Abstract
The Weibull model is one of the widely used distributions in reliability engineering. For the parameter estimation of the Weibull model, there are several existing methods. The method of the maximum likelihood estimation among others is preferred because of its attractive statistical properties. However, for the case of the three-parameter Weibull model, the method of the maximum likelihood estimation has several drawbacks. To avoid the drawbacks, the method using the sample correlation from the Weibull plot is recently suggested. In this paper, we provide the justification for using this new method by showing that the location estimate of the three-parameter Weibull model exists in a bounded interval.
1. Introduction
There are several existing methods for the location parameter estimation of the three-parameter Weibull model. One can use the method of the maximum likelihood estimation (MLE) which is preferred by many statisticians due to its attractive statistical properties. However, it is well known that the MLE method has several drawbacks for the case of the three-parameter Weibull model. For example, the global maximum can reach infinity at the singularity and this singularity can result in local maxima of the likelihood function when it is numerically computed. For more details, see Barnard [1] and Smith and Naylor [2].
This MLE method has a convergence issue and it can also have an unfeasible value so that the location estimate of the three-parameter Weibull model can be greater than the minimum value of the observations [3, 4]. Cheng and Amin [5], Cheng and Iles [6], and Liu et al. [7] also pointed out that the likelihood function has the unbounded likelihood problem and the location parameter tends to approach the smallest observation. Huzurbazar [8] also showed that no stationary point can yield a consistent estimator, which results in no local maximum. Thus, whether a global or a local maximum is sought, the MLE is bound to fail.
To avoid the above problems, several authors, including Gumbel [3] and Vogel and Kroll [4], suggest the method of estimating the parameters using an estimate for the minimum drought. However, in order to estimate the location parameter of the three-parameter Weibull model using the methods in Gumbel [3] or Vogel and Kroll [4], one has to use the special tables provided by Gumbel [3] which are available only for limited cases. Sirvanci and Yang [9] also recommend to estimate the location parameter with . However, it is reported that the performance of these methods is not satisfactory. For more details, the reader is referred to Park [10]. It should be noted that this Gumbel method is improved by Park [10]. He proposes to estimate only the location parameter using the ordinary Gumbel method and estimate the other shape and scale parameters using the MLE of the two-parameter Weibull model and shows that the parameter estimates are noticeably improved by the proposed method.
Park [10] also proposed a method which maximizes the sample correlation function from the Weibull plot to estimate the location parameter of the three-parameter Weibull model. Comparing the sample correlations, the p-values, and the Anderson-Darling test statistics, he shows that his method outperforms the afore-mentioned existing methods. His method is conceptually easy to understand, simple to use and convenient for practitioners. However, the existence of the location estimate is not yet proved.
In this paper, we show that the location parameter estimate of the three-parameter Weibull model should exist in the bounded interval. Thus, unlike the MLE case, the method by Park [10] does not suffer from nonconvergence, singularity, or infeasibility issues when we calculate the location parameter numerically.
2. Weibull Plot and Correlation Coefficient from the Plot
In this section, we briefly review the Weibull plot [11] and present the sample correlation coefficient from the Weibull plot. The Weibull distribution has the respective probability density function and cumulative distribution function (CDF) We let for convenience. Then we haveIt is immediate from (2) that we have It is observed that the plot of versus is ideally a straight line with slope and intercept if the data are from the Weibull distribution. The widely used Weibull probability paper in engineering reliability is based on this idea.
With real data, we need to estimate to draw the Weibull plot. It should be noted that the estimation of is often called the plotting position in the statistics literature. Let be the order statistics from the smallest to the largest. There are several methods of estimating in the literature. Let be the empirical CDF value at for convenience. In practice, the plotting positions such as are widely used due to Blom [12] and Wilk and Gnanadesikan [13].
The Weibull plot is constructed by plotting on the vertical axis and on the horizontal axis. It should be noted that the straightness of the Weibull plot can also be used to assess the goodness-of-fit of the Weibull model. See Park [10] along with the weibullness R package by Park [14]. The measure of the straightness in the Weibull plot can be evaluated by calculating the sample correlation coefficient of the paired points, We let and for convenience. Then the sample correlation coefficient from the Weibull plot is defined aswhere and .
3. Existence of the Location Parameter Estimate of the Three-Parameter Weibull Model
In many reliability applications, failures do not occur below a certain limit which is also known as a failure-free life (FFL) parameter in the engineering literature [15]. The three-parameter Weibull model with this FFL parameter has been widely used to describe the reliability of surface-mount assemblies due to wear-out failures, etc. For more details, see Wong [16], Clech et al. [17], Drapella [18], Mitchell et al. [19], and Lam et al. [20].
It is thus reasonable to consider a lower limit to the Weibull model. This Weibull model is called the three-parameter Weibull with its CDF given by where . This lower limit is often called a location parameter.
Replacing by in (6), we can obtain the sample correlation as a function of where , and . For more details, see Section 5 of Park [10].
It is quite reasonable to impose a condition that for practical applications. Then the estimate of is given by
Lemma 1. The function has the limit as
Proof. In the following, we use the Bachmann-Landau’s big notation. See de Bruijin [21] for more details. That is, if and are defined on the domain , then means that for all where is a constant.
For convenience, let and then . It is easily seen that as (that is, ), we haveandIt is immediate upon substituting (11) and (12) into (8) that as , we haveWe have as . Thus, we let for convenience and we then have as . Rewriting (13) using , we haveBy dividing both the numerator and denominator of (14) by , we haveWhen taking the limit of (15) as (that is, as ), we havewhich completes the proof.
Lemma 2. As , we havewhere is the sample covariance between and and a series of observations of and is given by and for .
Proof. Differentiating (8) with respect to , we havewhere and alsoAgain, we let for convenience so that we have and . Then we can rewrite , , , and as a function of as follows: and alsowhere It should be noted that , , , and do not include .
After some tedious algebra, when , we haveSince as , we have . The sample covariance between and is given byIt is easily shown thatwhere is the Bachmann-Landau’s little -notation in de Bruijin [21] for example. That is, implies that as .
Substituting , (26) and (27) into (25), we haveSimilarly, we can rewrite asSubstituting (28) and (29) into (18), we have which completes the proof.
Theorem 3. The global maximum of exists on with .
Proof. The function is continuous on . Considering the result of Lemma 1, we define Then is continuous on the closed bounded interval . Thus, the function has a global maximum and a global minimum on , due to Theorem 4.28 in Apostol [22]. Note that is a singularity point. Thus, it suffices to show that the global maximum of is not obtained at .
Since and , it is easily seen that the term in Lemma 2 is always positive. It is immediate from the L’Hôpital’s rule that we have Thus, using these with Lemma 2, we have as (that is, as ). Since is differentiable on , it is easily seen that cannot be a maximum from the intermediate value property of derivatives. For more details, see Lemma 6.2.11 of Bartle and Sherbert [23]. Thus, the global maximum exists on .
It is worthwhile to mention the lower bound of the sample correlation coefficient from the Weibull plot. It is well known that the sample correlation coefficient should be in in general. However, in the Weibull plot, the data and plotting positions are ordered and thus the sample correlation coefficient should be positive. Also, it should be noted that with the order statistics restriction, the sample correlation coefficient is bounded below by which is the best lower bound due to Hwang and Hu [24].
Finally, after the location parameter is obtained, we can estimate the other shape and scale parameters by several existing methods. We recommend the MLE method of the two-parameter Weibull for the estimation of shape and scale. For more details, see Section 5 of Park [10]. Unlike the MLE of the three-parameter Weibull, the MLE of the two-parameter Weibull guarantees the existence and uniqueness due to Farnum and Booth [25].
4. An Illustrative Example
The data in this example, published in Bilikam et al. [26], are the numbers of miles to failure of a type of vehicle. This data set has since then been often used for illustration of a three-parameter Weibull distribution [6, 20].
We can estimate the location parameter by maximizing the correlation function in (8) or solving in (18) as shown in Figure 1 which results in . As recommended earlier, we estimated the other shape and scale parameters using the MLE of the two-parameter Weibull.
In order to examine the performance of the proposed method, we compare it with other existing methods in Lam et al. [20]. They estimated the parameters using the constrained MLE (McLE) approach and the minimum SSE approach. The results are summarized in Table 1 with the corresponding , p value for Weibullness, and log-likelihood. Note that the p values for Weibullness testing were obtained using the weibullness R package by Park [14]. The results show that the proposed method outperforms the existing methods.
Data Availability
The article has no data used for study. So, it belongs to case 12 in the statement of the provided URL: that is, “12. No data were used to support this study”.
Conflicts of Interest
The author declares that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (No. NRF-2017R1A2B4004169). The author also wishes to dedicate this work to the memory and honor of Professor Byung Ho Lee in the Department of Nuclear Engineering at Seoul National University. The author’s interests in mathematics were formed under the strong influence of Professor Lee who passed away peacefully in Seoul on July 9, 2001.