1 Introduction
The Weibull distribution is one of the most commonly used distributions with a wide range of applications in some study fields such as: chemical engineering ([3], [22], and [41]), ecology [33], electrical engineering ([11] and [30]), food industry [4], mechanical engineering ([31] and [23]), telecommunications ([34] and [2]), wireless communications [28], economic ([27] and [7]), civil engineering ([26] and [1]), and seismology [15]. For further details on applications of the Weibull distribution, we refer the readers to Meeker and Escobar (1998), Murthy et al. (2004), and Dodson (2006). However, a comprehensive study has not been performed to compare the estimators. All comparative studies, to the best of our knowledge, have been devoted to compare the performance of the MLE with the estimators of another class. For example, Kanter (2015) made a comparison between least square estimators and the MLEs. The bias of the MLE for the Weibull distribution has been studied by Ross (1996), Watkins (1996) and Montanari et al. (1997). Seki and Yokoyama (1996) made a comparison between the MLE and a bootstrap estimator. Zhang et al.
(2007) compared the estimation methods based on the Weibull probability plot. We also refer the readers to
[10], [12], [17], [25], [36], and references therein. This is while estimators may have different appeals to different users. For example, the maximum likelihood estimator (MLE) that has attractive properties is biased and has not closedform expression. This is while the practitioners from some fields may looking for an estimator that is unbiased or has closedform expression. Also, user may prefer to use an estimator which works satisfactorily with sample of small size. Hence, a comparative study is needed to compare the performance of the known estimators under different situations. In this paper, we perform a comprehensive comparison study between ten class of estimators including: the generalized least square type 1 (GLS1), the generalized least square type 2 (GLS2), the moments (LM), the Logarithmic moments (MLM), the maximum likelihood estimation (MLE), the method of moments (MM), the percentile method (PM), the statistic, the weighted least square (WLS), and weighted maximum likelihood estimation (WMLE).The structure of the paper is as follows. In Section 2, we derive statistic for the shape and scale parameters of the Weibull distribution. The known estimation methods are reviewed briefly in Section 3. A comparison between proposed estimator and the known ones as well as a real data illustration are given in Section 4.
2 statistics for the Weibull distribution parameters
The probability density function (pdf) and cumulative distribution function (cdf) of twoparameter Weibull distribution are, respectively, given by (Nelson, 1982; Johnson
et al., 1994; Dodson, 2006):(2.1)  
(2.2) 
for , and . Here, and are known as the shape and scale parameters. In the following we give statistics for the shape and scale parameters of the Weibull distribution. For this, a lemma given by the following is necessary. Hereafter, we write to denote a Weibull distribution with pdf given in (2.1).
Lemma 2.1
Suppose . Then
(2.3) 
Proof: Define . It follows that
(2.4) 
On the other hand, define . We have,
(2.5) 
Comparing the righthand sides of (2.4) and (2.5), it turns out that ; for , and so the result follows.
Theorem 2.1
Suppose are independent realizations from Weibull distribution with pdf given in (2.1). Then,

in which
is statistic for .

in which
where is statistic for .
Proof:

By applying logtransformation to the both sides of (2.3), we have
(2.6) The righthand side of (2.6) can be used to construct a symmetric kernel as
(2.7) It is easy to see that . To guarantee the asymptotic normality of the introduced statistics for with kernel (2.7), it is necessary to show that is finite. For this, it suffices to show that is finite. To begin, we note that . Since , it follows that
(2.8) Also, suppose , , and
are given arbitrary random variables. Generally, we cannot conclude that if
, then . But, here, elementary statistical manipulations reveal that if we define , , and , then(2.9) Now, using (2.8), we can write
(2.10) Applying property (2.9) to the righthand side of (2.10), we obtain
It is easy to check that where and . The asymptotic normality of follows since

It is not hard to check that where . Define as
(2.11) It is easy to see that . Asymptotic normality of the introduced statistics for with kernel (2) holds if we prove or equivalently . We eliminate the proof since kernels and have similar structure.
3 Known estimators for the Weibull distribution
Here, we review briefly almost all of known estimation methods for the Weibull distribution.
3.1 Maximum likelihood estimation (MLE)
There is no closedform expression for MLEs of the Weibull distribution parameters. It is asymptotically normal and efficient for large sample sizes. Many attempts have been made to compute or modify the MLEs of the Weibull distribution parameters. Cohen and Whitten (1982) considered a modified MLE involving complicated numerical computations. Dodson (2006) derived the MLE for the shape parameter graphically. The MLE of the shape parameter is computed as the root of the equation, see [29]
and the MLE of the scale parameter is given by
It can be seen that depends on and also that must be computed numerically.
3.2 Weighted Maximum likelihood (WMLE)
It is known that MLEs are generally biased. To reduce the bias rate in the case of the Weibull distribution, the weighted maximum likelihood estimators (WMLE) have been proposed in [19]. Suppose is a random sample from cdf given in (2.2), then the WMLEs of the shape and scale parameters are given by
where the weights and are given by
Although the sampling distribution of the is gamma with shape parameter and scale parameter , but the sampling distribution of the is not known. In practice, both of random variables and
are replaced by their central quantities such as mean, median, or geometric mean. Here, we use the median of
and since they yield the best performance, see [5]. For this, in a comprehensive Monte Carlo simulation, we derive the median of and for different levels of (from 0.5 to 5 by 0.2) and small sample size (including 5, 10, 15, 30, 50, 100, …, 200). We note that as tends to , both WMLE and MLE approaches give the same results.3.3 Generalized and weighted least square (GLS and WLS)
The parameter estimation using least square approach is common in the statistical literature. For Pareto, loglogistic and Weibull distributions we refer the readers to [20], [18], [24], [42], [38], and [43]. Suppose are the ordered realizations from Weibull distribution with pdf given in (2.2). We can see that the following regression model holds.
(3.1) 
for where . The quantity , in the righthand side of regression model (3.1), is replaced by or , see [37], [38], and [21]. Since the sample is ordered, the dependent variable
is also ordered. Therefore the variance of dependent variable is not of the form
I, see [20]. To tackle this issue the generalized least square (GLS) technique is proposed, see [9]. The GLS estimate, i.e., is given by(3.2) 
where ,
and
for
The second type of GLS estimate, i.e.,
(3.3) 
can be constructed if we replace with as
We note that and . The weighted least square (WLS) estimate are also given by
(3.4) 
where and is a diagonal matrix whose entries are , see [20].
3.4 moment (LM)
The moments have their origin in works by Hosking (1990) and Elamir and Seheult (2003). By equating the sample moment to the population counterpart gives the moment estimate. The th moment of Weibull distribution with pdf (2.1) is given by:
where , , , and denotes the binomial coefficient , see [16]. So the first and the second moments are given by and , respectively. The first two sample moments are:
and
Now, equating and with and , respectively, the moments of and are obtained as:
and
3.5 Method of logarithmic moment (MLM)
The logmoment estimates of the shape and scale parameters of Weibull distribution with cdf (2.2) are given by (see [40], [29], and [8])
(3.5) 
and
(3.6) 
where and are the sample variance and the mean of logtransformed data, respectively. Also . It can be shown that (3.5) and (3.6) are both asymptotically unbiased and consistent, see [29].
3.6 Percentile method (PM)
The quantile of a Weibull distribution with cdf (
2.2) iswhere , see [29] and [8]. Using , one can construct percentilebased estimators for and as
(3.7) 
and
(3.8) 
respectively, where . The suggested values for are 0.15 (see [39]) and 0.31, see ([32] and [14]). Statistical tools show that percentilebased estimators are, in general, asymptotically normal and unbiased, see [40].
3.7 Method of moments (MM)
Momentbased estimators of a given population are obtained by equating the population moments to their sample counterparts and solving the resulting equations. The momentbased estimators for the Weibull distribution suffers from numerical computations, see [6]. Also, these estimators are not efficient. The th noncentral moment for the Weibull distribution is ([40]; [29]; [8]):
Equating the mean and variance ( and ) with the sample counterparts ( and ), the momentbased estimator of the shape parameter , is root of the equation
and the momentbased estimator of the scale parameter is
4 Performance comparisons
This section has two parts. In the first part, we compare the performances of estimators introduced in Section 2 and 3 through simulation. Second part devoted to an illustration in which all estimators are applied to a set of real data.
4.1 Simulation study
Here, we perform a Monte Carlo simulation to compare the performance of the statistic, MLE, WMLE, GLS1, GLS2, WLS, LM, MLM, PM, and MM. For this aim, we compare the bias and root of mean squared error (RMSE).
For computing the bias we adopt small sizes of sample including 5, 10, 30, and two levels for shape and scale parameters as: (0.5, 0.5), (2.5, 0.5), (0.5, 2.5), and (2.5, 2.5). The results after computing the bias are given in Tables 12. Also the bias of statistic, MLE, GLS1, GLS2, WLS, and LM are given for large sizes of sample including 1000 and 4000. The corresponding results are given in Tables 34 for shape and scale parameters, respectively.
For computing the RMSE, we choose the sample sizes as: 5, 10, 15, 30, 50, 100, and 200. Comparisons are performed for different levels of the shape (=0.5, 1, and 2.5) and the scale (=0.5, 2, and 5) parameters. We used a 7color scheme to distinguish between competitors through Figures 12 as follows. The brown for the statistic, green for the MLE, purple for the WMLE, dashed red for the GLS1, black for the GLS2, blue for the WLS, dashed purple for the MLM, dotted purple for the PM, solid red for the MM, and yellow solid curve for LM. The results for computing the RMSE are given in Figures 12.
4.1.1 Comparison results for the bias
According to the bias of the shape parameter estimator for small sizes 5, 10, and 30, the following conclusions can be made from Table 1.

GLS2 , WLS, and WMLE give the best performances for , , and , respectively.

WMLE shows the best performance next to the WLS and GLS1.

PM gives the worst performance.

When is small (say ), the MM gives the worst performance next to the GLS2.

When is large (say ) and , the GLS2 gives the worst performance.

WMLE outperforms the LM.

WMLE and statistic outperform the MLE.

statistic shows better performance than the MLE, MLM, MM, and PM.
The following observations can be made from Table 2 for bias of the scale parameter estimator for small sizes 5, 10, and 30.

WMLE and MLE give almost the same performances.

When is small (say ), the MM gives the worst performance.

When is small, the LM gives the best performance.

The GLS2, GLS1, and WLS show the same performances.

MLM outperforms GLS2, GLS1, and WLS.

When is large (say ), the PM shows the worst performance.

The GLS2, GLS1, and WLS outperforms the statistic for .
The following observations can be made from Tables 34 for bias of the shape parameter estimator for large sizes 1000 and 4000.

statistic gives the best performance for estimating the shape and scale parameters.

GLS1 shows the worst performance for estimating the shape and scale parameters.
We note that for bias analysis when sample sizes are large, the MLM, PM, and MM have been eliminated by competitions since the show weak performances. Also, since MLE and WMLE show the same performances, the WMLE has been removed by competitions.
4.1.2 Comparison results for RMSE
The following observations can be made from Figure 1 for RMSE of the shape parameter estimator .

The PM gives the worst performance.

When the GLS2 gives the best performance.

When the GLS2 gives the best performance.

The WGLS gives the best performance next to the GLS1.

The WMLE outperforms the LM and statistic.

The MLM shows better performance than the MLE for sample size (say ).
The following observations can be made from Figure 2 for RMSE of the scale parameter estimator .

The PM gives the worst performance.

When is small (say ), the MM gives the worst performance.

When is not small (say ), the PM gives the worst performance.

When is small (say ) and , the LM gives the best performance.
4.2 Real data illustration
Here, we apply all reviewed methods introduced in Sections 2 and 3 to a set of real data involving by lifetimes in years reported by [13, p. 17]. Data are shown in Table 5. To implement these techniques, programs have been written in R
environment, see [35]. In order to compare the performance of estimators presented in the Section 2 and 3, we employed the KolmogorovSmirnov (KS) and CramerVon Mises (CVM) distances which are given by
and
where is the sample size, ; for , is the th ordered observed value and is the distribution function of twoparameter Weibull distribution defined in (2.2). The following observations can be made from Table 6.

The WLS shows the best performance in the sense of both criteria KS and CVM.

The MLM shows the best performance in the sense of CVM criterion next to the WLS.

The PM shows the best performance in the sense of KS criterion next to the WLS.
n=5  
parameters level  
Method  (, )  (, )  (, )  (, ) 
Statistic  0.346  0.383  1.890  1.415 
MLE  0.422  0.473  2.453  1.813 
WMLE  0.299  0.340  1.764  1.309 
GLS1  0.266  0.292  1.420  1.106 
GLS2  0.249  0.255  1.261  1.025 
WLS  0.252  0.283  1.449  1.082 
LM  0.331  0.359  1.798  1.353 
MLM  0.405  0.443  2.203  1.645 
PM  1.199  1.699  8.597  7.132 
MM  0.450  0.476  1.971  1.452 
n=10  
parameters level  
Method  (, )  (, )  (, )  (, ) 
Statistic  0.181  0.189  0.865  0.713 
MLE  0.193  0.195  0.914  0.776 
WMLE  0.161  0.162  0.756  0.636 
GLS1  0.159  0.164  0.765  0.608 
GLS2  0.196  0.195  0.966  0.749 
WLS  0.148  0.146  0.697  0.572 
LM  0.193  0.191  0.781  0.649 
MLM  0.205  0.216  0.999  0.817 
PM  0.546  0.553  2.491  2.334 
MM  0.279  0.278  0.806  0.664 
n=30  
parameters level  
Method  (, )  (, )  (, )  (, ) 
Statistic  0.071  0.074  0.383  0.301 
MLE  0.078  0.079  0.410  0.332 
WMLE  0.074  0.073  0.379  0.309 
GLS1  0.085  0.084  0.425  0.342 
GLS2  0.121  0.119  0.588  0.468 
WLS  0.077  0.076  0.385  0.311 
LM  0.099  0.110  0.393  0.315 
MLM  0.095  0.096  0.493  0.400 
PM  0.184  0.191  1.045  0.867 
MM  0.150  0.149  0.386  0.313 
n=5  
parameters level  
Method  (, )  (, )  (, )  (, ) 
Statistic  0.816  2.717  0.094  0.481 
MLE  0.674  2.318  0.091  0.460 
WMLE  0.673  2.315  0.091  0.459 
GLS1  0.809  2.686  0.094  0.481 
GLS2  0.795  2.641  0.093  0.477 
WLS  0.806  2.670  0.094  0.479 
LM  0.544  1.909  0.093  0.469 
MLM  0.723  2.431  0.092  0.468 
PM  0.903  2.943  0.099  0.508 
MM  0.927  2.983  0.092  0.463 
n=10  
parameters level  
Method  (, )  (, )  (, )  (, ) 
Statistic  0.424  1.752  0.067  0.337 
MLE  0.389  1.608  0.066  0.329 
WMLE  0.387  1.596  0.066  0.329 
GLS1  0.385  1.652  0.067  0.337 
GLS2  0.421  1.744  0.067  0.336 
WLS  0.423  1.752  0.067  0.338 
LM  0.349  1.455  0.067  0.332 
MLM  0.409  1.663  0.067  0.335 
PM  0.485  1.948  0.077  0.393 
MM  0.513  2.129  0.066  0.331 
n=30  
parameters level  
Method  (, )  (, )  (, )  (, ) 
Statistic  0.215  0.866  0.038  0.186 
MLE  0.207  0.841  0.037  0.187 
WMLE  0.207  0.842  0.037  0.188 
GLS1  0.214  0.869  0.038  0.191 
GLS2  0.215  0.869  0.038  0.191 
WLS  0.215  0.874  0.038  0.191 
LM  0.201  0.830  0.037  0.188 
MLM  0.218  0.857  0.038  0.195 
PM  0.257  1.060  0.045  0.227 
MM  0.284  1.110  0.037  0.202 
n=1000  
parameters level  
Method  (, )  (, )  (, )  (, ) 
GLS1  0.005193  0.003696  0.018966  0.008460 
WLS  0.005025  0.003351  0.017895  0.006642 
GLS2  0.005167  0.002230  0.016367  0.007867 
MLE  0.004958  0.002026  0.017401  0.005958 
LM  0.004229  0.002091  0.016958  0.005620 
UStatistic  0.003600  0.001065  0.013052  0.003600 
n=4000  
parameters level  
Method  (, )  (, )  (, )  (, ) 
GLS1  0.002892  0.976e03  0.009939  0.003492 
WLS  0.002632  0.764e03  0.009614  0.002632 
GLS2  0.002260  6.088e04  0.009609  0.002260 
MLE  0.002372  9.705e04  0.009305  0.002372 
LM  0.001498  7.164e04  0.007535  0.002498 
UStatistic  0.001253  2.018e04  0.004859  0.001253 
n=1000  
parameters level  
Method  (, )  (, )  (, )  (, ) 
GLS1  0.006450  0.013542  0.006873  0.014509 
WLS  0.004087  0.011946  0.006442  0.013873 
GLS2  0.004891  0.011840  0.005900  0.011489 
MLE  0.005883  0.011700  0.005236  0.010988 
LM  0.006123  0.010598  0.006468  0.012319 
UStatistic  0.003323  0.007526  0.005378  0.009353 
n=4000  
parameters level  
Method  (, )  (, )  (, )  (, ) 
GLS1  0.001854  0.005237  1.968e03  0.002354 
WLS  0.001263  0.003971  1.879e03  0.001263 
GLS2  0.001495  0.003879  1.807e03  0.001495 
MLE  0.001382  0.004028  1.748e03  0.001382 
LM  0.001545  0.003890  1.651e03  0.001845 
UStatistic  0.000552  0.002518  1.029e03  0.000552 
30.20  36.55  25.11  39.35  27.57  25.91  31.50  29.24  18.39  16.65  21.85  24.88 
31.61  18.74  19.63  28.98  11.10  21.66  22.41  26.04  25.07  23.48  28.21  25.21 
25.12  27.76  23.47  23.51  24.39  21.93  37.63  20.32  28.17  24.66  30.13  21.42 
17.21  19.98  33.09  16.04  17.96  19.57  22.91  25.69  23.47  16.91  27.20  27.23 
Estimated parameters  goodnessoffit measures  

Method  KS  CVM  
statistic  5.1575  26.8644  0.0934  0.0591 
MLE  4.5922  26.9452  0.0920  0.0713 
WMLE  4.5141  26.9370  0.0906  0.0744 
GLS1  4.7548  26.9926  0.0971  0.0721 
GLS2  4.3035  26.9788  0.0904  0.0926 
WLS  4.7099  26.6979  0.0777  0.0482 
LM  4.9512  26.9055  0.0939  0.0609 
MLM  5.3119  26.7771  0.0889  0.0529 
PM  5.9767  25.8461  0.0867  0.0622 
MM  4.9150  26.9169  0.0942  0.0621 
5 Conclusion
We have introduced statistics for shape and scale parameters of twoparameter Weibull distribution. Asymptotic normality and consistency of the new estimators have been proved. Furthermore, a comprehensive Monte Carlo study have been carried out to compare the performance of the known estimators of the twoparameter Weibull distribution parameters. Since different estimators may appeal different users for different levels of sample size and parameters levels, a list of comparisons have been made in the paper for choosing desired estimator. Many facts can be concluded from this study, among them our results are the followings.

for small sizes of samples the weighted least square (WLS) approach gives the best performance in the sense of bias.

shape estimator based on method of weighted least square (WLS) gives the best performance root of mean squared error (RMSE).

shape estimator based on method of percentile gives the worst performance in terms of RMSE.

shape and scale estimators based on statistic show the best performances in the sense of bias for large sample sizes.

shape and scale estimators based on generalized least square typeI (GLS1) approach show the worst performances in the sense of bias for large sample sizes.
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