FOR ESTIMATING THE WEIBULL DISTRIBUTION PARAMETERS – A COMPARATIVE STUDY

The Weibull distribution is one of the widely used distributions in engineering practice. It is named after Walodi Weibull (1887– 1979), who popularized its use in the theory of reliability, especially for metallurgical failure models. Moreover, the Weibull distribution is useful for description of the life time of the machine components, for description of mechanical properties of the materials as fatigue of materials and strength of materials.


Introduction
The Weibull distribution is one of the widely used distributions in engineering practice. It is named after Walodi Weibull , who popularized its use in the theory of reliability, especially for metallurgical failure models. Moreover, the Weibull distribution is useful for description of the life time of the machine components, for description of mechanical properties of the materials as fatigue of materials and strength of materials.
We consider the two parameter Weibull distribution. The probability density function of the Weibull distribution with parameters c Ͼ 0 and δ Ͼ 0, abbreviated W(c, δ), is given by where x Ͼ 0, c is the shape parameter and δ is the scale parameter.
The cumulative distribution function of the Weibull distribution is . (1) The mean μ and the variance σ 2 of the Weibull distribution are The failure rate function of the Weibull distribution is given by . Fig. 1 shows the effect of the shape parameter c on the density function for different values c and δ ϭ 1. Fig. 2. shows the effect of the scale parameter δ on the density function for different values δ and c ϭ 2.
In this paper we study the performance of the methods for estimating the Weibull distribution parameters c and δ. The estimates of the parameters c and δ can be obtained in more ways. The commonly used methods are the maximum likelihood method (MLM), the method of moments (MOM), the least square method (LSM) and the weighted least square method (WLSM). The MLM is the most popular for its efficiency and good properties, but the calculation is complicated. The estimates of the parameters can be obtained only numerically. Several authors have studied and compared performance of the methods for estimating the Weibull distribution parameters, e. g. Bergman [1], Chu and Ke [2], Faucher and Tyson [3], Lu, Chen and Wu [4], Trustrum and Jayatilaka [5], Wu, Zhou and Li [6], Zerda [7].
Here, we consider the least square method (LSM) and the weighted least square method (WLSM), each with three estimators of the cumulative distribution function F(x). In engineering practice these methods are commonly used due to their simplicity. The estimates of the parameters can be calculated easily by the closedform formula. The methods are compared using the Monte Carlo simulation. The comparison is based on the root mean square error (RMSE) and the sample size n. Based on the simulation study we recommend the methods which have better performance.

Estimation of the parameters of the Weibull distribution
In this section we introduce the methods for estimating the Weibull distribution parameters. The estimates of the parameters c and δ denote c^and δ^, respectively. Let X 1 , X 2 , …, X n be a random sample of size n from the Weibull distribution W(c, δ) and let x 1 , x 2 , …, x n be a realization of a random sample.

Least square method
Now, the cumulative distribution function (1) will be transformed to a linear function. From (1) by two logarithmic calculations we obtain Let Y ϭ ln[Ϫln(1 Ϫ F(x))], X ϭ c ln x, β 1 ϭ c and β 0 ϭ ϭ Ϫ c ln δ. Then the equation (2) can be written as Now let X (1) , X (2) , …, X (n) be the order statistics of X 1 , X 2 , …, X n and let x (1) Ͻ x (2) Ͻ … Ͻ x (n) be observed ordered observations. To estimate the values of the cumulative distribution function F(x) we can use the folloving methods (the mean rank) , (the median rank) (5) . .
The estimates β^0 and β^1 of the regression parameters β 0 and β 1 minimize the function Therefore, the estimates β^0 and β^1 of the parameters β 0 and β 1 are given by , .
The estimate δ^of the parameter δ is given by .

Weighted least square method
We suppose that the estimates β^0 and β^1 of the regression parameters β 0 and β 1 minimize the function  where w i is the weight factor, i ϭ 1, 2, … n. Therefore, the estimates β^0 and β^1 of the parameters β 0 and β 1 and are given by .
Then the estimate δ^of the parameter δ is given by .
In this paper we use the weight factor proposed by Bergman [1] , i ϭ 1, 2, …, n.

Monte Carlo simulation
We generate by the Monte Carlo simulation the random samples from the Weibull distribution and compare the performance of the methods for estimating the Weibull distribution parameters mentioned above. In simulation study we consider the LSM and the WLSM each with three methods for estimating the cumulative distribution function. Thus together we compare six methods. We denote the methods with the estimators of the cumulative distribution function (3) as LSM_1, WLSM_1, with (4) as LSM_2, WLSM_2 and with (5) as LSM_3, WLSM_3.
We consider sample sizes n ϭ 5 to 100, δ ϭ 1 and several values of the parameters c ϭ 0.5, 1.5, 2.5 representing decreasing, increasing and concave, increasing and convex failure rate functions respectively. All possible combinations of the parameters c, δ and sample sizes n are considered. For each combination c, δ and n we generate by the Monte Carlo simulation N ϭ 5000 random samples from the Weibull distribution. For each of six methods we obtain 5000 estimates c^1, c^2, …, c^5 000 of the parameter c and 5000 estimates δ^1, δ^2, …, δ^5 000 of the parameter δ. Then we compute for each method the sample means c , δ and the sample variances

Comparison of the methods
In this section we summarize the performance of the methods for estimating the Weibull distribution parameters. The methods are compared in terms of the RMSE and sample size n.
The results of the comparison for selected sample sizes n ϭ 5, 10, 30, 50, 100 are summarized in Tables 1, 2 It is evident that the RMSE of the least square method is in many cases much larger than the RMSE of the weighted least square method for the case studies in this paper. The weight factor improves the accuracy of the estimation the Weibull distribution parameters. When gets larger the RMSE of all methods tends to be smaller.
For the sample size n Ն 10 and for c ϭ 1.5, 2.5 the comparison shows that the RMSE of the LSM_1 is in many cases much larger than the other methods. The LSM_2 and the LSM_3 are comparable methods in many cases in terms of the RMSE. The RMSE of the LSM_3 is slightly larger than the RMSE of the LSM_2. The RMSE of the WLSM_2 is larger than the WLSM_1 and the WLSM_3. The RMSE of the WLSM_3 is slightly larger than the RMSE of the WLSM_1, both methods are comparable for n Ն 40. In general, the WLSM_1 provides the best estimates of the Weibull distribution parameters than the other methods in terms of the RMSE.
For the small sample size 5 Յ n Ͻ 10 and for c ϭ 1.5, 2.5 the comparison shows that in general the RMSE of the WLSM_1 outperforms the other methods. The RMSE of the LSM_1 is only slightly larger than the RMSE of the WLSM_1. The LSM_1 provides good results in these cases.
For the sample size n Ն 10 and for c ϭ 0.5 the comparison shows that in general the RMSE of the WLSM_2 outperforms the other methods. The RMSE of the WLSM_3 is slightly larger than the RMSE of the WLSM_2. The RMSE of the LSM_2 and the   For the sample size 5 Յ n Ͻ 10 and for c ϭ 0.5 the RMSE of the LSM_2 is only slightly larger than the RMSE of the WLSM_2. The RMSE of the WLSM_3 and the LSM_3 follow. The RMSE of the LSM_1 is much larger than the RMSE of the other methods.

Concluding remarks
In this paper we compared the performance of the methods for estimating the Weibull distribution parameters in terms of the RMSE and sample size n. The comparison was based on the Monte Carlo simulation. The comparison shows that the weight factor improves the accuracy of the estimation the Weibull distribution parameters. The WLSM_1 performs the best in terms of the RMSE than the other methods for majority cases studied in this paper and for all sample sizes, the WLSM_3 follows as the second good choice. Except the case when c ϭ 0.5, the WLSM_2 performs to be the best for all the sample sizes than the other methods. The good choice is in this case for middle and large sample sizes the WLSM_3.
The advantages of these recommended methods are: simple derivation, easy calculation of the estimates of the parameters by the closed-form formula. And so from this point of view these methods are very useful for engineering practice.