Simulation comparisons of point estimation methods in the 2-parameter weibull distribution

Microelectron. Reliab., Vol. 19, pp. 333 336 ,~O,Pergamon Press Ltd 1979. Printed in Great Britain

0026 2714/79/0801~)333 ~02.00/0

SIMULATION COMPARISONS OF POINT ESTIMATION METHODS IN THE 2-PARAMETER WEIBULL DISTRIBUTION SHOJI KUCHII, NAOTO KAIO and SHUNJI OSAKI Department of Industrial Engineering, Hiroshima University, Hiroshima 730, Japan

(Received fl~r publication 23 March 1979) Weibull distributions are widely used for the analysis of reliability failure data. The following three estimation methods in the 2-parameter Weibull distribution are well-known; a method of maximum-likelihood estimation, a method of coefficient of variation, and a method of Weibull probability paper. By simulation we discuss which of these methods is better and conclude that a method of Weibull probability paper is relatively better both for complete samples and Type I censored samples. Abstract

1. I N T R O D U C T I O N

Weibull distribution is widely applicable for reliability failure data analysis. In general, Weibull distribution has three parameters, and estimating these parameters is of great importance in reliability engineering. In this paper, we assume that the location parameter is 0 or prespecified and restrict ourselves to the 2-parameter Weibull distribution. The 2parameter Weibull distribution is given by F(x) = 1 - exp (-x"/to),

(x > 0),

obtain the failure time data xl, xz,..., x,. Cohen [2] gave the method of MLE by solving the following nonlinear equations: n

x'~ l n x , / . ~ .~ - 1 / r h = ~ lnxi/n, i=1

2. ESTIMATIONBASED ON COMPLETE SAMPLES

(2)

i=1

{o = ~ ~x~/n.

(3)

i=1

Equation (2) can be solved by using a method of Newton-Raphson. Cohen [2] also gave the method of CV. Noting the coefficient of variation

(1)

where m is the shape parameter and to is the scale parameter. In the past, several methods of estimating m and to were proposed. In this paper we consider the following three well-known methods; a method of Maximum Likelihood Estimation (MLE), a method of Coefficient of Variation (CV), and a method of Weibull Probability Paper (WPP). Using these methods, we can estimate m and to both from complete and Type I (fixed time) censored samples by simulation, and compare the three methods of MLE, CV and W P P in the case of complete samples and the two methods of MLE and W P P in the case of type I censored samples. In the earlier contribution, Gross and Lurie [3] discussed Monte Carlo comparisons of the 2-parameter Weibull distribution where they applied the method of MLE and the two methods of Bain and Antle [1]. In this paper we compare the three methods of MLE, CV and WPP, where W P P is fairly easy to apply by reliability engineers in practice. In this paper, n is the sample size, xi is the ith failure time, x~ is a prespecified time at which the test terminates, k is a number of samples failed up to x 7', rh and/'0 are estimates of m and to, and F(') is a gamma function.

t=l

CV = x/F(1 + 2/m) - F2(1 + 1/m)/F(1 + 1/m), (4) we can estimate rn from the sample CV (i.e. the sample standard deviation/the sample mean) directly if a table between m and CV is provided. The scale parameter to can be obtained by to = b , / r ( 1 + 1/,h)] "~,

(5)

where ~t is the sample mean. W P P is easily applicable for estimating m and to by plotting the failure data. That is, noting the fact that In In [1 - F(x)] - 1 = rn In x + In to,

(6)

we can draw a figure with the axis of abscissae In x and the axis of ordinates In In [1 - F ( x ) ] - 1 in which the Weibull distribution is linear with slope m. Readers who are not familiar with W P P should consult an introductory book on W P P (for example, see [4-6]). 3, E S T I M A T I O N

BASED ON

TYPE I CENSORED

SAMPLES

The test for type I censoring terminates at xr. Therefore, the number of censored samples is k ( < n). For this sample, Cohen [2] gave the method of MLE. That is, by solving the following nonlinear equations:

Consider the complete samples of the sample size n. Executing the test until all samples fail, we can

k

Z,x~lnx,/Z,.~333

1/rh = E lnx,/k, i=1

(7)

334

SHOJI KUCHII, NAOTO KAIO a n d SHUNJI OSAKI

(8)

~o = E*¥~/k, where k

E*@ ln\'i =

E xi"~ In xi + (n - k ) . @ In x.r,

(9)

i=1 k

~ * x'[' = Z ?fi + (n - k)x~..

(10)

i=1

F o r this s a m p l e , we h a v e to u s e W P P by n o t i n g t h a t the failure law never c h a n g e s after t h e test t e r m i n a tion x.,,. In general, it is n o t so g o o d to e s t i m a t e m a n d to with little d a t a until x r , so we correct m a n d to by u s i n g t h e m e t h o d of M L E . T h e detailed d i s c u s s i o n of this c o r r e c t i o n will be f o u n d in reference [5]. W e a p p l y this c o r r e c t i o n in this paper.

4. SIMULATION

T o c o m p a r e the three m e t h o d s for c o m p l e t e d a t a a n d t h e t w o m e t h o d s for c e n s o r e d data, we exccute t h e following : (i) W e specify t h e p a r a m e t e r m a n d to: the p a r a m eters m are 0.5 (a d e c r e a s i n g h a z a r d rateh 1.0 ta c o n s t a n t h a z a r d rate) a n d 5.0 (an increasing h a z a r d rate), a n d the p a r a m e t e r to is 1.0. T h e s a m p l e sizes n are 5, 10 a n d 15. T h e prespecified t i m e x j is 1.0. (ii) By specifying t h e p a r a m e t e r s m a n d to, we can o b t a i n t h e W e i b u l l r a n d o m variable (r.v.) by the following e q u a t i o n : xi = t~:"( - I n yi) L'm,

w h e r e y~ is u n i f o r m l y d i s t r i b u t e d o n t h e interval (0, 1). F o r each c o m b i n a t i o n , we o b t a i n the W e i b u l l r.v.'s over 100 times.

Table 1. Simulation results of the Weibull 2-parameters based on 100 random samples tsubscript 1 denotes the estimate for the method of MLE, 2 for CV, and 3 for WPP) Bias n

Am I

Am 2

Am 3

At~

At~

5 10 15

0.221 0.073 0.066

0.465 0.240 0.184

0.248 0.t59 - 0.051

0.187 0.090 0.217

0.521 0.319 0.337

-0.049 0.091 0.231

{m = 0.5, to = 1.0)

5 10 15

0.392 0.148 0.100

0.482 0.215 0.142

0.180 - 0.081 0.051

-0.038 0.031 0.015

0.031 0.086 0.072

0.005 0.096 -0.119

(m = 1.0, to = 1.0)

5 10 15

1.885 0.713 0.492

1.864 0.743 0.454

-0.409 0.217 - 0.012

-0.081 -0.002 0.00002

0.033 0.063 0.084

0.031 -0.028 0.00008

(m = 5.0, to = 1.01

At~

Standard deviation

,

m~D

m~D

5 10 15

0.578 0.232 0.144

5 l0 15 5 10 15

(1 l)

m3D

th.s,

tS.sD

3 tO,SI)

0.529 0.204 0.160

0.394 0.457 0.175

1,136 0.517 0.836

1.480 0.543 0.641

0.325 0.546 0.966

[m = 0.5, to = 1.01

1.077 0.421 0.292

1.154 0.425 0.308

0.726 0.305 0.493

0.410 0.261 0.310

0.622 0.340 0.400

0.325 0.449 0.305

(m = 1.0, to = 1.0)

3.300 2.183 1.476

3.385 2.321 1.506

1.598 1.576 1.307

0.084 0.055 0.059

0.545 0.395 0.419

0.329 0.221 0.295

(m = 5.0, to = 1.0)

Rms error

5 l0 15

0.617 0.243 0.158

0.702 0.314 0.243

0.453 0.470 0.176

1.146 0.523 0.860

1.562 0.627 0.721

0.317 0.535 0.959

(m = 0.5, to = 1.0)

5 10 15

1.140 0.444 0.308

1.245 0.475 0.338

0.722 0.305 0.478

0.41/) 0.261 0.309

0.620 0.349 0.404

0.313 0.443 0.316

( m = 1.0, to = 1.0)

5 I0 15

3.786 2.286 1.549

3.850 2.426 1.566

1.584 1.524 1.255

0.085 0.055 0.059

0.543 0.398 0.425

0.317 0.213 0.284

(m = 5.0, to = 1.01

Point estimation methods in the 2-parameter Weibull distribution

335

Table 2. Simulation results of the Weibull 2-parameters based on 100 random samples (subscript 1 denotes the estimate for the method of M L E and 3 for W P P Bias #7

A/H I

Am 3

Ato1

At 3

5 10 15

- 0.084 -0.200 - 0.150

0.149 - 0.019 0.010

- 0.438 -0.322 4.650

- 0.112 0.264 - 0.056

(m = 0.5, to = 1.0)

5 10 15

-0.181 -0.058 - 0.021

0.230 0.055 -0.035

- 0.190 -0.068 0.004

-0.062 0.125 0.556

(m= 1.0, to = 1.0)

5 10 15

- 0.318 -0.814 - 0.072

- 0.012 - 0.813 - 0.481

0.049 - 0.023 - 0.008

0.009 0.057 -0.008

(m = 5.0, to = 1.0)

Standard deviation

5 10 15

1.167 0.667 0.534

0.345 0.274 0.175

0.596 0.578 48.370

0.410 0.847 0.315

(m = 0.5, to = 1.0)

5 10 15

2.044 1.053 0.784

0.695 0.574 0.388

0.510 0.411 0.489

0.446 0.851 7.120

(,,7 =

5 10 15

8.672 6.281 4.057

1.917 1.264 1.162

0.137 0.100 0.115

0.451 0.298 0.375

(m = 5.0, to = 1.0)

1.0, to =

1.0)

Rms error ,7

m~Ms

m3Ms

to,1 RMS

5 10 15

1.164 0.693 0.552

0.363 0.265 0.168

0.737 0.659 48.352

0.410 0.858 0.308

(m = 0.5, to = 1.0)

5 10 15

2.042 1.050 0.781

0.706 0.556 0.375

0.542 0.415 0.486

0.433 0.830 6.884

(m = 1.0, to = 1.0)

5 l0 15

8.634 6.302 4.037

1.828 1.458 1.216

0.144 0.103 0.114

0.430 0.291 0.361

(,,7 = 5.0, to =

(iii) U s i n g t h e W e i b u l l r.v.'s as failure data, we e s t i m a t e m a n d to by t h e t h r e e m e t h o d s for c o m p l e t e d a t a a n d t h e t w o m e t h o d s for c e n s o r e d data. (iv) Finally, for e a c h m e t h o d a n d 100 e s t i m a t o r s of m a n d to, we o b t a i n the Bias (B) (the s a m p l e m e a n t h e t r u e value), t h e S t a n d a r d D e v i a t i o n (SD), a n d the R o o t M e a n S q u a r e E r r o r (RMSE). T h e results B, S D a n d R M S E for e a c h c o m b i n a t i o n o f m, to a n d n are s h o w n in T a b l e 1 for c o m p l e t e s a m p l e s a n d T a b l e 2 for c e n s o r e d s a m p l e s .

5. D I S C U S S I O N

P r i o r to d i s c u s s i o n , we s h o u l d p a y a t t e n t i o n to the i m p o r t a n c e o f e a c h p a r a m e t e r . It is n o t e d t h a t t h e p a r a m e t e r m is m u c h m o r e i m p o r t a n t t h a n the MR 19:4

t)

3 t0.RMS

1.0)

p a r a m e t e r to since m c a n c h a r a c t e r i z e t h e s h a p e of t h e d i s t r i b u t i o n a n d to c a n o n l y c h a r a c t e r i z e t h e scale of time. T h i s is t h e r e a s o n w h y we specify to = 1.0 for s i m u l a t i o n . H o w e v e r , we c h a n g e m = 0.5, 1.0 a n d 5.0 for s i m u l a t i o n . W e c a n c o n c l u d e t h e following: (i) F o r c o m p l e t e data, t h e m e t h o d of W P P is best o f all in e s t i m a t i n g t h e p a r a m e t e r m. I n e s t i m a t i n g the p a r a m e t e r to, t h e m e t h o d o f M L E is best, b u t t h e m e t h o d o f W P P is better t h a n t h a t o f M L E w h e n t h e s a m p l e size n = 5. T h e m e t h o d o f C V is n o t g o o d in e s t i m a t i n g m a n d to. W h e n t h e s a m p l e size n = 5, n o m e t h o d s are g o o d in e s t i m a t i n g m a n d to. Therefore, all m e t h o d s n e e d at least 10 d a t a with e n o u g h confidence. (ii) F o r c e n s o r e d data, the m e t h o d o f W P P is better

336

SHOJI KUCHII, NAOrO Kalo and SH{INJI OSAK/

than that of M L E in estimating m. This is obvious from SD and RMSE in Table 2. In estimating to, the method of M L E and the method of W P P are equally good, but, as with complete data, the method of W P P is better than that of M L E f o r , = 5. (iii) The conclusions (i) and {it} above are described neglecting time and labor for estimation. Let us consider time and labor for each method. The method of M L E can be executed by solving the nonlinear equations (2) and (3) for complete data ((7) and (8) for censored data) using computers. The method of CV can be executed by using a table of CV, m and to. The method of W P P can be executed by the WPP. where we assume that we are familiar with this method. If we assume that there are computers available, a program of M L E provided, and a table used in the method of CV, the easiest is the method of M L E and the next the method of CV. (iv) Let us focus on the bias of the estimate {0 = 4.650 for m = 0.5, to = 1.0 and n = 15 for censored data in Table 2. This undesirable result may be caused by the use of a bad initial value in the method of N e w t o n - R a p h s o n . In this paper, the initial value is the estimate th by the method of CV. If the value of parameter m can be forecasted relatively properly in advance, the method of MLE is very useful. But if it

can not be forecasted in advance, the method of MLE can not be used in estimating m and to. Let us describe the method of WPP. In this method, we draw a straight line after plotting the thilure data on WPP. Therefore, even if the failure data are the same, those estimates arc not equal. This method has a small possibility of obtaining a quite wrong estimate by misuses of WPP. According to the conclusions above, we should recommend the method of W P P if we are familiar with WPP. But even this method is not good for the sample size n = 5. Therefore, it seems that the n u m b e r of failure data required is at least 10 for WPP.

REFERENCES

1. L.J. Bain and C. E Antle, Techmnnetrics 9, 621 (1967). 2. A. C. Cohen, Techmmletrics 7,579 (1965). 3. A. J. Gross and D. Ltlrie, IEEE Trans. Reliab. R-26, 356 (1977). 4. G. J. Hahn and W. B. Nelson, Statistical Models in E,gineeri,~l. Wiley, New York (1967). 5. JUSE (ed.), Analysis ,?] Reliclbility Data: How to Use JUSE Weibull Probability Paper, (in Japanese), Tokyo, JUSE {1967~. 6. N. R. Mann, R. E. Schafer and N. D. Singpurwalla, Methods/or Statistical Analysis ~?f Reliability and Lift" Dcmt, Wiley, New York (19741.