MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2018/6056975 6056975 Research Article A Note on the Existence of the Location Parameter Estimate of the Three-Parameter Weibull Model Using the Weibull Plot http://orcid.org/0000-0002-2208-3498 Park Chanseok 1 Onieva Enrique Applied Statistics Laboratory Department of Industrial Engineering Pusan National University Busan 46241 Republic of Korea pusan.ac.kr 2018 9102018 2018 02 06 2018 16 07 2018 09 08 2018 9102018 2018 Copyright © 2018 Chanseok Park. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

Ministry of Education NRF-2017R1A2B4004169
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 μ=min(x1,x2,,xn) and this singularity can result in local maxima of the likelihood function when it is numerically computed. For more details, see Barnard  and Smith and Naylor .

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 , Cheng and Iles , and Liu et al.  also pointed out that the likelihood function has the unbounded likelihood problem and the location parameter tends to approach the smallest observation. Huzurbazar  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  and Vogel and Kroll , 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  or Vogel and Kroll , one has to use the special tables provided by Gumbel  which are available only for limited cases. Sirvanci and Yang  also recommend to estimate the location parameter with x(1)-1/n. However, it is reported that the performance of these methods is not satisfactory. For more details, the reader is referred to Park . It should be noted that this Gumbel method is improved by Park . 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  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  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  and present the sample correlation coefficient from the Weibull plot. The Weibull distribution has the respective probability density function and cumulative distribution function (CDF) (1)fx=κθxθκ-1e-x/θκandFx=1-e-x/θκ. We let p=F(xp) for convenience. Then we have(2)log1-p=-xpθκ.It is immediate from (2) that we have(3)log-log1-p=κlogxp-κlogθ. It is observed that the plot of log{-log(1-p)} versus logxp is ideally a straight line with slope κ and intercept -κlogθ 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 p=F(xp) to draw the Weibull plot. It should be noted that the estimation of F(x(i)) is often called the plotting position in the statistics literature. Let x(1),x(2),,x(n) be the order statistics from the smallest to the largest. There are several methods of estimating F(x(i)) in the literature. Let pi=F^(x(i)) be the empirical CDF value at x(i) for convenience. In practice, the plotting positions such as (4)pi=i-3/8n+1/4forn10andpi=i-1/2nforn11are widely used due to Blom  and Wilk and Gnanadesikan .

The Weibull plot is constructed by plotting log{-log(1-pi)} on the vertical axis and logx(i) 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  along with the weibullness R package by Park . The measure of the straightness in the Weibull plot can be evaluated by calculating the sample correlation coefficient of the paired points,(5)logxi,log-log1-pi. We let ui=logx(i) and vi=log{-log(1-pi)} for convenience. Then the sample correlation coefficient from the Weibull plot is defined as(6)R=i=1nui-u¯vi-v¯i=1nui-u¯2·i=1nvi-v¯21/2,where u¯=ui/n and v¯=vi/n.

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 . 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 , Clech et al. , Drapella , Mitchell et al. , and Lam et al. .

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 (7)Fx=1-exp-x-μθκ, where x>μ. This lower limit μ is often called a location parameter.

Replacing x(i) by x(i)-μ in (6), we can obtain the sample correlation as a function of μ(8)Rμ=i=1nui-u¯vi-v¯i=1nui-u¯2·i=1nvi-v¯21/2,where 0μ<x(1), ui=log(x(i)-μ) and u¯=ui/n. For more details, see Section 5 of Park .

It is quite reasonable to impose a condition that 0μ<x(1) for practical applications. Then the estimate of μ is given by(9)μ^=argmax0μ<x1Rμ.

Lemma 1.

The function R(μ) has the limit as(10)limμx1-Rμ=v¯-v11-1/ni=1nvi-v¯21/2.

Proof.

In the following, we use the Bachmann-Landau’s big O(·) notation. See de Bruijin  for more details. That is, if f(·) and g(·) are defined on the domain D, then f(x)=O(g(x)) means that fxKgx for all xD where K is a constant.

For convenience, let δ=x(1)-μ and then u1=logδ. It is easily seen that as δ0+ (that is, μx(1)-), we have(11)i=1nui-u¯vi-v¯=i=1nuivi-v¯=u1vi-v¯+i=2nuivi-v¯=u1v1-v¯+O1and(12)i=1nui-u¯2=i=1nui2-1ni=1nui2=u12+i=2nui2-1nu1+i=2nui2=1-1nu12+u1·O1.

It is immediate upon substituting (11) and (12) into (8) that as δ0+, we have(13)Rμ=u1v1-v¯+O11-1/nu12+u1·O11/2·i=1nvi-v¯21/2.We have u1- as δ0+. Thus, we let u1=-u1 for convenience and we then have u1 as δ0+. Rewriting (13) using u1, we have(14)Rμ=u1v¯-v1+O11-1/nu12+u1·O11/2·i=1nvi-v¯21/2.By dividing both the numerator and denominator of (14) by u1, we have(15)Rμ=v1-v¯+O1/u11-1/n+O1/u11/2·i=1nvi-v¯21/2.When taking the limit of (15) as u1 (that is, as δ0+), we have(16)limδ0+Rμ=v¯-v11-1/ni=1nvi-v¯21/2,which completes the proof.

Lemma 2.

As δ0+, we have(17)dRμdμ=-nn-22n-1·covU,Vi=1nvi-v¯21/2·1δlog2δ+o1δlog2δ,where cov(U,V) is the sample covariance between U and V and a series of n observations of U and V is given by ui=log(x(i)-μ) and vi=log{-log(1-pi)} for i=1,2,,n.

Proof.

Differentiating (8) with respect to μ, we have(18)dRμdμ=A·B-C·DEwhere (19)A=i=1nwi-w¯vi-v¯,B=i=1nui-u¯2,C=i=1nui-u¯vi-v¯,D=i=1nui-u¯wi-w¯,E=B3/2·i=1nvi-v¯21/2,ui=logxi-μ,and also(20)wi=-1xi-μ.

Again, we let δ=x(1)-μ for convenience so that we have u1=logδ and w1=(-1)/δ. Then we can rewrite A, B, C, and D as a function of δ as follows: (21)Aδ=-v1-v¯1δ+K1,Bδ=1-1nlog2δ-2ni=2nuilogδ+K2,Cδ=v1-v¯logδ+K3,Dδ=-1-1nlogδδ+1ni=2nui1δ-1ni=2nwilogδ+K4,and also(22)Eδ=Bδ3/2·i=1nvi-v¯21/2,where (23)K1=i=2nwivi-v¯,K2=i=2nui2-i=2nui2n,K3=i=2nuivi-v¯,(24)K4=i=2nuiwi-1ni=2nuii=2nwi.It should be noted that K1, K2, K3, and K4 do not include δ.

After some tedious algebra, when δ0+, we have(25)Aδ·Bδ-Cδ·Dδ=n-1ni=2nuivi-1n-1i=2nuii=2nvi1δlogδ+O1δ+Olog2δ+Ologδ.

Since logδ<0 as δ0+, we have logδ=-logδ. The sample covariance between U and V is given by(26)covU,V=1n-2i=2nuivi-1n-1i=2nuii=2nvi.It is easily shown that(27)O1δ+Olog2δ+Ologδ=o1δlogδ,where o(·) is the Bachmann-Landau’s little o-notation in de Bruijin  for example. That is, f(x)=o(g(x)) implies that f(x)/g(x)0 as xc.

Substituting logδ=-logδ, (26) and (27) into (25), we have(28)Aδ·Bδ-Cδ·Dδ=-n-1n-2n·covU,V·1δlogδ+o1δlogδ.Similarly, we can rewrite E(δ) as(29)Eδ=1-1nlog2δ+Ologδ3/2i=1nvi-v¯21/2.

Substituting (28) and (29) into (18), we have (30)dRμdμ=-n-1n-2/n·covU,V·1/δlogδ+o1/δlogδ1-1/nlog2δ+Ologδ3/2i=1nvi-v¯21/2=-nn-22n-1·covU,Vi=1nvi-v¯21/2·1δlog2δ+o1δlog2δ, which completes the proof.

Theorem 3.

The global maximum of R(μ) exists on [0,x(1)) with n3.

Proof.

The function R(μ) is continuous on [0,x(1)). Considering the result of Lemma 1, we define(31)Rx1=v¯-v11-1/ni=1nvi-v¯21/2. Then R(μ) is continuous on the closed bounded interval [0,x(1)]. Thus, the function R(μ) has a global maximum and a global minimum on [0,x(1)], due to Theorem 4.28 in Apostol . Note that μ=x(1) is a singularity point. Thus, it suffices to show that the global maximum of R(μ) is not obtained at μ=x(1).

Since u1<u2<<un and v1<v2<<vn, it is easily seen that the term cov(U,V) in Lemma 2 is always positive. It is immediate from the L’Hôpital’s rule that we have(32)limδ0+1δlog2δ=. Thus, using these with Lemma 2, we have(33)dRμdμ- as δ0+ (that is, as μx(1)-). Since R(μ) is differentiable on (0,x(1)), it is easily seen that R(x(1)) cannot be a maximum from the intermediate value property of derivatives. For more details, see Lemma 6.2.11 of Bartle and Sherbert . Thus, the global maximum exists on [0,x(1)).

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 [-1,1] 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 1/(n-1) which is the best lower bound due to Hwang and Hu .

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 . 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 .

4. An Illustrative Example

The data in this example, published in Bilikam et al. , 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 dR(μ)/dμ=0 in (18) as shown in Figure 1 which results in μ^=139.33. As recommended earlier, we estimated the other shape and scale parameters using the MLE of the two-parameter Weibull.

Correlation function, R(μ), and dR(μ)/dμ.

In order to examine the performance of the proposed method, we compare it with other existing methods in Lam et al. . 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 R(μ^), p value for Weibullness, and log-likelihood. Note that the p values for Weibullness testing were obtained using the weibullness R package by Park . The results show that the proposed method outperforms the existing methods.

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

Method μ ^ κ ^ θ ^ R ( μ ^ ) p-value log-likelihood
Proposed 139.33 1.29174 1032.2 0.98135 0.6066 - 156.4056
McLE 123.46 1.15000 1081.7 0.98075 0.5870 - 156.8671
Min. SSE 125.23 1.15000 1081.7 0.98086 0.5906 - 156.8455
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.

Barnard G. A. The use of the likelihood function in statistical practice 1 Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 1965 Berkeley, Calif, USA University of California Press 27 40 Smith R. L. Naylor J. C. A comparison of maximum likelihood and Bayesian estimators for the three-parameter Weibull distribution Journal of the Royal Statistical Society: Series C (Applied Statistics) 1987 36 3 358 369 10.2307/2347795 MR918854 Gumbel E. Statistical forecast of droughts Hydrological Sciences Journal 1963 8 1 5 23 2-s2.0-0008220262 10.1080/02626666309493293 Vogel R. M. Kroll C. N. Low-flow frequency analysis using probability-plot correlation coefficients Journal of Water Resources Planning and Management 1989 115 3 338 357 2-s2.0-0024405005 10.1061/(ASCE)0733-9496(1989)115:3(338) Cheng R. C. Amin N. A. Estimating parameters in continuous univariate distributions with a shifted origin Journal of the Royal Statistical Society Series B (Methodological) 1983 45 3 394 403 MR737651 Cheng R. C. Iles T. C. Embedded models in three-parameter distributions and their estimation Journal of the Royal Statistical Society Series B (Methodological) 1990 52 1 135 149 MR1049306 Liu S. Wu H. Meeker W. Q. Understanding and addressing the unbounded "likelihood" problem The American Statistician 2015 69 3 191 200 10.1080/00031305.2014.1003968 MR3391639 Huzurbazar V. S. The likelihood equation, consistency and the maxima of the likelihood function Annals of Eugenics 1948 14 185 200 MR0028000 Zbl0033.07703 Sirvanci M. Yang G. Estimation of the weibull parameters under type I censoring Journal of the American Statistical Association 1984 79 385 183 187 10.1080/01621459.1984.10477082 MR742867 Park C. Weibullness test and parameter estimation of the three-parameter weibull model using the sample correlation coefficient International Journal of Industrial Engineering : Theory, Applications and Practice 2017 24 4 376 391 2-s2.0-85039447037 Nelson W. Applied Life Data Analysis 1982 New York, NY, USA John Wiley and Sons MR646615 Blom G. Statistical Estimates and Transformed Beta-Variables 1958 New York, NY, USA John Wiley & Sons MR0095553 Wilk M. B. Gnanadesikan R. Probability plotting methods for the analysis of data Biometrika 1968 55 1 1 17 2-s2.0-0014254846 Park C. Weibullness: Goodness-Of-Fit Test for Weibull Distribution (R Package) 2018 https://cran.r-project.org/web/packages/weibullness 10.15372/AUT20180114 Reliability HotWire Reliability Basics – Location Parameter of The Weibull Distribution May 2002 15 http://www.weibull.com/hotwire/issue15/relbasics15.htm Wong K. L. When reliable means failure free Quality and Reliability Engineering International 1987 3 3 145 145 2-s2.0-84984028976 10.1002/qre.4680030302 Clech J.-P. M. Noctor D. M. Manock J. C. Lynott G. W. Bader F. E. Surface mount assembly failure statistics and failure free time Proceedings of the 1994 IEEE 44th Electronic Components & Technology Conference May 1994 487 497 2-s2.0-0027961266 Drapella A. An improved failure-free time estimation method Quality and Reliability Engineering International 1999 15 3 235 238 2-s2.0-0346498326 10.1002/(SICI)1099-1638(199905/06)15:3<235::AID-QRE212>3.3.CO;2-J Mitchell D. Zahn B. Carson F. Board level thermal cycle reliability and solder joint fatigue life predictions of multiple Stacked Die chip scale package configurations Proceedings of the 54th Electronic Components and Technology Conference June 2004 USA 699 703 2-s2.0-10444242503 Lam S.-W. Halim T. Muthusamy K. Models with failure-free life—applied review and extensions IEEE Transactions on Device and Materials Reliability 2010 10 2 263 270 10.1109/TDMR.2010.2045758 2-s2.0-77953273783 de Bruijn N. G. Asymptotic Methods in Analysis 1981 3rd New York, NY, USA Dover Publications MR671583 Apostol T. M. Mathematical Analysis 1974 2nd Reading, Mass, USA Addison-Wesley Publication MR0344384 Bartle R. G. Sherbert D. R. Introduction to Real Analysis 1982 4th New York, NY, USA John Wiley and Sons MR669021 Hwang T. Y. Hu C. Y. The best lower bound of sample correlation coefficient with ordered restriction Statistics and Probability Letters 1994 19 3 195 198 10.1016/0167-7152(94)90104-X MR1278650 Farnum N. R. Booth P. Uniqueness of maximum likelihood estimators of the 2-parameter weibull distribution IEEE Transactions on Reliability 1997 46 4 523 525 2-s2.0-0031382496 10.1109/24.693786 Bilikam J. E. k-sample maximum likelihood ratio test for change of weibull shape parameter IEEE Transactions on Reliability 1979 R-28 1 47 50 10.1109/TR.1979.5220471 2-s2.0-0018454733