Department of Mathematics

wtrnumber2013-1

Control Charts for Monitoring Weibull Distribution

Chuanping Sze and Francis Pascual

March 2013

Postal address: Department of Mathematics, Washington State University, Pullman, WA 99164-3113Voice: 509-335-3926 • Fax: 509-335-1188 • Email: info@math.wsu.edu • URL: http://www.sci.wsu.edu/math

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Control Charts for Monitoring Weibull Distribution

Chuanping Sze and Francis Pascual, Department of Mathematics

Washington State University, Pullman WA 99164

ABSTRACT Tables of constants are provided for computing the control limits for monitoring processes described by the Weibull distribution. The smallest extreme value distribution (SEV) is used in the simulation process, and we obtain results for Weibull by taking exponentials. We monitor the quantiles of the Weibull distribution and the SEV mean. When monitoring the quantile, we considered the cases when is known and is unknown. Monitoring the SEV mean is done using the 0.4296 quantile estimate or the sample mean of SEV samples. Tables and control limits equations are given for each case. Average run length (ARL) charts are constructed for the problem of detecting the shift of a percentile of a Weibull distribution. An illustrative example is presented concerning the tensile strength of carbon fibers from Nichols and Padgett (2005).

1. Introduction In this paper, we provide tables of constants (quantiles of pivotal quantities) that practitioners can use for control charts with the Weibull distribution. We consider the two-parameter Weibull distribution with the probability density function

xx

xf exp)(1

, x>0(1.1)

where is the scale parameter and is the shape parameter. Because of its flexible shape and ability to model a wide range of failure rates, Weibull has been used successfully in many applications as a purely empirical model, for example, in biomedical applications and in modeling ball bearing, relay and material strength failures. The Weibull distribution includes as special cases known distributions. If X has a Weibull distribution with , it is identical to the exponential distribution. When , it is the Rayleigh distribution. If X is Weibull, then Y= ln(X) has an SEV distribution. This relationship between the Weibull and SEV is similar to the relationship between normal and lognormal. Hence, any statistical results we get from the Weibull distribution can be applied easily to SEV and vice versa. The Weibull distribution is equivalent to the SEV with parameter log(and where log is natural logarithm. The probability density function of SEV is written in the form

yy

yf expexp1

)( -∞ < y < ∞ (1.2)

2

The mean and variance of SEV are, respectively, )(YE (1.3)

6

)(22YV

where =0.5772 is Euler’s constant. Since the parameters of the SEV are location-scale parameters, it is more convenient to develop control chart techniques with SEV. The 100pth percentile or p quantile of the SEV distribution is )]1log(log[ pyp . The 100pth

percentile of the Weibull distribution is exp (yp). Note that when p=0.4296, yp = E (Y), that is, the SEV mean is also its 0.4296 quantile. Limits for control charts are obtained from the distributions of pivotal quantities. In Section 2, we introduce the basic definition of control limits and ARL. We consider two cases when simulating the tables of the percentiles of pivotal quantities. In Section 3, we monitor the quantile of the SEV when is known and is unknown. In Section 4, with known, we monitor the mean of SEV using the maximum likelihood estimate (MLE) of the 0.4296 quantile or, alternatively, the sample mean. We discuss chart performance using ARL in Section 5. We also provide one-sided and two-sided charts to see how the ARL changes with varying sample size and how fast a shift is detected. In Section 6, we use the carbon fiber tensile strength data from Nichols and Padgett (2005) and construct one and two-sided control charts for monitoring

the 0.01 quantile. We also compare the performance between charts using y and the MLE of E(Y)

when is known. Ramalhoto and Morais (1999) study the design and performance of the Shewhart control charts for monitoring the scale parameter of the three-parameter Weibull distribution. They assume that the Weibull shape and threshold parameters are known and use a pivotal quantity for to compute control limits. They also perform an ARL study to assess how fast the charts detect changes in . Nichols and Pagett (2005) also study control charts for the Weibull distribution by simulation. However, they fail to take the advantage of relevant pivotal quantity that they have to perform the simulation process every time data are gathered to generate the control limits. In addition, Ryan (2000), Jobe and Vardeman (1998), Montgomery (2004), and Wetherill and Brown (1991) discuss control charts in quality control almost exclusively for the normal distribution.

2. Probability limits and Average Run Length

2.1 Probability Limit

Suppose a quantity is being monitored using a sample statistic ^ . Assume that the process is

stable and let 1

^^

, where 0<<0.5, be the and (1-) quantiles of the distribution ^ ,

respectively.

Let CL, LCL, and UCL denote the center line, lower control limit and upper control limit, respectively. 100(1-2)% probability limits for monitoring are given by

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1

^

^

UCL

LCL

CL

^LCL is a 100(1-)% lower probability limit, while 1

^

UCL is a 100(1-)% upper

probability limit. A process consists of regular samples of size n. ^ is computed for each sample,

and an out of control (OOC) signal occurs when ^ <LCL or

^ >UCL.

2.2 ARL ARL is the average number of samples until the first OOC point occurs. Let L be the run length which is the number of periods until the first OOC signal. Then the ARL is the expected value of L and, because L is a geometric random variable, ARL = 1/P(OOC). To find the probability limits, we first solve for in 1/2RLfor two-sided charts or in 1/RL for one-sided charts For example, if we want to get all-OK(stable process) ARL of 370, we have =0.00135 for two-sided or =0.0027 for one-sided. Table 1 shows some values of and the corresponding all-OK ARL. Table 1: and all-OK ARL values

0.0005 0.001 0.00135 0.0027 0.005 0.01

ARL 2-sided= 1/2 1000 500 370 185 100 50

1-sided= 1/ 2000 1000 740 370 200 100 The practitioner decides whether to use one-sided or two-sided charts and picks the all-OK ARL from which is determined.

3. Monitor the quantile of the Weibull distribution

In this section, we provide tables of constants that practitioners can use for Weibull control

charts. Suppose Y ~ SEV. If the SEV p quantile is yp, then the Weibull p quantile is exp(yp), and

we can monitor either quantile. Furthermore, the exponential of the SEV control limits can be

used as Weibull control limits. We first tabulate percentiles of pivotal quantities when is

unknown and when is known. Note that when is unknown, we estimate both and and we

only need one table when finding the control limits because there exists a pivotal quantity for yp

that is free of μ and . However, when is known, we will need several tables based on the

specified value because the pivotal quantity for yp depends on . In other words, we need to do

the simulation every time for a different value. We give details in the following subsections.

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3.1 Monitor the quantile when is unknown.

Suppose that is unknown. We use ))1log(log(

^^^ p

V for the pivotal quantity for μ and

where ^ and

^ are the MLEs of and respectively From Bain and Engelhardt (1992),

^

and ^ are pivotal quantities for and that is, they have distributions free of unknown

parameters. It follows that ^

V is also a pivotal quantity for μ and . The 100pth percentile of the

SEV is estimated by

)]1log(log[^^^

py p (2.1)

and the 100pth percentile for Weibull distribution is )exp(^^

pp yx .

3.1.1 Simulations

The following steps are used to estimate the percentiles of the pivotal quantity: 1. From a stable process, generate a random sample of size n from Weibull with shape

and scale WEI (, where exp(and 2. Find the MLE of

^ and ^ .

3. Repeat steps 1-2 a large number of times T. We should have T values of ^ and

^ .

4. Estimate the 100pth percentile of SEV by )]1log(log[^^^

py p .

5. Compute the pivotal quantity: py

V

^^

.

6. Estimate the quantile v of ^

V for = 0.0005, 0.001, 0.00135, 0.0027, 0.005, 0.01, 0.99,

0.995, 0.9973, 0.99865, 0.999, and 0.9995.

We generated T=500,000 random samples of size n=4, 5, 6, 7, 8, 9, 10. Note that we can assign

and because ^

V is pivotal. When is unknown, we need to perform simulations for different values of p. For example, if we want to monitor the mean, we use p=0.4296; if we want to monitor the median, we use p=0.5, etc. In Appendix I, we list tables for p=0.01, 0.005, 0.1, 0.4296, 0.5, 0.6321. The Weibull 0.6321 quantile is =eμ.

To check the stability of the quantiles of ^

V estimated from the simulations, we plot the estimated quantile against the total number of simulations. The plots in Appendix II is for n = 4 when is unknown. We can see that at T=100,000, the curves appear to stabilize which signifies stable estimates or estimates with very low variance. Hence, T=500,000 is more than sufficient for n≥4.

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Section 3.1.2 Probability Limits

To find the two-sided 100(1-2)% probability limits, we isolate )]1log(log[^^^

py p in

1

^

21 vy

vP p

so that

1

^

21 vyvP p. Then, we plot

py^

with

1

))1log(log(

vUCL

vLCL

pCL

(2.2)

where v values are found in the tables of Appendix I.

The process is said to be in control when py

^

falls between the UCL and LCL. Otherwise, the

process is OOC. If the standards and are not known, we replace and with

MLEs^^

, based on prior samples assuming an in-control process. Using the relationship

between Weibull and SEV, the 100(1-2% probability limits for )exp(^^

pp yx are

)exp( vLCL and )exp( 1 vUCL with center line )))1log(log(exp( pCL .

For one sided 100(1-)% probability charts,

vLCL is a one sided lower control limit (2.3)

1vUCL is a one sided upper control limit (2.4)

If a point falls above the UCL or below the LCL, the process is said to be out of control. Otherwise, the process is said to be in control.

3.2 Monitor the quantile when is known.

Suppose that is known from prior information. The 100pth percentile of the SEV distribution is estimated by the MLE

)]1log(log[

^^

py p (2.5)

Hence, the 100pth percentile for Weibull distribution has MLE )exp(^^

pp yx .

From Brain and Engelhardt (1992), ^

is a pivotal quantity for . This implies that

pyW^^

is pivotal for when is known.

3.2.1 Simulations

The simulation procedure is similar to that in Section 3.1.1, except in step 2 where we do not

have to find the MLE of ^ because is known and

^ is replaced with elsewhere.

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Because ^

is pivotal for , we can assume =0 for our simulations. The following steps are

used to estimate the percentiles of ^

W : 1. From stable process, generate n observations from WEI (exp(. 2. If =1/ is known, the MLE of μ is

n

i

xn 1

1^ 1log (2.6)

3. Repeat steps 1-2 a large number of times T. We should have T values of ^ .

4. Estimate the 100pth percentile of the SEV distribution using

)]1log(log[^^

py p .

5. Find the pivotal quantity: pyW^^

.

6. Compute the quantile w of^

W for = 0.0005, 0.001, 0.00135, 0.0027, 0.005, 0.01,

0.99, 0.995, 0.9973, 0.99865, 0.999, and 0.9995.

Since the distribution of ^

W depends on and n, we consider 0.5, 1, 1.5, and 2 and n = 4, 5, 6, 7, 8, 9, 10. Again, T=500,000 is adequate for determining stable (i.e. small variance) estimate of

w . The results are shown in Appendix III. Notice that we simulated only for the 0.5 quantile. It

is because the 0.5 and p quantile differ by )]}1log(log[))]5.0log(log({[5.0 pyyp (2.7)

Thus, w values for another p are obtained by adding (2.7) to the w values in the Tables B1-B4.

For example, in Table B4 (Appendix III) with n=4 and =0.0005 we have 578299.5w .

If we want to monitor the SEV mean or the p=0.4296 quantile, let p=0.4296 in (2.7) and add –5.578299 to obtain 99985.5w .

3.2.2 Probability Limits As mentioned above, ^

is pivotal for . Hence, the probability limits for the pivotal quantity

pyW^^

can be calculated by isolating ^

py in

1

^^

21 wyWwP p (2.8)

Hence, the 100(1-2)% two-sided control limits for py^

are

1

))1log(log(

wUCL

wLCL

pCL

(2.9)

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If the standard is not known, replace it with the MLE ^ based on prior samples assuming an

in-control process. The 100(1-2% two-sided probability limits for px^

=exp(^

py ) are

)exp( wLCL and )exp( 1 wUCL with ))]1log(log(exp[ pCL

100(1-)% SEV one-sided limits are given by

wLCL for one sided lower control limit (2.10)

1wUCL for one sided upper control limit (2.11)

Take exponentials to obtain limits for Weibull.

4. Monitor the mean of the SEV distribution

4.1 Monitor the mean using the MLE

In the previous section, we discussed how to monitor the p quantile of the SEV or, equivalently,

the Weibull distribution. Since monitoring the SEV mean is the same as monitoring the 0.4296

quantile of either SEV or Weibull, we can simply use Tables A4 and B1-B4 with p = 0.4296 to

monitor the mean of SEV if we use^ and/or

^ . The pivotal quantity is

^^^

V when is

unknown and ^^

W when is known. For monitoring the SEV mean 4296.0y ,

the control limits of Sections 2.1 and 2.2 become:

(i) When is unknown,

1

5772.0

vUCL

vLCL

CL (3.1)

(ii) When is known,

1

5772.0

wUCL

wLCL

CL (3.2)

We can compute the control limits using Table A4 in Appendix I for unknown. For known,

add the constant (2.7) with p=0.4296 to values in Tables B1-B4 to derive appropriate constants

for computing control limits.

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4.2 Monitor the mean using the sample mean when is known

Suppose Y is SEV with location μ and known scale . An alternative to using the MLE ^ - to

monitor the SEV mean is to use the sample mean y because )(yE . Note that ^

U = y

is pivotal for So it suffices to let and to estimate the quantile of ^

U by simulation.

The following steps are used to compute the percentiles of )(^ y

U :

1. From an in-control process, generate n observations from WEI(, where exp(and 2. Find the mean y based on the n observations and compute

yU

^

.

3. Repeat steps 1-2 a large of times T.

4. Compute the quantile u of ^

U based on T values of ^

U for = 0.0005, 0.001, 0.00135,

0.0027, 0.005, 0.01, 0.99, 0.995, 0.9973, 0.99865, 0.999, and 0.9995. We generate T=500,000 random samples of size n= 4, 5, 6, 7, 8, 9, 10. The results are given in Table 2.

Table 2 Quantiles of )(^ y

U

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -3.3427851 -3.0088191 -2.7364958 -2.5695896 -2.4020134 -2.2783465 -2.1740657

0.0010 -3.1233805 -2.8176579 -2.5766133 -2.4058664 -2.2666685 -2.1490363 -2.0553671

0.00135 -3.0251365 -2.7348354 -2.5027405 -2.3382345 -2.1998603 -2.0930153 -2.0086182

0.0027 -2.7996759 -2.5306489 -2.3236362 -2.1692552 -2.0556217 -1.9632122 -1.8844210

0.0050 -2.5843925 -2.3553426 -2.1667165 -2.0274336 -1.9232048 -1.8426854 -1.7653132

0.01 -2.3399903 -2.1363632 -1.9811287 -1.8576874 -1.7669577 -1.6954580 -1.6314431

0.99 0.6610439 0.5582045 0.4732000 0.4071637 0.3503619 0.3026617 0.2631834

0.995 0.7622628 0.6535874 0.5633896 0.4916677 0.4317036 0.3789236 0.3404710

0.9973 0.8422290 0.7291895 0.6347943 0.5600745 0.5000745 0.4411480 0.4014530

0.99865 0.9316362 0.8063098 0.7079003 0.6315932 0.5688921 0.5062692 0.4575382

0.999 0.9707879 0.8397927 0.7373685 0.6629332 0.5958446 0.5288839 0.4825122

0.9995 1.0425379 0.9097128 0.8027029 0.7268452 0.6453780 0.5891650 0.5442317

To find the 100(1-2)% two-sided probability limits, we isolate y in

121 u

yuP

9

to obtain

1uUCL

uLCL

CL

(3.3)

for monitoring y .

If the standards and are not known, we replace and with estimates based on prior

samples assuming an in-control process. The 100(1-)% one sided probability limits for y are:

uLCL for one sided lower control limit.

1uUCL for one sided upper control limit.

4.3 “3” limit for y The SEV distribution’s mean and standard deviation are shown in (1.3). Hence, the “3” limits

for y arenyy 6

33

. Note that the “3” limits do not necessarily cover 99.7%

of all values of y because Y is not normal and sample sizes may be too small for the Central Limit Theorem to be adequate. 3 limits are closely related to the idea of ARL. In particular, 99.7% corresponds to an all-OK ARL of 370)997.01(1 for the normal distribution. This is not true here. We estimate

n

YPn

YPp6

3

6

3

and the ARL using 1/p. In this section, we estimate the constant m so that n

m

6

will

have an all-OK ARL approximately equal to 370. We estimate ARL in the following way:

1. Choose sample size n with =0, =1, and =0.5772. 2. Generate a random sample { nyyy ,..,, 21 } from SEV (. Compute the sample

average, y .

3. Determine if y falls outside the control limits

n

m

6

4. Repeat step 2 and 3 a large number of times, T.

5. Count the number of times N out of T that y falls outside the limits. The estimated ARL

is NT .

This procedure is done for sample sizes n=4, 5, 6, 7, 8, 9, 10 and m = 1, 1.5, 2, 2.5, 3, 3.1, 3.2, 3.3, 3.4, 3.5, 4. The results are given in Table 3.

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Table 3: Estimated in-control ARL for monitoring y with control limits n

m

6

for the SEV distribution.

n=4 n=5 n=6 n=7 n=8 n=9 n=10 m=1 3.3160 3.2234 3.2663 3.1943 3.2501 3.2117 3.1758

m=1.5 8.1899 8.0488 7.9352 7.8505 7.8269 7.6499 7.6670

m=2 23.2942 23.0560 22.8089 22.7252 22.7255 22.2776 22.1887

m=2.5 62.4241 65.7865 67.2104 68.5544 69.4314 70.0565 72.3680

m=3 156.4156 172.3722 184.9014 197.7326 209.6261 220.2534 221.9824

m=3.1 186.4370 206.7739 228.4273 242.7023 258.5726 273.2072 283.2133

m=3.2 222.5172 250.7131 275.2727 304.4280 320.2734 345.2510 362.3936 m=3.3 271.7492 312.2501 338.4171 365.8132 405.7135 429.5998 454.3955 m=3.4 324.2117 374.6649 426.3955 457.7909 518.7033 540.2367 574.9990

m=3.5 392.9365 460.1705 522.1974 575.4789 629.6129 680.7155 717.2218

m=4 1040.0585 1275.0456 1510.5131 1732.2402 1942.7158 2160.9364 2367.0300

Note from Table 3 that any ARL value below 370 indicates that the control limits may be too narrow; on the other hand, any ARL value above 370 indicates that the control limits may be too wide. We use interpolation to estimate the constant m that results in an ARL close to 370. o get the all-OK ARL close to 370 with n=4, we might use m=3.for the control limits. For n=5, we might choose m=3.39. For n=6, we might use m=3.35, and so on. Based on Table 3, it is clear that “” limits (m=3) for sample size between 4 and 10 give the ARL values below 370 which suggests that these limits are too narrow for small sample sizes. Moreover, from Table 3 with m=3 and increasing sample size, ARL gets larger. Therefore, the “” limits are appropriate only for sample size much larger than 10.

5. Average run length (ARL) properties of the charts.

Here, we discuss how quickly the control charts presented above detect a shift in the percentile of SEV or Weibull distribution.

5.1 Detecting a shift in the Weibull or SEV quantile When a process is on target, one can expect a long periods between signals. However, when the process mean is off target by a substantial amount, the detection of that change will be faster. Let X have a WEI( distribution. If changes to then will change to log (). Note that

if { nyyy ,..,, 21 } is a random sample from SEV (, then {y1+log(y2+log(yn +log()} is a random sample from SEV(log(), Let zp=log(-log(1-p)). This suggests that

if pzQ^^

1

^ is the MLE of the SEV p quantile based on { nyyy ,..,, 21 }, then

pzQ^^

2

^

)log( is the MLE based on {y1+log(y2+log(yn +log()}.

The probability of out of control can be calculated using

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)log()log(

1)( 1

^^

vz

vPOOCP p (4.1)

where v and v1- are contants discussed in Section 3.1.2.

If )log( is known, we can use simulations to approximate P(OOC). A shift of means

that has changed by a factor of exp(). We study how varying sample size will affect the ARL in the presence of a shift For ARL computations, Ramalhoto and Morais (1997) considered 0<<2 and 0.5<Thus,

)2log(2 . We use the latter range in the following procedure to estimate ARL given the

value when both μ and are estimated: 1. Use the simulation results we already have from previous sections and compute:

))1log(log(^^

pyp for known.

))1log(log(^^^

pyp for unknown.

We should have T= 500,000 values for each py^

in each case.

2. Use the following rules to determine OOC points:

i. One-sided lower: OOC when py^

< y

ii. One-sided Upper: OOC when py^

> 1y

iii. Two-sided: OOC when py^

< y or py^

> 1y

3. An estimate of P(OOC) is the number OOC signals from Step 2 divided by 500,000. 4. Estimate ARL by 1/ P(OOC).

Consider monitoring the p=0.4296 quantile. The graphs of ARL versus for all-OK ARL=100 (=0.01 for one-sided and =0.005 for two-sided) are given in Figures 7 - 15 in Appendix IV.

5.2 Sample size and ARL

Suppose we are given and an all-OK ARL = 100. We want to know what sample size is needed

to detect with OOC ARL = 50. Or, suppose we want to quickly detect a decrease of 10% in .

All these questions can be addressed by the ARL charts created in Section 5.1 and given in

Appendix IV.

Let be unknown. Consider a one-sided lower ARL chart for monitoring a process that has shifted by =-0.5, i.e. has changed by a factor of exp(-/2). It is desired to detect this shift on average by 50 periods. Figure 7 plots versus ARL for different sample sizes and an all-OK (=0) ARL of 100. Draw a vertical line through =-0.5 and ARL = 50. We see that the intersection point is above all the curves. This means all sample sizes 4n are sufficient, i.e.,

12

they yield OOC ARL <50. Furthermore, if =-0.5 but now the OOC ARL changes to 15, then 8n will be sufficient. Use the same approach for the one-sided upper ARL chart where we

have a positive value of . For two-sided charts, if we are given |we will choose the sample size whose ARL curve falls below the desired ARL on both sides of κ=0. For example, in Figure 9 with OOC ARL = 20 and || = 0.5, sample size n=5 or 6 is sufficient.

For known, we can use the ARL charts generated using either^ or y . See Figures 10-12

and 13-15 for =1. To detect a 100a% decrease in we use 1- a) and = )1log( a

. To

detect a 100a% increase in we use 1+a) and = )1log( a

. Since we conducted the ARL

chart for equal to 1, the value of indicates the amount change in . Suppose decreases by

0.5, i.e. =-0.5, Figure 10 plots verses ARL for monitoring the SEV mean using ^ with

sample size n=4. Figure 13 is for charts using y with sample size n=4. ARL values indicate that

the ^ charts detect shifts faster than y charts.

6. Application 6.1 Monitoring the 0.01 quantile of the Weibull distribution In this section, we use the carbon fiber tensile strength data from Nichols and Padgett (2005) as an example. They have the objective of monitoring Weibull quantiles. However, they fail to take the advantage of relevant pivotal quantiles that they have to perform Monte Carlo simulations each time data are gathered to generate control limits. With our approach, the practitioner can use our tables to compute control limits without performing simulations. Table 4 shows the WEI( data from Nichols and Padgett (2005).

Table 4 Carbon fiber tensile strength data from WEI(

Breaking stress of carbon fibers (GPa) 3.70 2.74 2.73 2.50 3.60 3.11 3.27 2.87 1.47 3.11 4.42 2.41 3.19 3.22 1.69 3.28 3.09 1.87 3.15 4.90 3.75 2.43 2.95 2.97 3.39 2.96 2.53 2.67 2.93 3.22 3.39 2.81 4.20 3.33 2.55 3.31 3.31 2.85 2.56 3.56 3.15 2.35 2.55 2.59 2.38 2.81 2.77 2.17 2.83 1.92

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Suppose we do not know the standards. We assume that the first 10 periods are stable and obtain

MLEs for and using these periods. We obtain 20.3^ and 78.4

^ . Thus ^ and

^ Using results in Section 3.1.2, the two-sided control limits for monitoring the Weibull 0.01 quantile are

]208333.016315.1exp[

]208333.016315.1exp[

))]01.01log(log(208333.016315.1exp[

1

vUCL

vLCL

CL

(2.2)

For an all-OK ARL of 370, we need =0.00135 and In particular, if n=5, use Table A1 from Appendix I to get 05772045.1100135.0 v and 27147116.099865.0 v . Hence, CL=1.224813,

LCL=0.3175420, and UCL=3.027339. Table 5 shows the next 10 samples of Weibull data from a shifted process with and

The MLEs of the Weibull 0.01 quantile, ))01.01log(log(exp(^^^

01.0 x , are also given.

Table 5 New Weibull samples with Subgroup Breaking stress of carbon fibers

(GPa) Weibull 0.01 quantile:

))01.01log(log(exp(^^^

01.0 x 1 1.41 3.68 2.97 1.36 0.98 0.2784472 2 2.76 4.91 3.68 1.84 1.59 0.5856856 3 3.19 1.57 0.81 5.56 1.73 0.1739993 4 1.59 2.00 1.22 1.12 1.71 0.7054395 5 2.17 1.17 5.08 2.48 1.18 0.2231035 6 3.51 2.17 1.69 1.25 4.38 0.4444314 7 1.84 0.39 3.68 2.48 0.85 0.1120410 8 1.61 2.79 4.70 2.03 1.80 0.4456006 9 1.57 1.08 2.03 1.61 2.12 0.7901942 10 1.89 2.88 2.82 2.05 3.65 1.0785641

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Figure 1: Two-sided control chart for monitoring the Weibull 0.01 quantile using

))90.0log(log(exp(^^^

01.0 x with an all-OK ARL of 370.

Figure 1 is the control chart with all-OK ARL of 370 for monitoring the 0.01 quantile for the data in Table 5. The control limits in Nichols and Padgett (2005) are incorrect. Their control limits should be the same as ours. In Figure 1, the first subgroup falls below the LCL. Hence, the process is immediately characterized as out of control. We can also see a general downward shift from the CL. We also construct the one-sided lower control chart for this data set using (2.3). Thus,

36712624.100027.0 v and LCL=0.3668586. The chart is shown in Figure 2. From Figure 2, we

can also see that the process is out of control.

15

Figure 2: One-sided control chart for monitoring the Weibull 0.01 quantile

using ))01.01log(log(exp(^^^

01.0 x with an all-OK ARL of 370.

6.2 Monitoring the mean of the SEV distribution Consider the strength data from Nichols and Padgett (2005) to monitor the SEV mean or, equivalently, the Weibull 0.4296 quantile. In this case, we can monitor the mean using either ^

or y when is known. We perform new simulations, since we don’t have a table for =1/4.8 like those in Section 2.2 with p=0.4296. The results are shown in Table 6 below:

Table 6: Quantiles of ^^

W when =0.208333 and p=0.4296

n=5 0.0005 -0.54825570 0.001 -0.51910035 0.00135 -0.50283515 0.0027 -0.46896470 0.005 -0.43844799 0.01 -0.40356201 0.99 0.05563039 0.995 0.07299378 0.9973 0.08619378 0.99865 0.10058787 0.999 0.10629327 0.9995 0.11933828

16

For an all-OK ARL=100, with ^ , we have 43844799.0005.0 w and 07299378.0995.0 w from

Table 6, and we apply equation (3.2) with p=0.4296 to get

237428.1

7259861.0

043772.1

UCL

LCL

CL

.

With y , we have 3553426.2005.0 u and 6535874.0995.0 u from Table 2 , and we apply

equation (3.3) to get

301064.1

6720576.0

043772.1

UCL

LCL

CL

Suppose has shifted from 3.2 to 2.6, but has remained at 4.8. Here, = -0.996669. Table 7

gives randomly generated data for the next 10 periods. It also gives values of ^

and y for those periods.

Table 7 Estimates of SEV mean after process shift with and Subgroup Breaking stress of carbon fibers

(GPa) ^ y

1 1.90 1.86 2.59 2.14 3.16 0.8093111 0.8247290 2 1.62 2.80 1.09 3.65 2.93 0.8804951 0.7928218 3 2.85 1.90 1.95 2.23 2.43 0.7718773 0.8159619 4 3.19 2.89 1.38 1.64 2.36 0.8137889 0.7780611 5 0.70 2.01 3.04 1.71 1.31 0.5638649 0.4503879 6 2.20 1.90 2.13 2.00 3.14 0.7824529 0.8053458 7 2.52 2.63 3.43 2.56 2.30 0.9319928 0.9789097 8 1.95 2.37 2.41 2.51 2.20 0.7434428 0.8233097 9 2.07 2.23 2.06 3.06 1.97 0.7771920 0.8093959 10 2.93 2.77 3.01 2.66 2.35 0.9253580 1.0054634

17

Figure 3: Two-sided control chart for the SEV mean Figure 4: Two-sided control chart for the SEV mean

using ^ using y

Figures 3 and 4 are the appropriate ^ and y charts. The process is judged to be out of control

in Figure 3 (^ chart) and Figure 4 ( y chart).

Our simulation study of ARL shows that the ^ chart detects downward shifts sooner than the

y chart on average. See Figure 5 where a vertical line at =-0.996669 is drawn to indicate the

shifts. In general, it appears that the ^ chart will be sufficient.

Figure 5: ARL ^ and y charts for

monitoring the SEV mean with sample size n=5 and known.

18

Conclusion The advantage of using pivotal quantities in monitoring Weibull quantiles or the SEV mean is that we may not have to simulate every time we gather data to obtain probability-based control limits as Nichols and Padgett (2005) did. Pivotal quantities which do not depend on any combination of unknown model parameters exist for MLE’s of the quantiles and the SEV mean. With our approach, the practitioner can use our tables to compute control limits, in particular, when both and are estimated. Note that only one table is needed when monitoring the Weibull or SEV quantile when is unknown and when monitoring the sample mean of SEV when is known. Further research should study other types of control charts, such as EWMA and CUSUM using pivotal quantities to monitor and detect shifts for the Weibull distribution. REFERENCES Bain & Engelhardt (1992). Introduction to Probability and Mathematical Statistics. Duxbury

Classic Series. Montgomery, D. C. (2004). Introduction to Statistical Quality Control. New York: John Wiley &

Sons Inc.. Nichols, M. D. & Padgett, W. J. (2005). A Bootstrap Control Chart for Weibull Percentiles.

Quality and Reliability Engineering International, 22, pp. 141-151. Ramalhoto, M. F. & Morais, M. (1999). Shewhart Control Charts for the scale parameter of a

Weibull control variable with fixed and variable sampling intervals. Journal of Applied Statistics, Vol. 26, No. 1, 1999, pp. 129-160.

Ryan, T. P. (2000). Statistical Methods for Quality Improvement. New York: John Wiley & Sons

Inc. Vardeman, S. B. & Jobe, M. J. (1998). Statistical Quality Assurance Methods for Engineers.

New York: John Wiley & Sons Inc.. Wetherill, Barrie G. & Brown Don W. (1991). Statistical Process Control: Theory and

Practice.Chapman & Hall.

19

Appendix I – is unknown

Table A1 Quantiles of ))1log(log(

^^^ p

V for p=0.01

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -13.1197356 -11.95116746 -11.2945473 -10.6697184 -10.1654035 -9.8309147 -9.425233

0.0010 -12.2605390 -11.34582040 -10.6974864 -10.1674982 -9.7070584 -9.4052969 -9.051192

0.00135 -11.90815265 -11.05772045 -10.4240847 -9.9182182 -9.5120973 -9.1976061 -8.888573

0.0027 -11.09539554 -10.36712624 -9.8051680 -9.3522701 -9.0145648 -8.7659981 -8.489666

0.0050 -10.3546787 -9.73505545 -9.2440551 -8.8701593 -8.5811405 -8.3334188 -8.108398

0.01 -9.5221964 -9.00175107 -8.5974439 -8.3020462 -8.0531235 -7.8380470 -7.673273

0.99 -0.4645084 -0.80429303 -1.0690974 -1.2957829 -1.4801177 -1.6416562 -1.780662

0.995 -0.2617261 -0.58934909 -0.8574173 -1.0929151 -1.2722054 -1.4406651 -1.572437

0.9973 -0.10426837 -0.42666922 -0.7006281 -0.9249135 -1.1153579 -1.2844245 -1.417095

0.99865 0.04990769 -0.27147116 -0.5389921 -0.7612888 -0.9572014 -1.1245911 -1.247842

0.999 0.1099935 -0.20566920 -0.4689339 -0.6982400 -0.8993620 -1.0598766 -1.181335

0.9995 0.2446409 -0.07598204 -0.3298543 -0.5477519 -0.7518982 -0.9291917 -1.048684

Table A2 Quantiles of ))1log(log(

^^^ p

V for p=0.05

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -8.92754140 -8.16019013 -7.70187315 -7.2299451 -6.8924039 -6.6740705 -6.3486049

0.0010 -8.36827150 -7.73151020 -7.26202210 -6.8677787 -6.5764063 -6.3715640 -6.1117361

0.00135 -8.12591619 -7.53404105 -7.08816780 -6.7167052 -6.4400935 -6.2223209 -5.9883993

0.0027 -7.56303882 -7.03355776 -6.65301845 -6.3313811 -6.0976948 -5.9012216 -5.7071950

0.0050 -7.04688450 -6.5948461 -6.26254576 -5.9915682 -5.7759468 -5.6120168 -5.4449021

0.01 -6.47614587 -6.0996935 -5.81176004 -5.5922731 -5.4194776 -5.2672980 -5.1430434

0.99 -0.10254807 -0.3402266 -0.52670392 -0.6864904 -0.8171472 -0.9283509 -1.0204720

0.995 0.05943781 -0.1800438 -0.36902630 -0.5340246 -0.6653235 -0.7812537 -0.8742788

0.9973 0.19196755 -0.05689070 -0.24690827 -0.4155564 -0.5534547 -0.6626347 -0.7605220

0.99865 0.31629233 0.06952718 -0.12808278 -0.2945274 -0.4352780 -0.5492497 -0.6390998

0.999 0.36172934 0.1142596 -0.08131233 -0.2373782 -0.3875238 -0.5014300 -0.5875293

0.9995 0.47228942 0.2314238 0.03065043 -0.1296435 -0.2824749 -0.4069731 -0.4867712

20

Table A3 Quantiles of ))1log(log(

^^^ p

V for p=0.10

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -7.15343984 -6.50250785 -6.1221117 -5.73833417 -5.46629019 -5.2903151 -5.0233378

0.0010 -6.69237884 -6.15083474 -5.7760246 -5.44807372 -5.21712886 -5.0451911 -4.8197174

0.00135 -6.49016548 -5.99748955 -5.63611190 -5.32202428 -5.09550447 -4.9269481 -4.7241165

0.0027 -6.04414244 -5.59791842 -5.28689672 -5.01745740 -4.82195194 -4.6610400 -4.4980181

0.0050 -5.61353011 -5.24642724 -4.9675730 -4.73866985 -4.56158600 -4.4259242 -4.2904079

0.01 -5.15021268 -4.83610653 -4.5969712 -4.41832528 -4.27029359 -4.1470345 -4.0371942

0.99 0.08617411 -0.11188191 -0.2662189 -0.39616305 -0.50223568 -0.5959685 -0.6684522

0.995 0.23280219 0.02426769 -0.1310521 -0.26655008 -0.37571389 -0.4703507 -0.5447399

0.9973 0.34506186 0.13291583 -0.02628131 -0.16652751 -0.27754730 -0.3676236 -0.4473746

0.99865 0.45601210 0.24378821 0.07828072 -0.05842571 -0.17597331 -0.2679771 -0.3449488

0.999 0.50325243 0.29381319 0.1232784 -0.01679254 -0.13685807 -0.2309167 -0.3070368

0.9995 0.59579752 0.38546381 0.2109366 0.07674925 -0.04969978 -0.1512895 -0.2199523

Table A4 Quantiles of ))1log(log(

^^^ p

V for p=0.4296

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -3.3164757 -2.9471749 -2.7281941 -2.545246 -2.3619840 -2.2613017 -2.1216778

0.0010 -3.1014199 -2.7678991 -2.5675222 -2.372675 -2.2220312 -2.1220199 -2.0105267

0.00135 -3.0111547 -2.6814521 -2.4913414 -2.3059571 -2.1691075 -2.0720487 -1.9572124

0.0027 -2.7804659 -2.4955511 -2.3047547 -2.1523330 -2.0313855 -1.9386923 -1.8407955

0.0050 -2.5627431 -2.3145607 -2.1475939 -2.010808 -1.9019024 -1.8128607 -1.7334616

0.01 -2.3116752 -2.1032009 -1.9554241 -1.843145 -1.7470469 -1.6721102 -1.6048974

0.99 0.6729500 0.5576404 0.4736482 0.402458 0.3489147 0.3021865 0.2621191

0.995 0.7720288 0.6528880 0.5637629 0.489096 0.4310127 0.3791796 0.3380845

0.9973 0.8551015 0.7307631 0.6343493 0.5597845 0.4999286 0.4411823 0.4008992

0.99865 0.9351531 0.8045155 0.7062191 0.6347147 0.5646794 0.5052460 0.4640960

0.999 0.9667635 0.8413732 0.7352503 0.660738 0.5929371 0.5328954 0.4893393

0.9995 1.0367165 0.9101638 0.7981911 0.727071 0.6452355 0.5938498 0.5391143

21

Table A5 Quantiles of ))1log(log(

^^^ p

V for p=0.5

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -2.9768434 -2.5853739 -2.3683099 -2.2030594 -2.0352323 -1.9278472 -1.8089466

0.0010 -2.7666764 -2.4205185 -2.2359020 -2.0520702 -1.9100373 -1.8146580 -1.7074298

0.00135 -2.6623029 -2.3496775 -2.1610001 -1.9920506 -1.8541425 -1.7660259 -1.6594505

0.0027 -2.4560859 -2.1701334 -1.9907100 -1.8512757 -1.7354488 -1.6410015 -1.5550371

0.0050 -2.2506577 -2.0117699 -1.8455571 -1.7142202 -1.6123224 -1.5281563 -1.4518068

0.01 -2.0220294 -1.8194932 -1.6738767 -1.5608118 -1.4697805 -1.3994799 -1.3331317

0.99 0.7736012 0.6708933 0.5948495 0.5295783 0.4831558 0.4390108 0.4036249

0.995 0.8716108 0.7616066 0.6793410 0.6106286 0.5606909 0.5118756 0.4753558

0.9973 0.9471344 0.8345142 0.7444385 0.6800418 0.6229182 0.5704040 0.5337830

0.99865 1.0233523 0.9077800 0.8181156 0.7491249 0.6874706 0.6320210 0.5928418

0.999 1.0608015 0.9401001 0.8436216 0.7765776 0.7098408 0.6579656 0.6151707

0.9995 1.1269149 1.0069782 0.9072461 0.8382497 0.7648689 0.7129103 0.6598830

Table A6 Quantiles of ))1log(log(

^^^ p

V for p=0.6321

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -2.4860429 -2.0887355 -1.8591493 -1.7077677 -1.5446078 -1.4349447 -1.3323636

0.0010 -2.2888902 -1.9280524 -1.7343210 -1.5757958 -1.4355884 -1.3398501 -1.2403784

0.00135 -2.2015617 -1.8588769 -1.6803010 -1.5147198 -1.3839304 -1.2931286 -1.2024394

0.0027 -1.9848509 -1.7072653 -1.5260499 -1.3839813 -1.2667609 -1.1835070 -1.1031387

0.0050 -1.8060506 -1.5626489 -1.3906202 -1.2638828 -1.1620227 -1.0859823 -1.0108624

0.01 -1.5993731 -1.3809047 -1.2382306 -1.1233998 -1.0373195 -0.9684307 -0.9029131

0.99 0.9865502 0.8974841 0.8339783 0.7801263 0.7405912 0.7012896 0.6734038

0.995 1.0768732 0.9837554 0.9129739 0.8557701 0.8137308 0.7687109 0.7404851

0.9973 1.1460667 1.0508508 0.9738037 0.9172644 0.8703516 0.8248249 0.7938204

0.99865 1.2203151 1.1183434 1.0406506 0.9786607 0.9255626 0.8819917 0.8446138

0.999 1.2535509 1.1458218 1.0675943 1.0042146 0.9469336 0.9054965 0.8634417

0.9995 1.3234264 1.2074486 1.1222827 1.0681778 1.0008799 0.9507587 0.9124514

22

Appendix II

Figure 6: Plot of the estimated quantile against the total number T of simulations for n=4 when is unknown.

23

Appendix III – Known

Table B1 Quantiles of ^^

W for = 0.5, p = 0.5

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -1.4022710 -1.2182854 -1.0975957 -1.0003111 -0.9347658 -0.8819503 -0.8303474

0.001 -1.3059286 -1.1456800 -1.0313313 -0.9432789 -0.8871479 -0.8309640 -0.7852535

0.00135 -1.2641686 -1.1132979 -0.9967075 -0.9161431 -0.8625456 -0.8079631 -0.7653249

0.0027 -1.1659271 -1.0329212 -0.9264800 -0.8583289 -0.8057207 -0.7571010 -0.7194680

0.005 -1.0783076 -0.9562748 -0.8641984 -0.7991691 -0.7528352 -0.7095664 -0.6758557

0.01 -0.9742774 -0.8675724 -0.7888756 -0.7326593 -0.6909430 -0.6524418 -0.6234508

0.99 0.2753074 0.2375582 0.2066781 0.1849165 0.1627232 0.1461126 0.1312813

0.995 0.3190844 0.2791264 0.2443819 0.2203841 0.1976675 0.1781220 0.1624858

0.9973 0.3537133 0.3126279 0.2762641 0.2527345 0.2245768 0.2054095 0.1872496

0.99865 0.3921773 0.3473081 0.3100486 0.2826297 0.2538528 0.2326264 0.2142533

0.999 0.4065707 0.3611718 0.3233239 0.2938702 0.2661136 0.2429593 0.2250406

0.9995 0.4387860 0.3887112 0.3525577 0.3181975 0.2906899 0.2686299 0.2501285

Table B2 Quantiles of ^^

W for = 1.0, p = 0.5

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -2.7992485 -2.4308058 -2.1814967 -2.0232181 -1.8844372 -1.7632417 -1.6614876

0.001 -2.5964105 -2.2739898 -2.0584386 -1.8913619 -1.7690006 -1.6736604 -1.5721866

0.00135 -2.5208173 -2.2008818 -1.9971538 -1.8324979 -1.7177312 -1.6276396 -1.5358328

0.0027 -2.3266140 -2.0375625 -1.8574432 -1.7073577 -1.6033075 -1.5164884 -1.4387701

0.005 -2.1460468 -1.8995608 -1.7283015 -1.5933592 -1.4951634 -1.4215735 -1.3493332

0.01 -1.9456486 -1.7294830 -1.5786573 -1.4633046 -1.3748976 -1.3078818 -1.2444983

0.99 0.5540803 0.4752601 0.4146435 0.3660400 0.3267866 0.2935502 0.2629908

0.995 0.6405487 0.5579843 0.4886067 0.4374954 0.3972123 0.3593339 0.3260825

0.9973 0.7083977 0.6261208 0.5494690 0.4959449 0.4538520 0.4107882 0.3770750

0.99865 0.7832215 0.6923288 0.6138220 0.5572038 0.5097916 0.4649810 0.4272412

0.999 0.8069829 0.7190586 0.6415718 0.5847517 0.5352399 0.4873201 0.4499928

0.9995 0.8726649 0.7833175 0.6969718 0.6371127 0.5838082 0.5344619 0.4963528

24

Table B3 Quantiles of ^^

W for = 1.5, p = 0.5

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -4.1697458 -3.6612552 -3.2771110 -3.0134750 -2.8073849 -2.6260597 -2.4966236

0.001 -3.8769157 -3.4096448 -3.0910308 -2.8201111 -2.6538889 -2.4984362 -2.3621826

0.00135 -3.7560556 -3.3007294 -2.9994950 -2.7410403 -2.5809665 -2.4291580 -2.3040461

0.0027 -3.4704000 -3.0668480 -2.7908219 -2.5566860 -2.4086788 -2.2715692 -2.1623859

0.005 -3.2157150 -2.8524573 -2.6022528 -2.3942549 -2.2561950 -2.1255137 -2.0322876

0.01 -2.9147790 -2.6001537 -2.3797515 -2.1973552 -2.0742230 -1.9611745 -1.8775469

0.99 0.8335160 0.7134033 0.6226225 0.5475977 0.4940064 0.4368137 0.3958892

0.995 0.9649032 0.8330087 0.7377994 0.6534692 0.5971777 0.5342635 0.4908068

0.9973 1.0692366 0.9294155 0.8255802 0.7411660 0.6800003 0.6110223 0.5682765

0.99865 1.1783460 1.0320496 0.9219517 0.8320989 0.7648961 0.6906019 0.6512290

0.999 1.2232228 1.0718786 0.9564689 0.8702445 0.8007793 0.7252708 0.6811290

0.9995 1.3212283 1.1645970 1.0416511 0.9555308 0.8865026 0.7932084 0.7542641

Table B4 Quantiles of ^^

W for = 2.0, p = 0.5

n=4 n=5 n=6 n=7 n=8 n=9 n=10 0.0005 -5.578299 -4.8754998 -4.3596799 -4.0387251 -3.7615830 -3.5119281 -3.3369744

0.001 -5.183856 -4.5595480 -4.1048844 -3.7981305 -3.5432861 -3.3234075 -3.1566375

0.00135 -5.036683 -4.4126723 -3.9863109 -3.7021159 -3.4360352 -3.2345637 -3.0760473

0.0027 -4.661281 -4.0772168 -3.7163490 -3.4470620 -3.2111975 -3.0282605 -2.8845326

0.005 -4.313916 -3.7813849 -3.4680734 -3.2123417 -3.0008875 -2.8312896 -2.7068916

0.01 -3.903354 -3.4502207 -3.1571617 -2.9322657 -2.7570167 -2.6118614 -2.4964557

0.99 1.107951 0.9507813 0.8302713 0.7362205 0.6507695 0.5824954 0.5299619

0.995 1.288770 1.1167085 0.9859933 0.8745446 0.7870375 0.7178330 0.6549864

0.9973 1.429584 1.2521767 1.1112750 0.9909421 0.8953659 0.8185368 0.7564218

0.99865 1.577729 1.3911103 1.2416205 1.1086174 1.0103344 0.9332097 0.8651528

0.999 1.631698 1.4414555 1.2881798 1.1608240 1.0540278 0.9714392 0.9057965

0.9995 1.749157 1.5661514 1.3969577 1.2758315 1.1581803 1.0795786 1.0055325

25

Appendix IV

A. is Unknown

Figure 7: ARL for one-sided lower charts when is unknown with all-OK ARL = 100 to detect the shift in for different sample size of 0.4296 quantile.

26

Figure 8: ARL for one-sided upper charts when is unknown with all-OK ARL = 100 to detect the shift in for different sample size of 0.4296 quantile.

27

Figure 9: ARL for two-sided charts when is unknown with all-OK ARL = 100 to detect the shift in for different sample size of 0.4296 quantile.

28

B. is Known (=1, p=0.4296)

Figure 10: ARL for one-sided lower charts using ^ when is known, all-OK ARL = 100 to detect the

shift in SEV mean for different sample size.

29

Figure 11: ARL for one-sided upper charts using ^ when is known, all-OK ARL = 100 to detect the

shift in SEV mean for different sample size.

30

Figure 12: ARL for two-sided charts using ^ when is known, all-OK ARL = 100 to detect the shift in

SEV mean for different sample size.

31

Monitor mean of SEV use ybar

Figure 13: ARL for one-sided lower charts using y when is known, all-OK ARL = 100 to detect the shift in SEV mean for different sample size.

32

Figure 14: ARL for one-sided upper charts using y when is known, all-OK ARL = 100 to detect the shift in SEV mean for different sample size.

33

Figure 15: ARL for two-sided charts using y when is known, all-OK ARL = 100 to detect the shift in SEV mean for different sample size.