follows: MSE_1≤𝑖≤9𝜃_𝑖,MSE_1≤𝑖≤9𝜎_𝑖. (24)

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 𝑟, 𝑘, 𝑙, and 𝑠 for sample size 𝑁=15,25, 𝜎=1, and 𝜃=1. The study results are summarized in Tables 1 to 4.

Table 1 

MSEs of BLUE and WMEs of the scale parameter 𝜎 at 𝑁=15, 𝜎=1.0, and 𝜃=1.

For comparison results of scale estimators for 𝜎, Tables 1 and 2 indicate that (i) WME2 is the best estimator for 𝑁=15, 𝑙≠0, and all different censoring schemes except (𝑟,𝑘,𝑙,𝑠)=(2,5,3,1), (1,5,2,3), (1,3,7,1), (1,5,1,4); (ii) WME3 is the best estimator for 𝑁=25, 𝑙≠0, and all different censoring schemes except (𝑟,𝑘,𝑙,𝑠)=(3,5,2,1),  (3,5,1,2), (1,3,7,1); (iii) WME8 is the best estimator for 𝑁=15,25 and right censoring scheme; (iv) WME5(=WME6) is the best estimator for 𝑁=15,25 and left censoring scheme; (v) WME6 is the best estimator for 𝑁=15,25 and left censoring scheme. The magnitude of MSE decreases as sample size 𝑁 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 𝑁=15,25 and 𝑙≠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 𝑁=15,25 and 𝑙=0. 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 𝜎 at 𝑁=25, 𝜎=1.0, and 𝜃=1.

For comparison results of location estimators for 𝜃, Tables 3 and 4 show that (i) WME1 is the best estimator for 𝑁=15, (𝑟,𝑘,𝑙,𝑠)=(1,5,3,2), (1,5,2,3), (1,3,7,1), (1,5,1,4), and middle censoring scheme; (ii) WME1 is the best estimator for 𝑁=25 and all different censoring schemes except right censoring scheme; (iii) WME8 is the best estimator for 𝑁=15 and right censoring scheme; (iv) WME5 is the best estimator for 𝑁=25 and left and doubly censoring schemes. The magnitude of MSE decreases as sample size 𝑁 increases. The range of WME/BLUE is between 0.861855 and 1.048036 for 𝑁=15,25 and 𝑙≠0. The range of WME/BLUE is between 0.843873 and 1.02405 for 𝑁=15,25 and 𝑙=0.

Table 3 

MSEs of BLUE and WMEs of the location parameter 𝜃 at 𝑁=15, 𝜎=1.0, and 𝜃=1.

Table 4 

MSEs of BLUE and WMEs of the location parameter 𝜃 at 𝑁=25, 𝜎=1.0, and 𝜃=1.

4. Conclusion

The object of this paper is to propose WMEs of location parameter 𝜃 and scale parameter 𝜎 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 𝑁=15, our proposed WMEs are, in general, superior to BLUE and different censoring schemes except (𝑟,𝑘,𝑙,𝑠)=(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 schemes; for 𝑁=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 𝑗th (𝑁−𝑠<𝑗≤𝑁) ordered observation 𝑌_(𝑗) in a sample of size 𝑛 from the extreme value distribution based on multiply type II censored sample. Moreover, if observations 𝑇=exp(−𝑌) are from a Weibull distribution with a scale parameter 𝛼 and a shape parameter 𝛽, then the best estimators of Weibull distribution can be obtained through proposing the best estimators of the extreme value distribution, where 𝛼=exp(−𝜃) and 𝛽=1/𝜎. Moreover, by replacing 𝑌 by −𝑌, the corresponding distribution of −𝑌 is also called extreme value distribution with c.d.f. as given by 𝐹_−𝑌𝑦=1−exp−𝑒^{(𝑦−𝜇)/𝜎},−∞<𝑦<∞, (25) where −∞<𝜇=−𝜃<∞, 𝜎>0. Our method and the best linear unbiased method can be used to find the best estimator of 𝜇 and 𝜎 and predict the 𝑗th (𝑁−𝑠<𝑗≤𝑁) ordered observation 𝑌_(𝑗) 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).