Estimation of the Parameters and Expected Test Time of Exponentiated Weibull Lifetimes Under Type II Progressive Censoring Scheme With Random Removals

Based on progressive type-II censored sample with random removals, point and interval estimations for the shape parameters of the exponentiated Weibull distribution are discussed. Computational formula for the expected total test time are derived for different situations of sampling plans. This is useful in planning a life test experiment. The efficiency of the estimators are compared in terms of the root mean square error, the variance and the coverage probability of the corresponding confidence intervals. A simulation study is presented for several values of removal probability and different values of failure percentage. Also, numerical applications are conducted to illustrate and compare the usefulness of the different sampling plans in terms of expected test times for different patterns of failure rates.


Introduction
In reliability tests involving massive numbers of items and in situations where exceptionally large times to failure are expected, the use of censored data is economically more realistic.The procedure of removing good items from the test before the termination of experiment is known as progressive censoring.In a progressive type-II censoring scheme n identical units are placed on a life-test experiment and removals of still operating units are carried out at observed failure times.That is, after the i-th failure some preassigned number   ( i = 1, 2, . .., m) of surviving units are randomly removed from the test.This process continues until the m-th failure, t m , is obtained and the remaining   =  − ( + ∑   −1 =1 ) units are all removed at the end of the experiment.Therefore, the observed failure times; t 1 , t 2 , … , t m are called progressively type-II censored order statistics.We denote to the set of observed failure-times and survival withdrawn by {t i , R i }, i = 1, 2, . .., m ; t 1 < t 2 < … < t m .If R i = 0 for i=1, 2, . .., m-1 and R m > 0, then the progressive type-II censored sample reduces to a conventional type-II censored of t 1 , t 2 , … , t m , m<n.The importance and usefulness of progressive censoring schemes and its applications in industrial life testing and medical survival analysis can be seen by referring to Balakrishnan and Aggarwalla (2000).
In the last two decades, the statistical inferences for various distribution using progressive censored samples get special attention and are studied by several authors among of them, Viveros and Balakrishnan (1994), Balakrishnan and Sandhu (1995), Childs and Balakrishnan (2000), Mousa and Jaheen (2002), (Balakrishnan et al. (2003), Soliman (2005), Basak et al. (2009), Madi and Raqab (2009), Cheng et al. (2010) and Kim et al. (2011).In these studies the value of R i at the time of the ith failure t i are fixed and preassigned.In practical situations, R i may occur at random.As an example, in some reliability studies in industerial field, an experimenter may decide that it is inapproperiate or too dangerous to carry on the test on some of the tested units even though these units have not failed completely.The number of patients drop out from a clinical test at each stage is random and cannot be prefixed, as another example (See Tse et al. (2000)).In such cases, the experimenter may need to change the censoring numbers during the experiment and so adaptation process of progressive censoring scheme is needed.Ng et al. (2009) proposed an adaptive censoring scheme in which the censoring scheme (R 1, R 2, , … , R m ) may change during the experiment according to a prefixed threshold time T> 0 to keep the effective sample size m of failures.Before the time T the conventional progressive censoring scheme is applied.
After passing time T withdrawing surviving units is stopped at all until the time of mth failure where all remaining surviving units are removed.Yuen and Tse (1996) proposed progressive censoring scheme in which the number R j of units removed at each failure is random and is chosen according to a probability distribution with support {0, . .., n-m} and the prbability mass function of this distribution is independent of failure time.Recently, some authors have discussed inference problem for various distribution when the data are progressively censored with random removals, among of them: Tse et al. (2000), Wu and chang (2003), Wu et al. (2007), Sarhan et al. (2008) and Kumar et al. (2016).
The purpose of this paper is concerned with inferences using progressively type-II censored data with random removals from exponentiated Weibull distribution.The probability density functon, cumulative distribution function, and reliability function of this distribution are, respectively, () =  −1  −  (1 −  −  ) −1 , t> 0; , > 0 (1) and (2) where  and are the two shape parameters.This distribution is an extension of the well known Weibull distribution.It accommodates unimodal, bathub-shaped and broader classes of monotone failure rates.Mudholkar et al. (1995) showed that, for the exponentiated Weibull distribution, the failure rate is (a) unimodal for α < 1 and αθ > 1, (b) bathtub-shaped for α > 1 and αθ < 1, (c) decreasing for α ≤ 1 and αθ ≤ 1, and (d) increasing for α ≥ 1 and αθ ≥ 1.In practice, the exponentiated Weibull distribution as a failure model is more realistic than that of monotone failure rates and plays an important role to represent several lifetime data that are of bathtub-shaped or upside-down bathtub shape failure rates.The distribution includes exponential, generalized exponential, Burr type X and reyligh sub-models.See Mudholkar and Hutson (1996) and Nassar and Eissa (2003).However, The exponentiated Weibull distribution as a failure model gives a good idea about using the progressive censoring schemes with random removals.
The rest of this paper is organized as follows: Section 2 presents the model under consideration and the likelihood function of the model parameters.Section 3 derives the maximum likelihood equations and discusses the asymptotic properties of the maximum likelihood estimators (MLEs).Also, the confidence intervals of  and  are obtained.In Section 4, simulation results is made to assess the performance of the estimators.Analytical formulas for expected test time are derived in Section 5. Numerical analyses for the expected test time are presented in Section 6.

Model
Suppose a progressive type-II censored sample {t i , R i }, i=1,2,. .., m, is obtained when n units are put on a life test terminated at prespcified number m (≤n) of failures are observed.The number of surviving units that are removed from the test at the ith failure, R i , is a random variable.The sample of failure times are considered to have an exponentiated Weibull distribution with density and distribution functions given, respectively, by equations (1) and (2).Let R 1 =r 1 , R 2 =r 2 ,. .., R m =r m and denote the set of lifetimes t = *t 1 , t 2 , … , t m + and r = *r 1 , r 2 , … , r m +, say.The conditional likelihood function defined by Cohen (1963) is . (3) For progressive type-II censoring with predetermined R's and applying equations (1) and (2) the likelihood function can be obtained as where . The form of equation ( 4) is conditional on  .We consider the case of progressive type-II censoring in which   is independent of  for all i and is assumed to follow a binomial distribution with parameters n-m-∑ −1 =1 and .The parameter is the probability of removal of each unit and is assumed to be the same for all removals.Therefore, we have where 0 ≤ r i ≤ − − ∑ r j i−1 j=1 , i=1, 2, . . ., m-1.Since the random mechanism producing the removals is independent of the data, i.e.R i is independent of lifetimes for all i, the likelihood function can be written as where the second term in the right hand side of equation ( 7), is Substituting the forms of equation ( 6) in equation ( 8), we get where C1=∑ .

Maximum Likelihood Estimation
In this section, we will derive the MLEs of α and θ.It is clear that the joint probability distribution of R's does not involve the parameters  and .The MLEs  ̂ and  ̂ of  and , respectively, can be obtained directly by maximizing the function 1 (, ;  | = ) given by equation ( 4).The log-likelihood function, ln 1 (, ; The maximum likelihood estimators α ̂ and θ ̂ are the values that solves the first order conditions where Equations ( 10) gives the necessary conditions for a maximum.In addition, the Hessian matrix, (), of the exponentiated Weibull log-likelihood function is The elements  , , = 1, 2 are given by − 3 (11) where  =   (1 +  ).It easy to see that 11 , 22 and 12 are all negative where the terms  < 0,  > 0,  > 0 and 3 > 0 for  > 0 and  > 0 as well as   < 0 and    can be positive or negative according to   is greater or less than one.Fluctuating values of    between positive and negative does not affect on the ultimate value of the 11 , 22 and 12 to be negative.Therefore, the Hessian matrix of the log-likelihood function l(, ) is negative definite at every value of  > 0 and  > 0. We can say that l(, ) is strictly concave and hence any local maximizer that we find will be the unique global maximizer, i.e. the estimated parameters  ̂ and  ̂ solving the first-order conditions given by equations ( 10) are the unique maximum likelihood solution.At this point, we conclude that the system of likelihood equations (10) admits a unique solution in the region  + ×  + and that the log-likelihood function l(, ) is strictly concave but we can not say that this system always has a unique solution.The conditions, by which the solutions are always exist, need further and deep discussion in other work.Now, searching about a solution ( α ̂, θ ̂) of equations ( 10) at which the Hessian matrix is negative definite must be cosidered.It is difficult to obtain such solution (analytically) in closed form.Thus we resort to obtain the solution by using an iterative technique.For more details about the existence and uniqueness of maximum likelihood estimates, see Makelainen et al. (1981).
To evaluate the performance of the estimators and studying the variation of the MLEs of the parameters of our lifetime distribution, the asymptotic variances and covariances of  ̂ and  ̂ are obtained.It is clear that is only a parameter of binomial removal distribution and does not provide information about the lifetime distribution.On the basis of the asymptotic normality property of the tow-dimentional ML estimators, the asymptotic distribution of the vector √( ̂−   ̂− ) approaches (0,  −1 ) as n approaches infinity and for large value of effective sample size m, m<n (See Ng et. al. (2009), Rao (1973), and Mood et. al. (1974)).The Fisher information matrix  = (, ) is associated with the parameters α and θ .Using the observed information, we evaluate the approximate variance-covariance matrix, (C ij ), say, The elements ij can be obtained as follows: where  =   (1 +  );  and  are given in equation ( 10).
Computing the values of the elements given by equations ( 12 The approximate 100(1-τ)% confidence intervals (CIs) for the parametrs  and  from proressively type II censored data with binomial removals are respectively given by where 2 ⁄ is the (1 −  2 ) ⁄ quantile of the standard normal distribution.

Simulation Results
We derive, in the above Section, the ML estimators and CIs for the two shape parameters  and  of the exponentiated Weibull distribution.The likelihood equations (10) are solved, numerically.In order to assess the statistical performances of these estimators under progressive type-II censored sample with random removals, a simulation study is conducted by using generated random samples of different sizes.Samples were generated from exponentiated Weibull distribution with α=0.8 (α < 1) and θ=1.7 (αθ > 1) (unimodal failure rate case).One thousand samples are generated by applying the algorithm of Balakrishnan and Sandhu (1995).Different values of removal probability and failure percentage (denoted by) Fp, are considered where Fp=(  ⁄ )% .We employ MATHEMATICA7 program to perform the computations.For different sample sizes, n=20 (small), 30 (moderate), 50 (large), different values of removal probability, =0.01, 0.05, 0.10, 0.25 and 0.50 and different failure-percentages, Fp= 50%, 60%, 70% and 80%, the results are presented in Tables 1-3.The results based on 1000 replications include the root mean square error (RMSE), the mean width of confidence interval, and the average coverage probability (CAP) for each parameter at =0.05.
We discuss in light of the results presented in Tables 1-3 how the estimates affected by censoring schemes in our case of all surviving units are randomly removed from the experiment.The removals are binomially distributed with probability .
Our observations about the results are sumerized in the following points: 1.The RMSEs of α ̂ and θ ̂ decrease as Fp increases for all sample sizes.This is due to large Fp would imply relatively more failure times being observed, more information about the life times distribution are available and so more efficient estimation of the parameters under consideration.The effect of the removal probability is also important to decrease the RMSEs of α ̂ and θ ̂.For a fixed Fp, as the increases, the RMSEs of these estimators reduce appreciably, specially for small and moderate sample sizes.This means that more withdrawals should be occurred during (not at the end of) the test experiment.The results also reveal that, even for small sample size, the perfomance of ML estimators α ̂ and θ ̂ improve in terms of their variances for increasing .The RMSEs and variances are decreasing, as a rule, from table to table as the sample sizes increase.
2-The mean width of the CIs for  and  decrease when Fp increases.For a fixed Fp, the mean-width of CIs for  are typically decreasing when increases for all sample sizes; whilst θ ̂ is relatively severe in decreasing of reduction of its mean-width of CIs.The effect of on the coverage probability, CAP, of the CIs is evident.For small sample size, increasing of improves the coverage probability to be satiafactory at Fp=50% and 60% and at Fp=50% for moderate sample size.The coverage probabilities are much upper than the nominal level at Fp>60% and/or increasing for all sample sizes.We considered other values of failure percentages and the results are not reported since they have a similar pattern to the cases listed in Tables 1-3.Other values of the distribution parameters: α=0.8 (α < 1) and θ=0.75 (αθ < 1) (decreasing failure rate case), were also considered and the results are not reported since they have a similar pattern to the cases listed in Tables 1-3.The expected time of a life test is the expected time required to complete the test experiment.In practice, an experimenter may have aim to explore the time duration of his experiment to choose an approperiate sampling plan.The cost and the time duration to complete a test experiment are related.In the following we derived the expected test time when the data come from exponentiated Weibull distribution for some csaes of sampling schemes.
In case of complete sample test, the required time to complete the test is the time at the nth, T , failure in a sample of n test units.The expected value of T to observe the nth failure in this sample is given by ( ) =  ∫  () −1 () .In our case, ( ) is given by (15) Now, for any real number a, we have the expansion, (See J. Stewart; page 742), where This series always converges when |x|<1 as well as it converges at the end points 1 if ≥ 0. If a is positive integer the series reduces to the Binomial Theorem and then the last term of the expansion is at i=a.Applying the form ( 16) to the term (1 − e −t α ) −1 , E(T ) given by equation ( 15) can take the form When m=n we get the expected time of test of complete sampling.
Similarly, under the case of progressive type-II censoring with a fixed number of removals, the expected time required to completean experiment is dfined as the expected value of  conditioning on R, and given as follows: and B= (  ) − 1.
For progressive type-II censoring with random removals, R terms are random.The expected test time, (  ), to complete the experiment is given by taking the expectation of equation ( 19) with respect to R. That is, where P(R, p) is given by equation ( 9).
As a comparison scale, the ratio of the expected test times (RETT) can be computed between any two of progressive type II censoring with binomial removals, type II censoring, and complete sample by using the following ratios: .( 21)

Conclusion
In this study, we present some statistical inferences when the observed data are progressively type II censored with random removals.The results demonstrate that the classical point and interval estimations of shape parameters  and  of the exponentiated Weibull model are most significant by the removal probability .The significance is in terms of the reduction of RMSE, variance, the reduction of the width of CI and the satisfaction of coverage probability of the CIs.
Although the CI based on the asymptotic distribution of the MLEs do not perform well for small sample size, the coverage probabilities improve to close the nominal level at suitable values of the removal probability .However, increasing the removal probability produces censoring scheme as  1 approaches n-m and  2 ,  3 , … ,   approach zero.This choice of censoring scheme provides the smallest RMEs and variances for the estimates and improve the value of coverage probability of the CIs for all sample sizes.Balakrishnan et. al. (2003) reveals a similar result.
On the other hand, under profressively type II censored data with random removals, the required time to complete the test experiment increases significantly for various shapes of failure rates.The bathtub-shaped failure rate is of a shortest time of the test compared to other types of failure rates.However, in case of bathtub-shaped or increasing failure rate and the removal probability not too large, the test time can be reduced to that of type II censoring and the reduction is significant compared to complete sampling.
) at the estimated values  ̂ and  ̂ we get Var )In case of type-II censoring sampling test, The density function of the m-th oredr statistics is(  ) =  .  / ()  −1 (),1 − ()-− .The expected time, E(  * ), of type II censoring test can be derived as

Table 1 .
RMSEs, variances and average width of CIs and their coverage probabilities (CAP) for MLEs  ̂ and  ̂ for

Table 2 .
RMSEs, variances and average width of CIs and their coverage probabilities (CAP) for MLEs α ̂ and θ ̂ for different values of Fp and at n=30

Table 3
. RMSEs, variances and average width of CIs and their coverage probabilities (CAP) for MLEs  ̂ and  ̂ for different values of Fp and

Table 4 .
Expected test time under progressive type II censoring with binomial removals and RETT3 when α = 0 6 and