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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.

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

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 [

To avoid the above problems, several authors, including Gumbel [

Park [

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 [

In this section, we briefly review the Weibull plot [

With real data, we need to estimate

The Weibull plot is constructed by plotting

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 [

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

Replacing

It is quite reasonable to impose a condition that

The function

In the following, we use the Bachmann-Landau’s big

For convenience, let

It is immediate upon substituting (

As

Differentiating (

Again, we let

After some tedious algebra, when

Since

Substituting

Substituting (

The global maximum of

The function

Since

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

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 [

The data in this example, published in Bilikam et al. [

We can estimate the location parameter by maximizing the correlation function in (

Correlation function,

In order to examine the performance of the proposed method, we compare it with other existing methods in Lam et al. [

Parameter estimates, correlations, p value, and log-likelihood under consideration.

Method | | | | | p-value | log-likelihood |
---|---|---|---|---|---|---|

Proposed | 139.33 | 1.29174 | 1032.2 | 0.98135 | 0.6066 | |

McLE | 123.46 | 1.15000 | 1081.7 | 0.98075 | 0.5870 | |

Min. SSE | 125.23 | 1.15000 | 1081.7 | 0.98086 | 0.5906 | |

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”.

The author declares that they have no conflicts of interest.

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.