Control Charts for Simultaneous Monitoring of ...

Control Charts for Simultaneous Monitoring of Parameters of a Shifted Exponential Distribution

A. Mukherjee XLRI - Xavier School of Management, Production, Operations and Decision Sciences Area, Jamshedpur-831001, India [email protected]

A. K. McCracken J.P. Morgan Chase & Co., Columbus, OH, USA

[email protected]

S. Chakraborti Department of Information Systems, Statistics and Management Science University of Alabama Tuscaloosa, AL, USA [email protected]

Abstract Since their introduction in the 1920’s, control charts have played a key role in process monitoring and control in a variety of areas, from manufacturing to healthcare. Many of these charts are designed to monitor a single process parameter, such as the mean or the variance, of a normally distributed process, although recently, a number of charts have been developed for jointly monitoring the mean and variance. In practice, however, there are processes that follow multi-parameter non-normal distributions, but the joint monitoring of parameters of non-normal distributions remains largely unaddressed in the literature. This paper proposes several control charts and monitoring schemes for the origin and the scale parameters of a process which follows the two-parameter (or the shifted) exponential distribution. This distribution arises in various applications in practice, particularly with time to an event data, such as in reliability studies, and has been studied extensively in the statistical testing and estimation literature. Exact derivations and computer simulations are used to study performance properties of the proposed charts. An illustrative example is provided along with a summary and some conclusions.

Key Words: Average Run Length (ARL); Origin and Scale; In Control (IC); Joint Monitoring; Out of Control (OOC); Probability Integral Transform; Maximum Likelihood.

1.

Introduction The exponential distribution plays an important role in quality monitoring studies where

the data are positively skewed. For example, one common assumption in practice is that time to

2

certain events, such as failure of an item, follows an exponential distribution. In high yield processes, with very low defect rates, it is recommended that one monitors the times between consecutive failures or non-conforming items. This elapsed or inter-arrival time can be shown to follow an exponential distribution under the assumption that the failures or the number of defects follows a Poisson process. It is however well recognized that in many situations, although the failure times follow an exponential distribution, the origin of the distribution may not be from zero. Thus, there is an origin (or threshold) parameter 𝜃 > 0, in addition to the scale parameter 𝜆 > 0, and the resulting lifetime distribution is a two-parameter exponential distribution given by the pdf 𝑓(𝑥; 𝜃, 𝜆) = 𝜆−1 𝑒 −(𝑥−𝜃)/𝜆 , 𝑥 > 𝜃 and 𝜆 > 0. This distribution is also called the shifted exponential because the corresponding random variable (lifetime) 𝑋 may be seen as distributed as 𝑌 + , where 𝑌 follows an exponential distribution with origin parameter 0 and scale parameter 𝜆; the latter of course is the more wellknown exponential distribution with mean 𝜆. Thus, the lifetime 𝑋 has a mean 𝜆 +  and variance 𝜆2. Note that we refer to 𝜃 as the origin parameter rather than the location parameter, since the location (the mean) is actually given by the sum of the two parameters 𝜃 and 𝜆. Also note that in the literature, see for example, Johnson and Kotz (1970), the two-parameter exponential distribution has been defined for −∞ < 𝜃 < ∞, however, in our case 𝜃 > 0, since the variable being monitored is the time to an event. We now motivate the application of the two-parameter exponential distribution with several examples. One popular application of the exponential distribution in general, and the two-parameter exponential distribution in particular, has been in the field of reliability analysis,

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in situations, where the origin parameter 𝜃 denotes a guarantee period (warranty time) so that the failures are assumed to occur only beyond a certain time 𝜃 (see for example, Epstein (1960)). For example, an automobile may come with a three-year warranty under which the manufacturer agrees to fix, for free, certain defects which may occur during that period. In order to set an appropriate warranty period, a manufacturer would utilize available failure time data and a relevant statistical distribution, such as the two-parameter exponential distribution to model the lifetime of the product. Lawless (1982; page 126) states “This model is employed in situations where it is thought that death cannot occur before some particular time 𝜃.” He then refers to an example discussed by Grubbs (1971), Engelhardt and Bain (1978) and others involving some data that are “mileages of military personnel carriers that failed in service and appear consonant with an exponential model.”

Parameter estimates and confidence intervals are obtained for

these data. We use these mileage data later to illustrate our procedures. The warranty time is also referred to as a location or a threshold time by some authors. Tobias and Trindade (2012; page 76) state “in some situations, it may be impossible for failures to occur before the end of a waiting period of length  hours. This period is sometimes called the threshold time, and  is a location or a threshold parameter. Assume that after that waiting period, an exponential model with parameter  is a good model for the population failure times. The distribution model for this population has the two parameter exponential density function…” In fact, they provide a dataset that fits this two-parameter distribution but not the one-parameter or the standard exponential distribution. However, the two-parameter exponential distribution does not only apply to time to events scenarios. Kao (2010) considered a situation where a high-voltage of current is applied in a P-type high-voltage metal oxide semiconductor (MOS) transistor (HPM) on a flash memory

4

wafer. When the amount of current required to break an insulator down is of interest, as in Kao (2010), it is plausible to assume that there is a threshold value of the current that all the wafers would withstand, and beyond this threshold, the additional amount of current needed for the breakdown follows an exponential distribution. Finally, Baten and Kamil (2009) noted that the two-parameter exponential distribution can be an appropriate model when considering inventory management of hazardous items. Researchers have studied various aspects of estimation and inference related to the twoparameter exponential distribution based on complete as well as censored data, both under the frequentist approach and the Bayesian paradigm. However, in addition to the classical problems involving estimation and hypothesis testing of parameters, from a quality control and monitoring perspective, there is the need for prospective monitoring of the lifetimes following this distribution, to determine whether the lifetime distribution (of the product) has remained unchanged at acceptable targets or whether some changes have occurred, in one or both of the parameters, which might cause concern or deserve attention. In such a situation, a control charting scheme capable of accurately determining whether each sample taken at specific time points comes from the specified shifted or two-parameter exponential distribution and/or detecting one that differs from the specified one in some way is needed. External factors, such as pollution or changes in the materials such as the ones used to pave roads, or changes in the engineering processes could alter the distribution of the product’s lifetime distribution by changing one or both of the parameters. A shift in the standard deviation 𝜆 changes the mean of the distribution and thus, simultaneous monitoring of the parameters 𝜃 and 𝜆 is an important problem, one we consider in this paper.

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Control charts have proved to be a useful tool in the process monitoring/control context. However, the majority of the common control charts are designed for situations where the data are assumed to be normally distributed. For a detailed overview of the area of monitoring the mean and the variance of a normal distribution, see the recent paper by McCracken and Chakraborti (2013). However, when the underlying distribution is non-normal, such as the case assumed in this paper, the use of normal theory based charts is inappropriate and can be quite misleading and costly. One possible approach in this case might be to use some transformation to normality and then using normal theory methods. We do not pursue this here. Instead, as many researchers have suggested, we recommend that the practitioners should use the correct control charts designed for the assumed distribution, such as in the present case, for the shifted exponential distribution. Unfortunately however, while parameter estimation, hypothesis testing, and prediction of future data points for the shifted exponential distribution are addressed in the literature, the research on control charts for data arising from this distribution has received scant attention. Ramalhoto and Morais (1999) developed a control chart for monitoring only the scale parameter, and Sürücü and Sazak (2009) presented a control scheme for this distribution in which moments are used to approximate the distribution. Neither method appears entirely satisfactory from a practical point of view. There is a need for an appropriate control charting scheme capable of jointly monitoring the origin and scale parameters. At first, it might seem tempting to construct a control chart using the mean and standard deviation of the samples as is typically done in the case of the normal distribution. This, however, would be inappropriate since these are not the proper estimators of the parameters of the two-parameter exponential distribution. As a result, these statistics can be inefficient and

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unreliable in this case, as noted by Ramalhoto and Morais (1999). They pointed out that “a more reliable alternative is to identify or fit a distributional model for the output that is more appropriate than the normal model, and then to construct control charts based on that model.” Here, we offer two proposals for one-chart schemes for jointly monitoring the two parameters of the shifted exponential distribution. First, we propose a scheme similar to Chen and Cheng’s (1998) max chart for monitoring the mean and variance of a normal distribution. This Shifted Exponential MLE Max (SEMLE-Max) chart is composed of a combination of the maximum likelihood estimators (MLEs) of the origin and the scale parameters. Note that there are other ways of constructing joint monitoring schemes, but the max-type charts have been popular in the literature and with practitioners, due to their relative simplicity and operational ease. Next, we consider a likelihood ratio-type chart, which we will refer to as the Shifted Exponential Likelihood Ratio (SE-LR) chart. Finally, taking a cue from what is typically done in practice under the assumption of a normal distribution, we consider a two-chart combination scheme, made up of one control chart for monitoring the origin and a separate control chart for monitoring the scale. Each of these component charts utilizes the MLE of the corresponding parameter that it monitors, and the control limits for the two charts are (adjusted) determined so that the overall IC ARL of the two-chart scheme is a desired nominal value. We refer to this as the SEMLE-2 joint monitoring scheme. 2.

Statistical Framework and Preliminaries Let 𝑉1 , 𝑉2 , … , 𝑉𝑛 be a random sample of size 𝑛 from the shifted exponential distribution,

with origin parameter 𝜃 and scale parameter 𝜆. The parameters 𝜃 and 𝜆 are unknown; however, when the process is in-control (IC), let 𝜃 = 𝜃0 and 𝜆 = 𝜆0 where the values 𝜃0 and 𝜆0 , are known or specified. These may reflect the manufacturer’s experience and costs of making the

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product. When one or both of these equalities is violated, the product no longer meets the requirements, and the process is said to be out-of-control (OOC). In such a case, the origin parameter, the scale parameter, or both differ from the specified values, 𝜃0 and 𝜆0 , respectively. Let 𝜃̂ = 𝑉(1) be the minimum or the first order statistic of the sample. This is known to be the MLE for the origin parameter 𝜃 (see Johnson and Kotz, 1970). One can also see from 1 Johnson and Kotz (1970) that 𝜆̂ = 𝑛 ∑𝑛𝑖=1(𝑉𝑖 − 𝜃̂) =

1 𝑛

∑𝑛𝑖=1 𝑉𝑖 − 𝜃̂ = 𝑉̅ − 𝑉(1) is the MLE of the

scale parameter 𝜆. Moreover, it has been shown that 𝜃̂ and 𝜆̂ are independent (see Johnson and Kotz, 1970; Govindarajulu, 1966; Tanis, 1964). The distribution of 𝜃̂ is well known to be the shifted exponential distribution with origin parameter 𝜃 and scale parameter 𝜆/𝑛. one can show that 𝐸1 =

̂ −𝜃) 2𝑛(𝜃 𝜆

As a result,

has a chi-square distribution with 2 degrees of freedom. This is

one of the key results used in the subsequent developments. The distribution of 𝜆̂ is less straightforward but can be obtained using standard distribution theory. First, observe that we can write ∑𝑛𝑖=1(𝑉𝑖 − 𝜃̂) = ∑𝑛𝑖=1(𝑉𝑖 − 𝑉(1) ) = ∑𝑛𝑖=2(𝑛 − 𝑖 + 1)(𝑉(𝑖) − 𝑉(𝑖−1) ). 2 𝜆

Now,

since

∑𝑛𝑖=2(𝑛 − 𝑖 + 1)(𝑉(𝑖) − 𝑉(𝑖−1) ) has a chi-square distribution with 2𝑛 − 2 degrees of freedom

(Tanis, 1964), it follows that 𝐸2 =

̂ 2𝑛𝜆 𝜆

2

= 𝜆 ∑𝑛𝑖=2(𝑛 − 𝑖 + 1)(𝑉(𝑖) − 𝑉(𝑖−1) ) has a chi-square

distribution with 2𝑛 − 2 degrees of freedom. 3.

The SEMLE-Max Chart First

for

the

SEMLE-Max

control

chart,

let

𝐵1 = Φ−1 {𝐺(𝐸1 , 2)} and

𝐵2 = Φ−1 {𝐺(𝐸2 , 2𝑛 − 2)} where Φ (Φ−1) denotes the cdf (the quantile function) of the standard normal distribution and 𝐺(𝑋, 𝜐) denotes the cdf of a random variable X following a chi-square distribution with 𝜐 degrees of freedom. Using the probability integral transform (see e.g.,

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Gibbons and Chakraborti, 2010), 𝐺(𝐸1 , 2) and 𝐺(𝐸2 , 2𝑛 − 2) each follows a uniform distribution over (0, 1) under the IC set up (when 𝜃 = 𝜃0 and 𝜆= 𝜆0 ). Using the probability integral transform one more time, it follows that 𝐵1 and 𝐵2 each follows the standard normal distribution when the process is IC. Further, it can be shown that 𝐵1 and 𝐵2 are mutually independent. We will use these two statistics to construct the SEMLE-Max chart following Chen and Cheng’s (1998) idea for the max chart for joint monitoring in the case of the normal distribution. 3.1.

Proposed Charting Procedure for the SEMLE-Max chart We propose the following method for constructing the SEMLE-Max charts when 𝜃0 and

𝜆0 are specified: Step 1. Let 𝑌𝑖,𝑛 = (𝑌𝑖1 , 𝑌𝑖2 , … , 𝑌𝑖𝑛 ) be the i-th sample of size 𝑛, 𝑖 = 1,2, … Step 2. Identify the 𝑉’𝑠 with the 𝑌’𝑠. Calculate the statistics 𝐵1𝑖 and 𝐵2𝑖 for the 𝑖th sample, 𝑖 = 1,2, … Step 3: Calculate the plotting statistic ℳ𝑖 = max{|𝐵1𝑖 |, |𝐵2𝑖 |},

𝑖 = 1,2, ….

Step 4. Plot ℳ𝑖 against an UCL 𝐻ℳ . Note that ℳ𝑖 ≥ 0 by definition so that the LCL is 0 and that larger values of ℳ𝑖 suggest an OOC process. Tables for 𝐻ℳ are given later. Step 5. If ℳ𝑖 exceeds 𝐻ℳ , the process is declared OOC at the ith test sample. If not, the process is considered to be IC, and testing continues to the next sample. Step 6. Follow-up: When the process is declared OOC at the ith test sample, compare each of |𝐵1𝑖 | and |𝐵2𝑖 | with 𝐻ℳ . (i)

If |𝐵2𝑖 | < 𝐻ℳ < |𝐵1𝑖 |, a shift in the origin parameter 𝜃 is indicated.

(ii)

If |𝐵1𝑖 | < 𝐻ℳ < |𝐵2𝑖 |, a shift in the scale parameter is indicated.

(iii)

If |𝐵1𝑖 | and |𝐵2𝑖 | both exceed 𝐻ℳ , it should be safe to conclude that 𝜆 has shifted, but a diagnosis about the origin 𝜃 will need further follow-up work. This is since the

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distribution of the origin estimator 𝜃̂ = 𝑉(1) depends on the scale parameter 𝜆 so a large shift in 𝜆 could create false signals in the origin chart. As pointed out by a reviewer, this situation is familiar to simultaneously monitoring the mean and the variance of a normal distribution. 3.2.

Run Length Distribution For the SEMLE-Max chart, the process is declared OOC when ℳ > 𝐻ℳ where the UCL

𝐻ℳ is obtained so that the IC ARL is some given nominal value. The cdf of the plotting statistic ℳ is given by 𝑃(ℳ ≤ 𝑔) = 𝑃(max{|𝐵1 |, |𝐵2 |} ≤ 𝑔) = 𝑃(|𝐵1 | ≤ 𝑔)𝑃(|𝐵2 | ≤ 𝑔) 2𝑛 𝜆0 2𝑛 𝜆0 = {𝐺 ( ( 𝐺 −1 (𝛷(𝑔), 2) + 𝜃0 − 𝜃) , 2) − 𝐺 ( ( 𝐺 −1 (𝛷(−𝑔), 2) + 𝜃0 − 𝜃) , 2)} 𝜆 2𝑛 𝜆 2𝑛 𝜆0 𝜆0 × {𝐺 ( 𝐺 −1 (Φ(𝑔), 2𝑛 − 2), 2𝑛 − 2) − 𝐺 ( 𝐺 −1 (Φ(−𝑔), 2𝑛 − 2), 2𝑛 − 2)}, 𝜆 𝜆 Where 𝐺 −1 (𝑢, 𝑣) denotes the quantile function of the chi-square distribution with 𝑣 degrees of freedom. Interested readers may see the Result A.0 of the supplementary file for detailed derivation. Since 𝐵1 and 𝐵2 are independent and identically distributed standard normal variables when the process is IC 2

𝑃(ℳ ≤ 𝑔|𝐼𝐶) = [𝑃(−𝑔 ≤ 𝐵1 ≤ 𝑔)]2 = {𝛷(𝑔) − 𝛷(−𝑔)}2 = {𝟐𝛷(𝑔) − 1} . It follows that as the SEMLE-Max chart is basically a Shewhart type chart, its IC run length distribution is geometric with success probability (false alarm rate) equal to 𝒑 = 𝟏 − {𝜱( 𝑯𝓜 ) − 𝜱(− 𝑯𝓜 )}𝟐 = 𝟏 − {𝟐𝜱( 𝑯𝓜 ) − 𝟏}𝟐 . 3.3.

Determination of 𝑯𝓜

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Noting that the run length distribution is geometric with success probability 𝑝 shown above, the IC ARL is: 𝑝−1 = [1 − {2𝛷( 𝐻ℳ ) − 1}2 ]−1 . Thus, for a nominal IC ARL, say ARL0, 1

the UCL of the chart 𝐻ℳ = Φ−1 [0.5 (1 + √1 − 𝐴𝑅𝐿 )]. 0

4.

The SE-LR Chart The likelihood ratio method is a popular method for hypothesis testing problems, for

finding test statistics that have attractive statistical properties. The joint monitoring of origin and scale over time may be viewed as repeatedly testing the null hypothesis that the sample at hand comes from a completely specified shifted exponential population (a simple null hypothesis) versus all alternatives (a composite alternative hypothesis), and as such we consider a chart based upon the perspective of a likelihood ratio statistic. It can be shown (see the Result A.1 of the supplementary material for further details) that the likelihood-ratio statistic for the shifted ̅ −𝑉(1) 𝑛 𝑉

exponential distribution is given by Λ = 𝑒 𝑛 (

) exp{−𝑛(𝑉̅ − 𝜃0 )/𝜆0 }. This will be a

𝜆0

plotting statistic used to define the SE-LR chart. Note that, writing in general 𝑉(1) =

𝜆𝐸1 2𝑛

+ 𝜃 and 𝑉̅ − 𝑉(1) =

𝜆𝐸2 2𝑛

, we can rewrite the

likelihood ratio criterion as Λ=

̅−𝑉(1) 𝑛 𝑉 𝑒𝑛 ( 𝜆 ) 0

𝑒

−

̅ −𝜃0 ) 𝑛(𝑉 𝜆0

𝑛

𝜆𝐸2

𝑛

= 𝑒 (2𝑛 𝜆 ) 𝑒

−

𝜆 𝑛(𝜃−𝜃0 )+ (𝐸1 + 𝐸2 ) 2 𝜆0

0

.

Further, since in the IC set up, Λ reduces to Λ=

̅−𝑉(1) 𝑛 𝑉 𝑒𝑛 ( 𝜆 ) 0

𝑒

−

̅ −𝜃0 ) 𝑛(𝑉 𝜆0

𝐸

𝑛

= 𝑒 𝑛 (2𝑛2 ) 𝑒 −

𝐸1 + 𝐸2 2

.

This shows that the SE-LR chart plotting statistic is also a function of the two statistics 𝐸1 and

𝐸2 used in the other charts, but in this case the function combining the two statistics is a more complex one with unequal (rather than any pre-assigned, including equal) weights. Moreover, note that even

11 though the charts are based on the same basic statistics, the combinations are different and that make them more sensitive to different types of shifts.

4.1

Proposed Charting Procedure for the SE-LR chart We propose the following steps for constructing SE-LR charts when the standard values

of the parameters 𝜃0 and 𝜆0 are specified: Step 1. Let 𝑌𝑖,𝑛 = (𝑌𝑖1 , 𝑌𝑖2 , … , 𝑌𝑖𝑛 ) be the i-th sample of size 𝑛, 𝑖 = 1,2, … Step 2. Identify the 𝑉’𝑠 with the 𝑌’𝑠. Calculate the likelihood ratio criterion, Λ 𝑖 , for the 𝑖th sample, 𝑖 = 1,2, … Step 3: Plot Λ 𝑖 against a LCL 𝐻Λ . The upper control limit (UCL) is 1. Note that Λ 𝑖 ≤ 1 by definition and that smaller values of Λ 𝑖 suggest an OOC process. Step 4. If Λ 𝑖 is lower than 𝐻Λ , the process is declared OOC at the i-th sample. If not, the process is considered to be IC, and testing continues to the next sample. Step 5. Follow-up: When the process is declared OOC at the i-th sample, the follow-up strategy in McCracken et al. (2013) may be applied using the statistics 𝐸1𝑖 and 𝐸2𝑖 and the associated pvalues. Also, as noted earlier, the statistics 𝐵1𝑖 and 𝐵2𝑖 are functions of 𝐸1𝑖 and 𝐸2𝑖 respectively and hence the follow-up procedure based on 𝐸1𝑖 and 𝐸2𝑖 for the SE-LR chart may be very similar to that of the SE-MLE chart. Now as mentioned in Subsection 3.1, note that the scale plotting statistic is unaffected if the shift occurs in the origin parameter. Consequently, if one gets a signal for a scale change, one can believe it regardless of what happens with the origin chart. But a scale shift changes the distribution of the plotting statistic for the origin chart and so if one may have a signal for the scale; one still needs to follow-up on the origin. As such, the p-value of the test for origin shift based on 𝐸1𝑖 is not relevant, since the p-value for 𝐸1𝑖 is based on the assumption that the scale parameter is fixed at its IC value.

12 𝐸1 /2

To this end, we calculate 𝐸1,modified = 𝐸

2

= /(2𝑛−2)

̂ −𝜃) (𝑛−1)(𝜃 ̂ 𝜆

replacing the fixed value

of the scale parameter by its estimator. This will be more appropriate when shift occurs in both origin and scale parameters. It can be shown that in the IC case this statistic has an F distribution with 2 and (2n-2) degrees of freedom. Accordingly, we suggest the modified follow-up strategy as: If 𝐸1i,modified < 0, this is a clear indication for a downward shift (decrease) in the

(i)

origin parameter 𝜃. If 𝐸1i,modified < 𝐹1−𝛼,

(ii)

2, 2𝑛−2

and 𝜒𝛼2, 2

2𝑛−2

2 < 𝐸2𝑖 < 𝜒1− 𝛼 , 2

2𝑛−2

, the OOC signal is

2 deemed a false alarm. Note that 𝜒𝛼,𝜐 denotes the 𝛼th percentile of a chi-square

distribution with 𝜐 degrees of freedom and 𝐹𝛼,𝜐1 , 𝜐2 denotes the 𝛼th percentile of a F with 𝜐1 , 𝜐2 degrees of freedom. (iii)

If 𝐸1,modified > 𝐹1−𝛼,

2, 2𝑛−2

and 𝜒𝛼2, 2

2𝑛−2

2 < 𝐸2𝑖 < 𝜒1− 𝛼 , 2

2𝑛−2

, it is safe to assume

that there is no shift in the scale parameter 𝜆 and only a shift in origin 𝜃 has taken place. (iv)

2 If 𝐸2𝑖 > 𝜒1− 𝛼 , 2

2𝑛−2

𝑜𝑟 𝐸2𝑖 < 𝜒𝛼2, 2

2𝑛−2

, a shift in scale is indicated as the statistic for

𝜆 remains unaffected by a shift in 𝜃 regardless of the value of 𝐸1𝑖 . In particular, if 2 𝐸2𝑖 > 𝜒0.975𝜒 2

𝛼 1− , 2𝑛−2 2

or 𝐸2𝑖 < 𝜒𝛼2, 2

2𝑛−2

along with 𝐸1,modified > 𝐹1−𝛼,

2, 2𝑛−2 ,

a

signal of shift in both the location and scale is indicated. In this context, readers may note that for a Shewhart type chart, the false alarm rate (FAR) is reciprocal of the target IC ARL. That is, if target IC ARL is 500, individual chart operate at a level of 0.002 or for a target IC-ARL 370, the chart operates with a FAR of 0.0027.

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Therefore, 𝛼 should be chosen carefully. In control chart context, 𝛼 is usually much lesser than common level of significance used in the testing of hypothesis literature (such as 1% or 5%). We may consider 𝛼 as 1/ ARL0. 4.2.

Determination of 𝑯𝚲 For the SE-LR chart, the process is declared OOC when Λ i < 𝐻Λ , where the LCL 𝐻Λ is

obtained so that the IC ARL is equal to some given nominal value. Note that in this case also run length distribution is geometric with success probability 𝑃 (Λ < 𝐻Λ |𝐼𝐶). However, the distribution of Λ is rather intractable so the success probability cannot be calculated easily. As an alternative, when 𝑛 is large, it is useful to consider the asymptotic properties of the likelihood ratio Λ. Using general theory (see for example, Hogg, McKean and Craig (2005)) it can be shown that −2ln Λ i converges in distribution to a chi-square random variable with 2 degrees of freedom. Hence it is easy to obtain 𝐻Λ approximately as long as 𝑚 and/or 𝑛 are appropriately large. However, when 𝑛 is small or moderate, as is nearly always the case in control charting, 𝐻Λ can easily be obtained through Monte Carlo simulation for a given ARL0. This is the approach used in this paper. 5.

The SEMLE-2 Charting Scheme While the one-chart schemes such as the SEMLE-Max chart and the SE-LR chart are

appealing in that they allow the practitioner to focus on a single chart (and a single charting statistic), some practitioners might find it convenient to monitor the origin and the scale parameters on two separate charts. For the normal distribution, this is typically accomplished using an 𝑋̅ chart for the mean and an 𝑆 chart or an R chart for the spread. Following this traditional paradigm, we consider a two-chart scheme for jointly monitoring the origin and scale parameters of a shifted exponential distribution based on the 𝐸1 and 𝐸2 .

14

Recall that when the process is IC, 𝐸1 and 𝐸2 have a chi-square distribution with 2 and 2𝑛 − 2 degrees of freedom, respectively and that they are independent since 𝜃̂ and 𝜆̂ are independent. These are the two statistics we use to construct the charts which make up the SEMLE-2 scheme. 5.1

Proposed Charting Procedure for the SEMLE-2 scheme We propose the following method for constructing the SEMLE-2 scheme when the origin

and scale parameters are known or specified: Step 1. Let 𝑌𝑖,𝑛 = (𝑌𝑖1 , 𝑌𝑖2 , … , 𝑌𝑖𝑛 ) be the i-th sample of size 𝑛, 𝑖 = 1,2, … Step 2. Identify the 𝑉’𝑠 with the 𝑌’𝑠. Calculate the plotting statistics 𝐸1𝑖 and 𝐸2𝑖 for the 𝑖-th sample, for 𝑖 = 1,2, …. Note that if the process is IC, both of these quantities should be positive. If 𝐸1𝑖 < 0, the process is declared OOC at the i-th sample. It immediately implies a leftward shift (decrease) in origin. If 𝐸1𝑖 ≥ 0, continue to step 3. Step 3. Plot 𝐸1𝑖 against an UCL 𝐻𝐸1 . The lower control limit (LCL) is 0. Step 4. On a separate chart, plot E2𝑖 against an UCL 𝐻𝐸2𝑈 and LCL 𝐻𝐸2𝐿 . Step 5. If E1𝑖 is greater than 𝐻𝐸1 and/or 𝐸2𝑖 is greater than 𝐻𝐸2𝑈 or less than 𝐻𝐸2𝐿, the process is declared OOC at the i-th sample and we go to the next step. If not, the process is considered to be IC, and testing continues to the next sample. Step 6. In a two chart scheme, the follow-up is traditionally done by using the signal(s) from the individual chart(s). However, a signal from the location chart should be dealt with more carefully as it may be the result of a shift in scale as discussed before. So, we recommend following up with the test for origin parameter using 𝐸1,modified as detailed in Subsection 4.1. 5.2.

Determination of 𝑯𝑬𝟏 , 𝑯𝑬𝟐𝑼 , and 𝑯𝑬𝟐𝑳

15

We find the control limits so that each of the two individual charts has the same (a common) IC ARL and the resulting two-chart combo scheme has a given nominal overall IC ARL. Denoting the IC ARL of the individual control chart for the origin, the scale and the combination SEMLE − 2 chart as 𝒜1 , 𝒜2 , and 𝒜, respectively, it is easy to show that 𝒜−1 = 1 − (1 − 𝒜1−1 )(1 − 𝒜2−1 ).

Now if since each individual (component) chart has the −1

−1

same (common) IC ARL, 𝒜1 = 𝒜2 = (1 − √1 − 𝒜−1 ) where 𝒜1 = (𝑃(E1 > 𝐻E1 |𝐼𝐶))

=

[1 − 𝐺( 𝐻E1 , 2)]−1 . Hence, for a given nominal value of 𝒜, the UCL is given by 𝐻E1 = 𝐺 −1 (√1 − 𝒜−1 , 2). 2) − 𝐺(𝐻𝐸2𝐿 , 2𝑛 − 2)}

Similarly, 𝒜2 = (𝑃(E2 > 𝐻𝐸2𝑈 or E2 < 𝐻𝐸2𝐿 |𝐼𝐶)) −1

−1

= {𝐺(𝐻𝐸2𝑈 , 2𝑛 −

and we choose the control limits 𝐻E2𝑈 and 𝐻E2𝐿 so that the

probability of a false alarm is the same in both the upper and the lower tails. Thus, setting 1

𝑃(E2 > 𝐻𝐸2𝑈 |𝐼𝐶) = 𝑃( E2 < 𝐻𝐸2𝐿 |𝐼𝐶), we get 𝐻E2𝑈 = 𝐺 −1 (1 − 2 (1 − √1 − 𝒜−1 ), 2𝑛 − 2) 1

and 𝐻E2𝐿 = 𝐺 −1 (2 (1 − √1 − 𝒜−1 ), 2𝑛 − 2), respectively. Clearly, these are probability limits and are flexible enough to be adapted in practice if the equal false alarm probability assignment does not fit the particular application. Further details about the probability of an OOC signal for the two chart scheme are given in Result A.2 of the supplementary material. 6.

Implementation In order to use the proposed charts we need to find and use the appropriate control limits

corresponding to a desired nominal IC ARL. Except for the SE-LR charts, this can be accomplished by using the analytical expressions given in the earlier sections. For the SE-LR chart, exact analytical calculations are not possible in a closed form. In that case, we obtain the control limits through simulation. Then for consistency, we used simulations in R to compute

16

all of the control limits, and where possible we used the analytical expressions to verify the results. As expected, we find that higher nominal IC ARL values require higher UCLs and lower LCLs. < Table 1 Here > Next we compare the performance of the proposed charts. 7.

Performance Comparisons The performances of the one-chart and two-chart schemes are evaluated in a simulation

study for various origin and scale shifts. Note that, for the proposed charts, the run length variable follows a geometric distribution which is completely characterized by its mean, the ARL. We have also provided the run length standard deviation (SDRL) as it shows the accuracy of the computational results.

We use 100,000 replications in the Monte-Carlo study and

therefore the standard error of the estimated ARL is approximately, 𝑆𝐷𝑅𝐿

.

√100,000

𝑆𝐷𝑅𝐿 √𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑟𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠

=

The SDRL values show that the results of the Monte-Carlo study are consistent and

robust. Table 2A provides the ARL and the SDRL values of the run length distribution when 𝑛 = 5. We use 𝜃 = 0 and 𝜆 = 1 for our IC case, which corresponds to the exponential distribution with mean 1. < Table 2A Here > From Table 2.A, it is seen that the SE-LR chart outperforms the other charts for downward shifts in the scale parameter 𝜆, especially when the shift in the origin parameter 𝜃 is not large. For upward shifts in the scale parameter, the SEMLE-2 chart is the best but the SEMLE-Max chart exhibits reasonably good performance as well.

Note that the standard

exponential distribution is used as the IC distribution in Table 2.A and so in that set up, we

17

cannot study the effect of a decreasing origin parameter. Therefore, in Table 2.B we consider the IC distribution as a two parameter exponential with mean 110 and standard deviation 100 (i.e., 𝜃0 = 10; 𝜆0 = 100) to study the comparative performance of the three proposed charts. We see from Table 2.B. that the results confirm the findings of Table 2.A for an increase in the origin parameter. Additionally, it shows that for a downward shift in origin, the SEMLE-Max and the SEMLE-2 charts behave in a similar fashion and are better than the SE-LR chart. < Table 2.B Here > In Table 2.C we summarize our findings in order to provide some guidance for choosing the appropriate chart under various situations. It is seen that no chart is best in all situations, but overall, the SEMLE-2 chart can be recommended in practice. < Table 2.C Here > Finally we discuss another property of the proposed charts. Intuitively, out-of-control ARL values for a control chart should be smaller than the in-control ARL. Control charts which have this property are called ARL-unbiased (see Acosta-Mejia, 1998) and charts for which this is not true are called ARL-biased. The unbiasedness of a control chart is similar to the idea of an unbiased test of hypothesis which is expected to be unbiased, that is, its power should be at least equal to its size. We see that all three of the proposed charts exhibit some ARL bias for certain shift sizes. The SE-LR chart, for example, is biased for decreases in the origin parameter that are accompanied by small or no decreases in the scale parameter. Similarly, the SEMLE-Max chart or the SEMLE-2 charts display bias for increases in the origin parameter, particularly, when there is also a small downward shift in the scale parameter. In general, the SE-LR chart is ARLunbiased for increases in the origin but it has worse performance than the SEMLE-Max chart whenever an upward shift in the scale occurs. The SEMLE-2 and the SEME-Max charts are

18

promising alternatives for the downward shift in origin where the SE-LR chart shows bias when there is no change in the scale parameter. For example, when 𝑛 = 5, 𝜃0 = 0, 𝜆0 = 1, and there is a shift of 0.1 in the origin but no scale shift, the resulting ARL is approximately 539.96, a value larger than the nominal IC ARL of 500. This suggests that when a small shift occurs only in the origin parameter, the chart is unable to detect it. To see this mathematically, note that when 𝑛 = 5, 𝜃 = 0.1, and 𝜆 = 1, the probability of a signal (exceeding the UCL 𝐻ℳ = 3.29) is 𝑃(ℳ > 3.29) = 1 − 𝑃(ℳ ≤ 3.29) = 0.00182695. Hence the OOC ARL is

1 𝑃(ℳ>3.29)

= 547.36, which is greater than the

nominal ARL0, 500, so that chart is ARL-biased. To investigate this issue more carefully, we took a closer look at the relationship between the sample size (𝑛), the origin shift, and the OOC ARL for the SEMLE-Max chart. Table 3 provides the ARL and the SDRL values for several small origin shifts (= 0.02(0.02) 0.20) at various sample sizes (n = 5, 10, and 30). From Table 3, we see that the larger the sample size, the smaller the shift size that the chart can detect. For example, when 𝑛 = 5, the chart is unable to detect shifts of size 0.1; however, a shift of this size can be detected when 𝑛 = 10 or 30. Moreover, when 𝑛 = 10, the chart is unable to detect shift of size 0.07 or smaller, though the chart can detect shifts as small as 0.03 when 𝑛 = 30. The bias can be investigated mathematically as well. It can be seen that when 𝜃0 = 0 and 𝜆0 = 1 and there has been a shift in the origin parameter but not in the scale parameter, bias may 1

occur when: 0 < 𝜃 < 2𝑛 𝐺 −1 (𝛷(𝑔), 2) − 𝐺 −1 {𝛷(𝑔) − 𝛷(−𝑔), 2}, where 𝑔 > 0 is the specified upper control limit. (For the detailed derivation, readers may be referred to Result A.3 of the

19

supplementary materials.) This confirms the relationship between the sample size and the detectable shift size observed in Table 3. < Table 3 Here > We next illustrate the application of proposed charts for jointly monitoring the origin and scale parameters of a shifted exponential distribution with an example. 8.

Illustrative Example Consider a data set referred to in the introduction and used in the literature by Grubbs

(1971), Engelhardt and Bain (1978), Lawless (1982), Roy and Mathew (2005), among others. The data are the mileages for some military personnel carriers that failed in service. The n = 19 observations are : 162, 200, 271, 320, 393, 508, 539, 629, 706, 777, 884, 1008, 1101, 1182, 1463, 1603, 1984, 2355 and 2880. The MLE of the two parameters, namely the origin and the scale, are obtained as 162 and 836, respectively, after rounding-off. Note that a two parameter exponential distribution provides a significantly better fit for these data; a Kolmogorov-Smirnov goodness of fit test yields a p-value greater than 0.99, whereas the pvalue is about 0.73 for a single-parameter exponential. To illustrate our control charts, we first simulate 10 samples of size 5 from the twoparameter exponential distribution with 𝜃0 = 162 and 𝜆0 = 836. The samples are (reading from left to right; each row is a sample) are given in Table 4. Again, a Kolmogorov-Smirnov test shows that for these data a two parameter exponential distribution is a better fit than a single parameter exponential (p-value of 0.33 against 0.14). Table 4. IC Data from an exponential distribution with origin 162 and scale 836 Sample No. 1

Observations 684

370

1874

961

272

20

2 3 4 5 6 7 8 9 10

1445 622 1038 214 529 2228 1395 1681 198

1942 1640 828 1845 637 1456 327 219 805

372 393 2606 760 1533 249 288 1546 769

413 723 244 595 705 337 795 222 3007

237 957 571 366 447 767 164 339 788

Next we simulate 10 more samples each of size 5 from a potentially out of control situation, namely an exponential distribution with 𝜃 = 324 (twice 𝜃0 ) and 𝜆 = 209 (one fourth of 𝜆0 ). This is a situation where the mean time to failure has decreased even though the origin parameter has increased, due possibly to a reduction in the variance (i.e., process improvement). The samples are obtained are presented in Table 5. Table 5. OOC Data from exponential distribution with origin 324 and scale 209 Sample No.

Observations

1 373 695 735 344 620 2 440 431 349 462 474 3 372 508 527 944 335 4 415 643 547 622 370 5 568 377 482 362 336 6 1130 453 988 501 349 7 499 949 329 481 524 8 413 431 389 395 541 9 368 589 748 563 367 10 328 386 571 481 348 Suppose that the military personnel management wishes to monitor the mileages of the vehicles that failed in service, which in turn reduces to monitoring the origin and the scale parameters of the exponential distribution. For this purpose we apply the three proposed charts using a nominal IC ARL of 500. We have, in this example, 𝑛 = 5 as before and therefore, the same control limits from Example-1 are used.

21

The resulting control charts are shown in Figures 1-3, respectively. Usually, the visibility of the SE-LR chart on a natural scale is not very good and we recommend using the log scale for the SE-LR chart. Figure 1 shows that an out of control signal is given at the same 18th test sample for the SE-LR chart on the log-scale. Figure-2 shows that the SEMLEMax chart also detects a shift at the 18th test sample. Whereas for the SEMLE-2 scheme, shown in Figure 3, we find no shift in the origin chart but a signal in the scale chart at the very same 18th test sample. < Figures 1-2-3 Here > Need some comments here. 9.

Summary and Conclusions Monitoring the origin and the scale parameters of an exponential distribution is an

important problem in certain practical situations. We consider three new control charts for this problem, based on the maximum likelihood estimators of the parameters but with different approaches of combining the two plotting statistics. However, much more work in this area is needed. For example, control charts based on some combinations of 𝐸1,modified and 𝐸2 will be worth considering in the future. Also, the proposed charts are suitable for sample sizes larger than 1, that is, for the rational subgroups setting, since they rely on statistics which are functions of order statistics. Many situations in today’s data rich environments involve individual data collected over time and the charts need to be adapted for that situation. Finally, the proposed charts have been developed for the known parameter case, Case K, and as such their performances are expected to be degraded if the parameters need to be estimated and the estimates are simply plugged into these charts for known parameter case. The adaptation of

22

these charts for the unknown parameters case, Case U, is not straightforward and will be studied elsewhere. Acknowledgements: The authors would like to thank the Editor, an anonymous reviewer and Prof. Bill Woodall of Virginia Tech for their comments on earlier drafts of the paper. The anonymous reviewer has helped us to enrich the paper in tuning our motivating examples and follow up procedures. Thanks are also due to Prof. Doug Hawkins of University of Minnesota for useful discussions and suggestions. The authors also wish to thank Prof. Shih-Chou Kao of Kao Yuan University, Taiwan who brought to our attention several examples from practice where the proposed chart can be used. All errors and omissions remain the responsibility of the authors. References Acosta-Mejia, C. A. (1998). “Monitoring Reduction in Variability with the Range,” IIE Transactions, 30, 515-523. Baten, M.A., Kamil, A. A. (2009). “Inventory Management Systems with Hazardous Items of Two Parameter Exponential Distribution,” Journal of Social Sciences 5(3), pp. 183-187. Chen, G. & Cheng S.W. (1998). “Max chart: combining X-bar chart and S chart,” Statistica Sinica, 8, 263-271. Engelhardt, M. & Bain, L. J. (1978). “Tolerance limits and confidence limits on reliability for the

two-parameter exponential distribution,” Technometrics, 20, 485-489. Epstein, B. (1960). “Estimation of the parameters of two parameter exponential distributions from censored Samples,” Technometrics, 2(3), 403-406. Govindarajulu, Z. (1966). “Characterization of the exponential and power distributions,” Scandinavian Actuarial Journal, 1966(3-4), 132-136. Gibbons, J. D. & Chakraborti, S. (2010). Nonparametric Statistical Inference, 5th ed. CRC Press, Boca Raton, FL. Grubbs, F. E. (1971). “Fiducial bounds on reliability for the two-parameter negative exponential distribution,” Technometrics, 13, 873-876.

23 Hogg, R.V., McKean, J.W. & Craig, A.T. (2005). Introduction to Mathematical Statistics, 6th Edition. Englewood Cliffs, NJ: Prentice Hall. Johnson, N.L. & Kotz, S. (1970). Distributions in Statistics, Volume 1: Continuous Univariate Distributions, Boston: Houghton Mifflin. Kao, S.C. (2010). “Normalization of the shifted exponential distribution for control chart construction,” Journal of Applied Statistics, 37(7), 1067-1087. Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data, John Wiley & Sons, New York. McCracken, A. K. & Chakraborti, S. (2013). “Control Charts for Joint Monitoring of Mean and Variance: An Overview,” Quality Technology and Quantitative Management, 10(1), 17-36. McCracken, A. K., Chakraborti, S. & Mukherjee, A. (2013). “Control charts for simultaneous monitoring of unknown mean and variances of normally distributed processes,” Journal of Quality Technology, Forthcoming. Montgomery, D. C. (2005). Introduction to Statistical Quality Control. Fifth Edition, John Wiley, New York. 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, 26(1), 129-160. Roy, A. & Mathew, T. (2005). “A generalized confidence limit for the reliability function of a two parameter exponential distribution,” Journal of Statistical Planning and Inference, 128, 509–517. Sürücü, B. & Sazak, H. S. (2009). “Monitoring reliability for a three-parameter Weibull distribution,” Reliability Engineering and System Safety, 94(2), 503-508. Tanis, E.A. (1964). “Linear forms in the order statistics from an exponential distribution,” The Annals of Mathematical Statistics, 35(1), 270-276. Tobias, P.A. and Trindade, D. C. (2012). Applied Reliability. Third Edition, CRC Press, Boca Raton, FL.

Table 1: Control Limits for the proposed charts when 𝜽 and 𝝀 are known and 𝒏 = 𝟓, for various nominal IC ARLs

Nominal IC ARL 125 250

SEMLE-Max Chart UCL 𝐻ℳ 2.88 3.09

SE-LR Chart LCL 𝐻Λ 0.000899 0.000365

SEMLE-2 Scheme (Origin Chart UCL/ Scale Chart UCL/ Scale Chart LCL) 𝐻𝐸1 , 𝐻𝐸2𝑈 , and 𝐻𝐸2𝐿 11.04/ 24.35/1.04 12.43/ 26.12/0.86

24 500 750 1000

3.29 3.40 3.48

0.000154 0.0000929 0.0000659

13.82/ 27.87/0.71 14.63/ 28.88/0.64 15.20/ 29.59/0.59

Table 2.A. Run length characteristics for the proposed charts for various values of 𝜽 and 𝝀 when In-Control Population is Two Parameter Exponential with 𝜽𝟎 = 𝟎, 𝝀𝟎 = 𝟏, and 𝒏 = 𝟓; SEMLE-Max Chart SDRL 16.62

SE-LR Chart SDRL 16.64

SEMLE-2 Scheme SDRL 17.30

θ 0

λ 0.25

ARL 17.12

0.25

0.25

17.86

17.31

8.04

7.50

17.87

17.39

0.5

0.25

17.76

17.26

4.12

3.60

17.77

17.21

1

0.25

17.81

17.36

1.49

0.86

17.90

17.35

1.5

0.25

1.44

0.79

1.01

0.12

1

0

0

0.5

141.33

140.78

121.93

121.66

165.16

165.39

0.25

0.5

164.64

163.13

49.17

48.60

165.49

166.29

0.5

0.5

163.90

162.96

20.58

20.10

161.38

160.97

1

0.5

85.95

85.46

4.21

3.68

35.79

35.10

1.5

0. 5

1.21

0.51

1.29

0.61

1

0

0

0.75

462.96

465.57

343.71

343.39

646.61

643.75

0.25

0.75

599.91

600.33

132.25

132.23

504.85

503.50

0.5

0.75

390.97

391.42

51.68

51.16

233.93

233.17

1

0.75

30.64

30.09

8.41

7.89

12.53

11.99

1.5

0.75

1.14

0.40

1.70

1.08

1

0

0

1

498.67

498.85

506.65

512.37

503.12

503.82

0.25

1

359.83

355.74

193.34

192.78

222.28

221.88

0.5

1

141.72

140.99

71.55

71.35

76.28

75.72

1

1

13.23

12.53

10.70

10.15

6.71

6.21

1.5

1

1.10

0.33

1.84

1.26

1

0

0

1.25

136.85

135.79

323.75

322.02

116.35

115.47

0.25

1.25

91.62

91.17

127.88

126.77

65.14

64.95

0.5

1.25

46.44

45.49

51.51

50.91

29.49

29.02

1

1.25

7.78

7.26

8.76

8.18

4.54

4.03

1.5

1.25

1.09

0.30

1.74

1.16

1

0

0

1.5

41.88

41.04

120.79

119.82

36.70

36.26

0.25

1.5

31.76

31.54

54.74

54.61

24.99

24.49

0.5

1.5

19.78

19.22

26.50

26.16

14.44

13.92

1

1.5

5.26

4.67

6.01

5.50

3.41

2.85

1.5

1.5

1.06

0.26

1.56

0.93

1

0

0

1.75

17.83

17.43

47.55

46.74

16.12

15.60

0.25

1.75

14.44

13.92

25.08

25.28

12.27

11.70

0.5

1.75

10.31

9.73

13.28

12.54

8.29

7.76

1

1.75

3.85

3.28

4.14

3.62

2.74

2.19

1.5

1.75

1.05

0.24

1.42

0.78

1

0

0

2

9.75

9.26

22.57

21.97

8.90

8.42

0.25

2

8.15

7.71

13.14

12.64

7.32

6.77

ARL 17.20

ARL 17.88

25 0.5

2

6.47

5.97

7.94

7.51

5.40

4.89

1

2

3.00

2.44

3.01

2.47

2.29

1.71

1.5

2

1.05

0.22

1.31

0.64

1

0

Table 2.B. Run length characteristics for the proposed charts for various values of 𝜽 and 𝝀 when In-Control Population is Two Parameter Exponential with mean=110; variance=110, i.e.; 𝜽𝟎 = 𝟏𝟎, 𝝀𝟎 = 𝟏𝟎𝟎, and 𝒏 = 𝟓;

SEMLE-Max Chart θ

λ

5 5 5 5 5 7 7 7 7 7 10 10 10 10 10 13 13 13 13 13 15 15 15 15 15

60 80 100 120 140 60 80 100 120 140 60 80 100 120 140 60 80 100 120 140 60 80 100 120 140

ARL

SE-LR Chart

SDRL

2.91 3.71 4.49 5.18 5.67 4.46 5.79 7.08 8.19 8.66 247.09 533.29 497.09 181.59 64.61 310.77 792.07 636.54 186.63 63.77 311.13 788.31 609.16 180.43 62.03

ARL

2.37 3.17 3.96 4.67 5.16 3.91 5.27 6.57 7.63 8.14 247.11 533.45 497.13 181.10 64.20 312.24 793.07 640.39 186.41 63.404 309.72 787.15 607.38 180.22 61.38

243.13 474.23 611.09 452.69 214.85 223.57 442.38 566.41 421.85 200.27 199.98 392.77 503.85 373.69 182.89 178.06 349.26 447.44 335.78 165.27 166.12 324.64 414.02 310.61 154.67

SEMLE-2 Scheme

SDRL

ARL

242.44 472.50 612.89 451.52 213.79 223.35 443.30 561.18 420.71 199.64 199.90 393.11 501.94 371.40 182.15 177.25 351.41 445.88 334.94 163.98 166.02 324.18 412.05 309.41 153.35

SDRL

2.91 3.73 4.48 5.18 5.59 4.46 5.79 7.11 8.13 8.52 311.79 743.16 499.82 154.19 55.87 310.58 725.02 465.38 144.26 53.44 309.81 714.42 436.83 138.51 51.93

2.35 3.21 3.94 4.65 5.08 3.94 5.29 6.57 7.58 8.02 310.93 745.58 498.84 152.82 55.47 311.68 725.68 466.13 143.38 52.88 308.27 717.94 437.02 137.76 51.43

Table 2.C. General Recommendation for Selecting the Best Chart Direction of Shift of 𝜆 Fixed

Increasing

Decreasing

Direction

Fixed

In-Control Set up

SEMLE-2 chart

SE-LR chart

of Shift

Increasing

SE-LR chart

SEMLE-2 chart

SE-LR chart

of 𝜃

Decreasing

SEMLE-Max chart or SEMLE-2 chart*

26

Table 3: Run length characteristics for the SEMLE-Max chart for small origin shifts not accompanied by scale shifts for various values of 𝒏 λ =1 θ 0

ARL 501.37

𝒏=𝟓 SDRL 494.70

ARL 500.53

𝒏 = 𝟏𝟎 SDRL 498.97

ARL 500.36

𝒏 = 𝟑𝟎 SDRL 511.14

0.02

644.45

636.15

618.29

619.69

513.70

507.97

0.04

621.79

613.15

581.69

585.36

376.58

371.10

0.06

598.68

593.15

518.57

519.68

249.18

246.86

0.08

580.14

584.88

477.05

482.54

151.56

152.43

0.10

562.28

558.75

426.39

426.90

90.15

90.19

0.12

520.79

522.69

374.68

369.67

52.25

51.72

0.14

495.53

494.49

334.09

336.77

29.42

2903

0.16

479.55

481.31

289.49

287.69

16.06

15.64

0.18

443.43

443.37

247.75

247.94

8.97

8.50

0.20

425.15

423.35

211.88

211.27

4.89

4.35

Figure 1: SE-LR chart with Log-Transform of Plot Statistics and LCL for Mileage data

-4 -6 -8

Natural Log of Plotting Statistic

-2

0

SE-LR CONTROL CHART FOR MILEAGE DATA IN LOG-SCLE

-10

LCL

5

10

15

20

Sample Number

Figures 2: SEMLE-max Chart for Mileage data

4

SEMLE-Max CONTROL CHART FOR MILEAGE DATA

2 1

LCL 0

Plotting Statistic

3

UCL

5

10 Sample Number

15

20

28

Figure 3: SEMLE-2 Scheme for Mileage Data LOCATION CHART FOR MILEAGE DATA BASED ON SEMLE-2 APPROACH

5

LCL

-5

0

Plotting Statistic

10

UCL

5

10

15

20

Sample Number

Location Chart

30

40

SCALE CHART FOR MILEAGE DATA BASED ON SEMLE-2 APPROACH

20 10

LCL

0

Plotting Statistic

UCL

5

10 Sample Number

Scale Chart

15

20