# On Progressive Type-II Censored Samples from Alpha Power Exponential Distribution

Authors: Mukhtar M. Salah
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper the two-parameterα-power exponential distribution is studied. We study the two-parameter α-power exponential μ,λ distribution with the location parameter μ>0 and scale parameter λ>0 under progressive Type-II censored data with fixed shape parameter α. We estimate the maximum likelihood estimators of these unknown parameters numerically since it cannot be solved analytically. We use the approximate best linear unbiased estimators μ∗ and λ∗, as an initial guesses to obtain the MLEs μ^ and λ^. We estimate the interval estimation of these unknowns’ parameters. Monte Carlo simulations are performed and data examples have been provided for illustration and comparison.


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## Body

## 1. Introduction

The censoring sample takes place in the life testing experiment, if we cannot notice the failure time of all items placed on this experiment. To see this, assume a life-testing experiment consisting froms units was kept under observation, and these units may be in industry manufactory, individuals in clinical queue, computers, and any system in a reliability study experiments so that if we preplanned to remove some units from this experiment in order to provide saving cost and time then after removing such units from this experiment, the resulting data are named censored data. There are three types of censoring schemes: Type-I, Type-II, and progressive censoring schemes. Before defining these types, assume n items are kept on a testing-life experiment; then,(1)
Assume we decided to stop the experiment at prefixed timeT, to get only the failure time of the items which fails prior to the time recorded, and the resulting sample from this experiment is defined as Type-I censoring sample.(2)
Assume that an experiment is stopped atRth failure then the resulting data is defined as Type-II censored sample; note here Rth failure is considered to be fixed, while XR:n denotes a random experiment duration. For more details about Type-I and Type-II censored samples, one can refer to Salah [1], Balakrishnan and Aggarwala [2], Pradhan and Kundu [3], and Lin et al. [4].(3)
Progressive censoring sample: assumes identical items are placed on a life-testing experiment, and it is decided to observe only k failures, and censor the remaining s−k items progressively as follows. At the time of the first failure, F1 of the s−1 surviving components are removed randomly from the experiment; at the time of the next failure, F2 of the remaining s−2−F1 surviving components are removed randomly from the experiment; repeat this censoring up to the final stage, the time of the kth failure, and we have all remaining Fk=s−k−F1−F2−⋯−Fk−1. The surviving components are randomly removed from the experiment and then the resulting censoring is called progressive censoring. More precisely, it is named as progressive Type-II censoring sample.Let the sampleZ1:k:s,Z2:k:s,…,Zk:k:s be a progressively Type-II censored sample and F1,F2,…,Fk be the progressive censoring scheme. Suppose that k, the number of observed failures, and F1,F2,…,Fk are all prefixed. Then, we have the following special cases:(1)
IfF1=F2=⋯=Fk−1=0 and Fk=s−k, then this scheme is reduced to the Type-II censoring scheme.(2)
IfF1=F2=⋯=Fk=0 and k=s, then no censoring happens (the complete data case). For progressive censoring and its inferences, one can refer to Balakrishnan [5], Balakrishnan et al. [6], Salah [7], Salah [8], Salah [9] and Khan et al. [10].Mahdavi and Kundu [11] present a new method to add a new parameter to a family of distribution. They named it as α- power transformation (APT) method. They applied the APT method to a specific class of distribution such as the exponential distribution, and they called this new distribution as the two-parameter α-power exponential (APE) distribution. In this paper, we study the statistical inferences of the APE μ,λ distribution under progressive Type-II censoring schemes. Here, we define the APE distribution as follows.Definition 1.
LetX be a random variable, then X follows a two-parameter APE distribution APE μ,λ, with location parameter μ>0 and scale parameter λ>0 if the probability density function (pdf) and the cumulative distribution function (cdf) of X for any x>0, respectively, are(1)fx;μ,λ=logαα−1λe−λx−μα1−e−λx−μ,ifα≠1,λe−λx−μ,ifα=1,(2)Fx;μ,λ=α1−e−λx−μ−1α−1,ifα≠1,1−e−λx−μ,ifα=1.
The rest of this paper is organized as follows. In Sections2 and 3, we present the maximum likelihood estimators MLEs μ^,λ^ and Fisher information matrix to estimate the interval estimation for μ and λ. In Section 4, we derive the approximate best linear unbiased estimators ABLUEs μ∗,λ∗ for μ and λ. In Section 5, numerical calculation and simulation are preformed. Finally, conclusion is given in Section 6.


## 2. Maximum Likelihood Estimator

This section presents the MLEμ^ of μ and the MLE λ^ of λ from the APE μ,λ distribution when the samples are progressively Type-II censored. To this end, assume X˜=X1:m:n,X2:m:n,…,Xm:m:n is a progressive Type-II censoring sample from APE distribution and R1,R2,…,Rm is the progressively censoring scheme. Then, we have likelihood function LX˜,μ,λ regarded to progressive Type-II censored sample as(3)LX˜,μ,λ=c∏i=1mfxi:m:n,μ,λ1−Fxi:m:n,μ,λRi,where(4)c=nn−1−R1n−2−R1−R2,…,n−m+1−R1−R2−⋯−Rm−1.For simplicity, we useL instead of LX˜,μ,λ and xi for xi:m:n. Now, by substituting equations (1) and (2) into equation (3) with some simplifications, we obtain(5)L=c∏i=1mλlnαα−1e−λxi−μα1−e−λxi−μα−α1−e−λxi−μα−1Ri,ifα≠1,c∏i=1mλe−λxi−μe−λxi−μRi,ifα=1.Since whenα=1 in equation (5), then we have the log-likelihood for the exponential distribution which is a special case from APE distribution and is studied before, one can refer to Balakrishnan et al. [6]. Now, from equation (5), the log-likelihood function is given by(6)lnL=lnc+mlnλlnαα−1−∑i=1mλxi−μ+mlnα−lnα∑i=1me−λxi−μ+∑i=1mRilnα−α1−e−λxi−μ−lnα−1.Differentiating equation (6) with respect to μ and λ, respectively, we obtain(7)∂lnL∂μ=mλ−λlnα∑i=1me−λxi−μ+λlnα∑i=1mRie−λxi−μαe−λxi−μ−1,(8)∂lnL∂λ=mλ+∑i=1mxi−μlnαe−λxi−μ−1−lnα∑i=1mRixi−μe−λxi−μαe−λxi−μ−1.The MLEsμ^ and λ^ can be estimated by solving system of equations: ∂lnL/∂μ=0 and ∂lnL/∂λ=0. It is observed that MLEs cannot be obtained in explicit forms analytically from the system, so one can use a numerical method to find these MLEs numerically. Here, we use the Newton method with ABLUEs as initial guesses for finding the MLEs μ^ and λ^.


## 3. Fisher Information Matrix

In this section, we develop the interval estimates for the parametersμ and λ of the APE distribution. To see this, we derive the Fisher information matrix and then derive the asymptotic Variance-Covariance matrix of these MLEs. Hence, we have to derive the second derivatives of the lnL functions with respect to μ and λ. From equations (7) and (8), we have(9)∂2lnL∂μ2=−λ2lnα∑i=1me−λxi−μαe−λxi−μ−12−Riαe−λxi−μ1−e−λxi−μlnα−1αe−λxi−μ−12,(10)∂2lnL∂λ2=−mλ2−lnα∑i=1mxi−μ2e−λxi−μ−lnα∑i=1mRixi−μ2e−λxi−μ1+αe−λxi−μe−λxi−μlnα−1αe−λxi−μ−12,(11)∂2lnL∂λ∂μ=m+lnα∑i=1mλxi−μ−1e−λxi−μ+∑i=1mRie−λxi−μαe−λxi−μ1−λxi−μ+λxi−μαe−λxi−μe−λxi−μlnα+1−1αe−λxi−μ−12.From equations (9)–(11), the observed information matrix can be inverted to obtain the asymptotic variance-covariance matrix of the MLEs as follows:(12)−∂2lnL∂μ2|μ^,λ^−∂2lnL∂μ∂λ|μ^,λ^−∂2lnL∂μ∂λ|μ^,λ^−∂2lnL∂λ2|μ^,λ^−1.The distribution of pivotal quantities(13)P1=μ^−μVarμ^,P2=λ^−λVarλ^,is considered to be standard normal regarded to the asymptotic properties of MLEs. From P1 and P2, one can construct the confidence interval for μ and λ, respectively. Hence, 95% confidence intervals for μ and λ based on the MLEs are given by(14)μ^−1.96Varμ^,μ^+1.96Varμ^,λ^−1.96Varλ^,λ^+1.96Varλ^.


## 4. Approximate Best Linear Unbiased Estimator

The BLUEs forμ and λ are so difficult due to the difficulty in solving the variance-covariance matrix Σ for the APE distribution and also its inverse Σ−1 for large sample. In order to that we derive the ABLUEs of μ and λ. To do this, suppose that the progressively Type-II censored sample(15)X1:m:nR1,…,Rm,X2:m:nR1,…,Rm,…,Xm:m:nR1,…,Rm,has been taken from APE distribution. By following the method described by Balakrishnan and Aggarwala [2], we compute the ABLUEs μ∗ and λ∗, respectively, as follows:(16)μ∗=∑i=1mAiXi:m:nR1,…,Rm,(17)λ∗=∑i=1mBiXi:m:nR1,…,Rm,where Ai and Bi are the coefficients of the ABLUEs. In order to find these coefficients, we developed an R-Code program. The values of Ai and Bi are given in Table 1 and checked for the conditions:(18)∑i=1mAi≃1,∑i=1mBi≃0,which is achieved. For more about the ABLUEs and the coefficients Ai and Bi, see Balakrishnan and Aggarwala [2] page 81 to page 110.Table 1
Coefficients of ABLUEs ofμ and λ from APE distribution.
Scheme/sumα=1.5α=2.5α=3.5AiBiAiBiAiBi[1]1.3300−1.51761.3522−1.30621.3716−1.2033−0.08230.4036−0.06250.3028−0.04570.2546−0.24771.1140−0.28981.0034−0.32590.9487Sum≈1≈0≈1≈0≈1≈0[2]1.1091−0.97091.1117−0.82531.1368−0.7565−0.01860.1851−0.00790.12250.00220.0939−0.02350.2153−0.02200.1803−0.02090.1655−0.02770.2416−0.03240.2204−0.03660.2101−0.03930.3289−0.04940.3022−0.05840.2870Sum≈1≈0≈1≈0≈1≈0[3]1.1651−2.79481.1633−2.19281.1613−1.8893−0.00640.12190.00040.04670.00760.0078−0.00700.1305−0.00190.06740.00340.0368−0.08051.3575−0.08001.0560−0.08020.9056−0.00940.1659−0.00820.1279−0.00710.1115−0.06181.0190−0.07360.8948−0.08500.8275Sum≈1≈0≈1≈0≈1≈0[4]1.0831−2.11491.0770−1.60141.0719−1.3558−0.04191.0740−0.03310.7273−0.02540.5638−0.00360.0980−0.00140.05650.00070.0383−0.00410.1085−0.00300.0784−0.00210.0662−0.00450.1176−0.00430.0958−0.00430.0871−0.00490.1254−0.00540.1094−0.00600.1026−0.00530.1320−0.00630.1198−0.00730.1137−0.01890.4594−0.02340.4141−0.02760.3870Sum≈1≈0≈1≈0≈1≈0[5]1.1089−4.49621.1086−3.46601.1079−2.9249−0.00150.06650.00030.01630.0023−0.0116−0.00150.06850.00010.02160.0018−0.0037−0.00160.0704−0.00020.02670.00140.0037−0.00160.0723−0.00040.03150.00100.0107−0.00180.0742−0.00060.03620.00060.0173−0.00170.0760−0.00070.04070.00030.0235−0.00180.0778−0.00090.0449−0.00040.0294−0.00180.0796−0.00110.0490−0.00040.0349−0.09573.9109−0.10563.1991−0.11492.8206Sum≈1≈0≈1≈0≈1≈0[6]1.0393−2.59531.0390−2.01211.0386−1.7159−0.00060.04160.00010.00970.0009−0.0069−0.00060.04300.00000.01330.0007−0.0017−0.00060.04440.00000.01680.00050.0032S10020071203049620.0457−0.00010.02000.00040.0078−0.00070.0470−0.00020.02310.00020.0120−0.00070.0483−0.00030.02610.00010.0160−0.00070.0495−0.00040.02890.00010.0198−0.00070.0507−0.00040.0316−0.00020.0233−0.00080.0519−0.00050.0342−0.00030.0266−0.00080.0531−0.00060.0366−0.00040.0297−0.00080.0542−0.00060.0389−0.00050.0326−0.00080.0552−0.00070.0411−0.00060.0353−0.00080.0563−0.00080.0433−0.00070.0379−0.03011.9544−0.03451.6485−0.03881.4803Sum≈1≈0≈1≈0≈1≈0[7]1.0405−3.34211.0388−2.51371.0370−2.0908−0.00990.8228−0.00820.5501−0.00660.4091−0.00030.03090.00080.00790.0005−0.0040−0.00030.03170.00030.00990.0005−0.0011−0.00040.03250.00000.01180.00040.0016−0.00040.0332−0.00040.01360.00030.0048−0.00040.0339−0.00080.01540.00020.0067−0.00040.0347−0.00010.01710.00020.0090−0.00040.0354−0.00020.01880.00000.0112−0.00040.0361−0.00020.02040.00000.0133−0.00040.0367−0.00020.02190.00000.0154−0.00040.0374−0.00030.02340.00000.0173−0.00040.0381−0.00030.0248−0.00010.0191−0.00040.0387−0.00030.0262−0.00020.0209−0.00050.0393−0.00030.0275−0.00020.0225−0.00050.0399−0.00040.0288−0.00030.0241−0.00050.0405−0.00040.0300−0.00030.0257−0.00050.0411−0.00040.0312−0.00040.0271−0.01150.9314−0.01300.7792−0.01460.6968−0.01251.0078−0.01440.8557−0.01640.7713Sum≈1≈0≈1≈0≈1≈0


## 5. Simulation Study and Numerical Calculations

Following the algorithm presented by Balakrishnan and Sandhu [12], one can generate progressively Type-II censored samples from the standard APE distribution taking μ=0 and λ=1. We apply the given algorithm below to perform the numerical computations and simulations proposed in the previous sections.Step 1:mindependent uniform0,1variablesW1,…,Wmare generatedStep 2:forR1,…,Rmprogressively censoring scheme,setVi=Wi1/i+∑j=m−i+1mRifori=1,2,…,mStep 3: setUi=1−VmVm−1…Vm−i+1fori=1,2,…,m, then Ui:m:n,…,Um:m:n is progressively Type-II censored sample of size m from uniform (0, 1)Step 4: finally, for some given values ofμ and λ, we have Xi=μ+1/λloglogα/logα−log1+α−1ui progressively generated from APE distributionAfter applying the algorithm and selecting sample sizes and censoring schemes from APE distribution given in Table2, we compute the ABLUEs of μ and λ from equations (16) and (17) and tabulate it in Table 1 for some chosen values of the shape parameter α=1.5,2.5,3.5. We also compute the biases and mean square error (MSE) of the ABLUEs of μ and λ in Tables 3–5. As we mentioned before, we use the ABLUEs of μ and λ to estimate the MLEs of μ and λ; these MLEs, biases, and MSEs of the parameters are tabulated in Tables 6–8. All the computations are computed using R program over 5000 Monte Carlo simulations. The computations are carried out for some selected samples size: n=5,10,20,30,50,80,100.Table 2
Sample sizes and censoring schemes from APE distribution.
nmSchemeCensoring scheme5311,0,110525,0,0,0,020632,0,0,8,0,4308410,10,0,0,0,0,0,2501050,0,0,0,0,0,0,0,0,408015[6]35,0,0,0,0,0,0,0,0,0,0,0,0,0,3010020[7]20,20,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20,20Table 3
ABLUEsμ∗, λ∗, biases, and MSEs for μ and λ from APE distribution when α=1.5.
Schemeμ∗λ∗Biasμ∗Biasλ∗MSEμ∗MSEλ∗10.04771.05750.00590.14390.07360.648220.13471.27110.00360.11720.01780.39893−0.00881.21990.00570.00980.00430.18434−0.03761.37690.00260.06940.00210.14875−0.01901.42040.00230.01770.00060.10486−0.00901.00300.00900.00300.00050.00527−0.01181.14810.00080.00970.00010.0041Table 4
ABLUEsμ∗, λ∗, biases, and MSEs for μ and λ from APE distribution when α=2.5.
Schemeμ∗λ∗Biasμ∗Biasλ∗MSEμ∗MSEλ∗1−0.19131.36000.02560.15360.11430.48222−0.13821.48430.01240.20850.03470.36723−0.08031.37270.00190.00370.00710.17544−0.04221.37330.00720.07550.00280.15595−0.02801.28740.00040.02330.00130.12976−0.01500.99500.00040.00220.00030.058270.00441.11360.00050.00090.00020.0439Table 5
ABLUEsμ∗, λ∗, biases, and MSEs for μ and λ from APE distribution when α=3.5.
Schemeμ∗λ∗Biasμ∗Biasλ∗MSEμ∗MSEλ∗1−0.04841.29760.06150.21540.18170.70382−0.22121.25300.03940.15330.04390.32483−0.10961.35580.00540.01310.01050.16924−0.07611.43500.01350.05370.00300.11775−0.01051.44350.00390.01320.00160.068160.04840.99500.00100.01460.00050.051770.00091.18700.00110.00080.00040.0403Table 6
MLEsμ^, λ^, biases, and MSEs for μ and λ from APE distribution when α=1.5.
Schemeμ^λ^Biasμ^Biasλ^MSEμ^MSEλ^1−0.08221.39600.01400.00080.01950.00012−0.08361.09890.01100.00080.01210.00013−0.09311.16210.01160.00090.01350.00024−0.01001.09140.01100.00100.01190.00005−0.07480.98600.00980.00070.00970.00006−0.05601.06000.00300.00050.00900.00007−0.06111.14800.00110.00000.00130.0000Table 7
MLEsμ^, λ^, biases, and MSEs for μ and λ from APE distribution when α=2.5.
Schemeμ^λ^Biasμ^Biasλ^MSEμ^MSEλ^1−0.05331.14930.01150.00050.01320.00012−0.12051.14300.01110.00120.01310.00023−0.18961.23410.01230.00190.01520.00044−0.18701.18380.01180.00180.01400.00035−0.17031.21740.01220.00170.01480.00026−0.16701.15600.00200.00020.00460.00007−0.11401.12360.00110.00010.00120.0000Table 8
MLEsμ^, λ^, biases, and MSEs for μ and λ from APE distribution when α=3.5.
Schemeμ^λ^Biasμ^Biasλ^MSEμ^MSEλ^1−0.13341.18400.01180.00130.01400.00022−0.09301.01470.01010.00090.01020.00013−0.19381.00480.01200.00190.01450.00034−0.24111.19200.01200.00240.01420.00055−0.16661.04490.01100.00170.01090.00036−0.22421.03450.00200.00020.00400.00007−0.14781.05000.00150.00010.00240.0000Example 1.
From the APE distribution withμ=0,λ=1, and α=2.5, we generate a progressively Type-II censored sample of size m=6, from n=20. The progressive censoring scheme and its observations are
Using the above censored sample and from the likelihood equations, (7) and (8), we compute the MLEs of μ and λ to be(19)μ^=−0.1764,λ^=1.0982.
Using the above MLEs, the95% confidence intervals for the location μ and scale λ parameters are −0.4812,2.2661 and 1.0255,2.3089, respectively.
Similarly, we obtain the ABLUEs ofμ and λ from Table 1 and equations (16) and (17) to be(20)μ∗=0.1992∗1.1633+0.2402∗0.0004+0.4857∗−0.0019+0.4920∗−0.0800+0.5762∗−0.0082+1.1805∗−0.0736,μ∗=0.0999,λ∗=0.1992∗−2.1928+0.2402∗0.0467+0.4857∗0.0674+0.4920∗1.0560+0.5762∗0.1279+1.1805∗0.8948,λ∗=1.2568.
Here, we notice that the ABLUEs are so closed to the MLEs of the parametersμ and λ, and this shows that ABLUEs give good initial guesses when estimating the MLEs of these parameters.Example 2.
The following data {2.247, 2.64, 2.908, 3.099, 3.126, 3.245, 3.328, 3.355, 3.383, 3.572, 3.581, 3.681, 3.726, 3.727, 3.728, 3.783, 3.785, 3.786, 3.896, 3.912, 3.964, 4.05, 4.063, 4.082, 4.111, 4.118, 4.141, 4.246, 4.251, 4.262, 4.326, 4.402, 4.457, 4.466, 4.519, 4.542, 4.555, 4.614, 4.632, 4.634, 4.636, 4.678, 4.698, 4.738, 4.832, 4.924, 5.043, 5.099, 5.134, 5.359, 5.473, 5.571, 5.684, 5.721, 5.98, 6.06} was taken from Bader and Priest [13] and represents the strength for single carbon fibers and impregnated 1000-cardon fiber tows, measured in GPa. The data is reported at gange length 1 mm. These data fitted the APE distribution, see Mahdavi and Kundu [11]. We found that the MLEs of μ and λ for the complete data are(21)μ^=2.3655,λ^=1.2571.
Using the above MLEs, the95% confidence intervals for the location μ and scale λ parameters are 0.9453,1.5688 and 2.1225,2.6085, respectively. Comparing the results in this paper to that in Mahdavi and Kundu [11], we see that these results are so close to each other.


## 6. Conclusion

The MLEs and ABLUEs of the location and scale parameters from APE distribution are derived and tabulated for some selected sample size and shape parameter. The ABLUEs are so closed to the MLEs which gives good initial guesses in estimating MLEs. AnR code program was performed to find the ABLUEs and MLEs. It will be interesting to study the Type-I and Type-II hybrid censoring scheme from the APE distribution in future And also Bayesian and non-Bayesian inference under adaptive type-II progressive censored sample from APE distribution.



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