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International Journal of Production Research
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The density control chart: a general approach
for constructing a single chart for simultaneously
monitoring multiple parameters
Hung-Chia Chen a , Arthur B. Yeh b , Chia-Ling Yen c & Lin-An Chen a a
Institute of Statistics , National Chiao Tung University , Hsinchu , Taiwan b
Department of Applied Statistics and Operations Research , Bowling Green State University , Bowling Green , OH 43403 , USA
c
Institute of Statistics , National Tsing Hua University , Hsinchu , Taiwan Published online: 23 Aug 2011.
To cite this article: Hung-Chia Chen , Arthur B. Yeh , Chia-Ling Yen & Lin-An Chen (2012) The density control chart: a
general approach for constructing a single chart for simultaneously monitoring multiple parameters, International Journal of Production Research, 50:14, 3904-3919, DOI: 10.1080/00207543.2011.604050
To link to this article: http://dx.doi.org/10.1080/00207543.2011.604050
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Vol. 50, No. 14, 15 July 2012, 3904–3919
The density control chart: a general approach for constructing a single chart for
simultaneously monitoring multiple parameters
Hung-Chia Chena, Arthur B. Yehb*, Chia-Ling Yenc
and Lin-An Chena
a
Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan;
b
Department of Applied Statistics and Operations Research, Bowling Green State University, Bowling Green, OH 43403, USA;cInstitute of Statistics, National Tsing Hua University, Hsinchu, Taiwan
(Received 31 May 2010; final version received 27 June 2011)
The majority of the existing literature on simultaneous control charts, i.e. control charting mechanisms that monitor multiple population parameters such as mean and variance on a single chart, assume that the process is normally distributed. In order to adjust and maintain the overall type-I error probability, these existing charts rely largely on the property that the sample mean and sample variance are independent under the normality assumption. Furthermore, the existing charting procedures cannot be readily extended to non-normal processes. In this article, we propose and study a general charting mechanism which can be used to construct simultaneous control charts for normal and non-normal processes. The proposed control chart, which we call the density control chart, is essentially based on the premise that if a sample of observations is from an in-control process, then another sample of observations is no less likely to be also from the in-control process if the likelihood of the latter is no less than the likelihood of the former. The density control chart is developed for normal and non-normal processes where the distribution of the plotting statistic of the density control chart can be explicitly derived. Real examples are given and discussed in these cases. We also discuss how the density control chart can be constructed in cases when the distribution of the plotting statistic cannot be determined. A discussion of potential future research is also given.
Keywords: hypothesis testing; likelihood function; multiple parameters; non-normal processes; Shewhart control chart; simultaneous control chart
1. Introduction
The statistical control chart is perhaps the most widely used tool among the set of techniques, such as the Pareto chart and cause-and-effect diagram, that form the core of statistical process control (SPC). First developed by Shewhart (1925), the use of control charts to monitor process quality has become a common practice in many industrial applications.
The development of control charts generally assumes that the quality characteristic to be monitored, be it univariate or multivariate, follows a certain distribution and that various population parameters of that distribution need to be monitored. For instance, under univariate normal processes, the X- and S-charts are specifically used to monitor the process mean and process standard deviation, respectively.
In the literature, there are two primary approaches for developing control charting schemes for simultaneously monitoring the process mean and variance. One approach is to combine two control charts, one for monitoring the process mean and the other for monitoring the process variance. These charts include (i) the combination of Shewhart X(or X ) and R (MR or S) charts, see, e.g., Jones and Case (1981), Rahim (1989), Saniga (1989, 1991), Costa (1993, 1998), Rahim and Costa (2000), Costa and Rahim (2000, 2002), Ohta et al. (2002), De Magalha˜es and Moura Neto (2005), He and Grigoryan (2006); (ii) the combination of EWMA-type charts, see, e.g., Sweet (1986), Domangue and Patch (1991), Gan (1995), Reynolds and Stumbos (2001); (iii) the combination of CUSUM-type charts, see, e.g., Takemoto et al. (2003); and (iv) the combination of the generally weighted moving average (GWMA)-type charts, see, e.g., Sheu et al. (2009), which developed a combined scheme consisting of a two-side GWMA mean chart and a two-side GWMA variance chart for monitoring the process mean and variance, respectively.
*Corresponding author. Email: [email protected]
ISSN 0020–7543 print/ISSN 1366–588X online ß 2012 Taylor & Francis
http://dx.doi.org/10.1080/00207543.2011.604050 http://www.tandfonline.com
The other approach is to use omnibus-type test statistics, i.e. combining charting statistics designed for different parameters into a single control chart for simultaneously monitoring both process mean and variance. Cheng and Thaga (2006) gave an overview of the existing studies which use a single chart to simultaneously monitor process parameters. They summarised the control charts that have been developed into two categories. One is to simultaneously plot two quality characteristics on the same chart, called the simultaneous control chart. Examples of the simultaneous control chart include the Shewhart-type charts (White and Schroeder 1987, Spiring and Cheng 1998); the EWMA-type chart (Domangue and Patch 1991); and the synthetic control chart with two-stage testing (Costa et al. 2009). The other is to use a plotting variable to represent the process mean and variance, called the single control chart. Existing control charts that fall into the single control chart category include (i) the Shewhart-type charts (Cheng and Li 1993, Chao and Cheng 1996, Chen and Cheng 1998, Yeh and Lin 2002, Hawkins and Zamba 2005, and Hawkins and Deng 2009); (ii) the EWMA-type charts (Xie 1999, Gan 2000 and Chen et al. 2001, 2004); and (iii) the CUSUM-type charts (Thaga 2003 and Wu and Tian 2005). Other charts that also rely on a single plotting statistic include the EWMA control chart based on a non-central chi-square statistic (called NCS chart) proposed by Costa and Rahim (2004), a self-starting control chart based on the likelihood ratio test (LRT), the EWMA procedure (SSELR) proposed by Li et al. (2010), an EWMA control chart based on the generalised likelihood ratio test (GLRT) proposed by Zhang et al. (2010), and a control chart which integrates the EWMA procedure with the GLRT statistic proposed by Zhou et al. (2010).
In addition, the semicircle control chart (Chao and Cheng 1996) plots two statistics against each other on a semicircle, as indicated by the positions of the mean and the standard deviation. When an out-of-control signal occurs, the chart would show which parameter has shifted from its target value. However, a disadvantage of this chart is that it loses track of the time sequence of the samples. Hawkins and Deng (2009) developed a single chart for simultaneously monitoring the mean and variance of a normal process based on the GLRT for testing H0: ¼ 0 and ¼ 0. The proposed chart, which they called the GLR chart, was shown to outperform the
conventional X- and S-charts.
Zhang et al. (2010) mentioned that all of the aforementioned single charts are shown to have satisfactory performance compared with the combination-type charts. Nevertheless, most of the existing control charts under univariate normal processes, whether a single chart or two separate charts are used, make explicit use of the property that the sample mean and sample variance are independent statistics, thus making it relatively easy to control the overall type-I error probability. However, in many applications where the process is not normal or when other non-independent statistics are used even for normal processes, it is sometimes difficult to control the overall type-I error probability. Moreover, it is usually non-trivial to extend a control charting mechanism designed for a specific underlying distribution to other types of distribution. Our main motivation in this article is to develop a general control charting mechanism which can be used to combine the monitoring of multiple parameters into a single chart. The proposed charting mechanism is general in that it can be applied to normal as well as non-normal processes, while maintaining the overall type-I error probability. In essence, the construction of the proposed chart is based on recognising that when the process is in control, the quality characteristic to be monitored follows a specified probability density function. The proposed chart is then derived from the likelihood for testing the hypothesis that the probability density function of a given sample is equal to the specified one. We will call the proposed chart the density control chart.
The rest of the article is organised as follows. We first discuss in detail the general methodology of the density control chart. A density control chart is specifically derived for normal processes. Hoping to demonstrate the flexibility of the proposed density chart, we also apply the density control charting mechanism to a negative exponential distribution which has been used to model certain processes commonly found in the semiconductor manufacturing industry. Two real examples are given for the normal and negative exponential processes. There are cases in which the distribution of the plotting statistic used in the density chart is too complex to derive. We propose using simulations to approximate the control limit in these cases. This is further discussed using a Gamma and an exponential distribution as examples. We conclude the article with some discussion of potential future research.
2. The general methodology
Let X denote the quality characteristic of interest. When the process is in control, we assume that X follows a certain distribution whose density function is denoted as f (x; 1, 2, . . . , p), where 1, 2, . . . , pare population parameters.
These population parameters are assumed to be either known or can be reasonably estimated by b1, b2, . . . , bp
using data obtained from Phase I control when the process was in control. When the Phase II monitoring begins, independent samples of size n each are repeatedly taken from the process.
When the process is in-control, assuming that the true in-control parameter values are 1, 2, . . . , p. For any
given sample of size n, x1, x2, . . . , xn from the in-control process, the joint likelihood is Lðx1, x2, . . . , xn;
1, 2, . . . , pÞ ¼Qni¼1f ðxi; 1, 2, . . . , pÞ: Suppose that xð1Þ1 , xð1Þ2 , . . . , xð1Þn are considered to be a sample from the
in-control process, and xð2Þ1 , xð2Þ2 , . . . , xð2Þ
n are another sample. If Lðx
ð1Þ 1 , x
ð1Þ
2 , . . . , xð1Þn ; 1, 2, . . . , pÞ
Lðxð2Þ1 , xð2Þ2 , . . . , xð2Þ
n ; 1, 2, . . . , pÞ, then from the perspective of testing the hypothesis that H0: the process is in
control, the sample ðxð2Þ1 , xð2Þ2 , . . . , xð2Þ
n Þ provides an even stronger evidence of supporting that the process is in
control. In other words, if ðxð1Þ1 , xð1Þ2 , . . . , xð1Þ
n Þcomes from the in-control process, there is no reason not to support
the same conclusion when ðxð2Þ1 , xð2Þ2 , . . . , xð2Þn Þis observed. This line of thinking leads us to consider LðX1, X2, . . . , Xn; 1, 2, . . . , pÞ
as the charting mechanism. Specifically, given a random sample x1, x2, . . . , xn, one calculates the joint likelihood
L(x1, x2, . . . , xn; 1, 2, . . . , p) and plots it against the sampling sequence. The control chart will signal when the
plotting statistic falls below the lower control limit (LCL), where the LCL is to be chosen such that
PðLðX1, X2, . . . , Xn; 1, 2, . . . , pÞ LCLÞ ¼ , ð1Þ
where is a pre-determined probability of type-I error. For example, in a typical 3- limit X-chart for normal processes, the is about 0.0027. Note that in the case when 1, 2, . . . , pare unknown and have to be estimated
using Phase I data, the parameter estimates b1, b2, . . . , bpare treated as known in Phase II monitoring. Therefore, we
will use LðX1, X2, . . . , Xn; b1, b2, . . . , bpÞinstead of L(X1, X2, . . . , Xn;1, 2, . . . , p). Subsequently, in the case when the
parameters have to be estimated from Phase I data, we will continue to use b1, b2, . . . , bpto denote the true parameter
values in Phase II monitoring, if no confusion is caused.
We will call the proposed chart the density control chart since it relies on calculating the joint density of the sample observations. The density control charting mechanism is quite general in that it can be applied to any density function f (x; 1, 2, . . . , p). However, in order to explicitly identify the LCL, one needs to know the
distribution of L(X1, X2, . . . , Xn; 1, 2, . . . , p) (or LðX1, X2, . . . , Xn; b1, b2, . . . , bpÞ). In some cases, it is possible to
derive the distribution of the joint density. In the subsequent sections, we will discuss two such cases. One is the most commonly used normal process and the other is based on a negative exponential distribution which has been used to model certain processes in semiconductor manufacturing applications. As pointed out by one referee, the LðX1, X2, . . . , Xn; b1, b2, . . . , bpÞ is only an estimate of the true L(X1, X2, . . . , Xn; 1, 2, . . . , p), and
that the statement in (1) is only approximately true when the true parameters are replaced by their corresponding estimates. Further, the likelihood estimate LðX1, X2, . . . , Xn; b1, b2, . . . , bpÞ could be poor if the
parameter estimates b1, b2, . . . , bp obtained from Phase I data are poor, which will affect the performance of the
density control chart. For a recent review of the effects of parameter estimation on the control chart properties, please see Jensen et al. (2006).
3. The density control chart under a normal process
We assume that when the process is in control, the quality characteristic X is distributed as Nð0, 02Þ, where 0and
0are unknown. We also assume that m samples of size n each are available for estimating 0and 0and that
b
0¼m1Pmj¼1Xjandb02¼m1
Pm
j¼1s2j, where Xjand s2j are, respectively, the sample mean and sample variance of the jth
sample, j = 1, 2, . . . , m.
When the Phase II monitoring begins, it is customary, as in most of the existing literature, to assume that the true in-control parameters 0and 20 are equal tob0andb20, respectively. For each sample of size n, x1, x2, . . . , xn, one
calculates Lðx1, x2, . . . , xn;b0,b20Þ ¼ 1 ð2Þn=2ðb2 0Þ n=2exp Pni¼1ðxib0Þ2 2b2 0
and plots Lðx1, x2, . . . , xn;b0,b20Þagainst the sampling sequence. The chart signals when
Lðx1, x2, . . . , xn;b0,b20Þ LCL: ð2Þ
The LCL ¼ l ðb0,b20Þis chosen such that
Pb0,b02
ðLðX1, X2, . . . , Xn;b0,b02Þ l ðb0,b20ÞÞ ¼1 , ð3Þ
where is pre-determined.
Note that givenb0 andb20, Xi’s are independent and identically distributed (i.i.d.) as Nðb0,b02Þand therefore
Pn
i¼1ðxib0Þ2=b02 is distributed as 2n, a chi-square distribution with n degrees of freedom. Straightforward
calculations show that the restriction in (3) leads to l ðb0,b02Þ ¼ 1 ð2Þn=2ðb2 0Þ n=2exp 2n,1 2 ( ) , ð4Þ
where Pð2n2n,1Þ ¼1 . It should also be noted that under the normality assumption, Lðx1, x2, . . . , xn;b0,b20Þ LCL if and only if Pn i¼1ðxib0Þ2 b2 0 2n,1: ð5Þ
Therefore, one can use the plotting statisticPni¼1ðxib0Þ2=b20 and the chart signals if it exceeds 2n,1. Note that
although the plotting statistic does depend onb0 andb02, the control limit does not.
Example 1: A vane operating example
The process control of vane operating, which is an important functional parameter of a component part in jet aircraft engines, has been studied by Montgomery et al. (2004) who used statistical control charts to assess the statistical stability of this manufacturing process. Twenty samples each of size 5 were collected in Phase I and used to construct the X- and R-charts. The X-chart indicated that samples 6, 8, 11 and 19 were out of control, while the R-chart indicated that sample 9 was out of control. Assuming that assignable causes can be identified and corrected for these out-of-control samples, updated X- and R-charts were calculated using the 15 remaining samples, and no additional samples were found to be out of control. For Phase II monitoring based on the 15 in-control Phase I samples, the upper and lower control limits are 36.1 and 30.33, respectively, for the X- chart, and 10.57 and 0, respectively, for the R- chart.
We use this example to demonstrate how the density chart can be used in real life applications. For ease of presentation, we construct the density chart based on the logarithm of the likelihood function. Based on the 15 Phase I samples,b0andb02were equal to 33.2133 and 3.3595, respectively. For Phase II monitoring, we generated 20
new samples, each of size 5. Of the 20 samples generated, sample 3 was generated from Nðb0þ1:75b0,b02Þ, sample
10 from Nðb0, ð2b0Þ2Þ, and sample 18 from Nðb0þ1:5b0, ð1:5b0Þ2Þ. The other 17 samples were generated from the
in-control process Nðb0,b20Þ. The X- and R-charts for 20 newly generated samples are shown in Figure 1. As can be
seen from Figure 1, only sample 3 was out of control on the X- chart and no sample was out of control on the R-chart.
Setting the type-I error probability equal to 0.0027, the corresponding density control chart calculates, for each sample x1, x2, . . . , x5, the plotting statistic
ln 0:0004885 exp P5 i¼1ðxi33:2133Þ2 2 3:2595 ( )! , and the chart signals if the plotting statistic falls below
LCL ¼ 5 2lnð2b 2 0Þ 1 2 2 5,0:9973¼ 16:7267:
The density chart of the 20 generated samples is shown in Figure 2, which shows that samples 3, 10, and 18 were out of control. It is interesting to point out that samples 10 and 18 were out-of-control in the density chart but not in the X- and R-charts.
In the case of normal processes, one can actually use the statistic Pn
i¼1ðxib0Þ2
b2 0
for diagnosing whether the mean or the variance or both are out of control. Rewrite the statistic as Pn i¼1ðxiX þ X b0Þ2 b2 0 ¼ðn 1Þs 2 b 2 0 þnðX b0Þ 2 b2 0 , ð6Þ
where X ¼Pni¼1xi=nand s2¼
Pn
i¼1ðxiXÞ2=n 1. Givenb0,b02 and , the region
ðX, s2Þ:ðn 1Þs 2 b 2 0 þnðX b0Þ 2 b2 0 2n,1 ð7Þ can be represented by a semi-circle whose center is at X ¼ 0 and S2¼0 as shown in Figure 3.
Any sample which produces ðX, s2Þ that lies outside of the semi-circle indicates that the sample is out
of control. If s2is closer to 0 while X is away from b0, this is an indication that there is a possible mean shift.
On the other hand, if X is closer to b0 while s2 is far away from 0, it is an indication that the variance
has changed. As can be seen from Figure 3, sample 3 was likely the result of an upward mean shift and sample 10 was mostly due to an increase in variance. Only sample 18 was likely due to changes both in mean and variance. Note that the semi-circle discussed here is very similar to that studied in Chao and Cheng (1996). They will be identical if 0and 0are assumed to be known. However, our proposed density chart has a much broader
appeal as it can be readily applied to other non-normal distributions.
0 2 4 6 8 10 12 14 16 18 20 30 31 32 33 34 35 36 37 UCL = 36.1 CL = 33.32 LCL = 30.33 3 ¯ X chart Sample number ¯ X 0 2 4 6 8 10 12 14 16 18 20 −2 0 2 4 6 8 10 12 UCL = 10.57 CL = 5.8 LCL = 0 R chart Sample number R
Figure 1. The X- and R- charts of the vane operating example for 20 new generated samples.
In a recent article, Hawkins and Deng (2009) proposed a simultaneous chart that is constructed based on the generalised likelihood ratio test for testing the hypothesis that H0: ¼ 0and ¼ 0. The plotting statistic to be
calculated for each sample, assuming 0and 0are known under H0, is given as
G ¼ U2þW nlnðWÞ þ nlnðnÞ n, where U ¼ ffiffiffi n p ðX 0Þ 0 , W ¼ðn 1Þs 2 2 0 :
They showed that the proposed chart, which they named the GLR chart, compared favourably to the conventional X- and S-charts. Note that if the parameters 0 and 0 are replaced by the estimates obtained
from the Phase I data, then the test statistic proposed in (6) for the density control chart is equal to U2þW. However, the two plotting statistics, (6) and G, are intrinsically different. For any given sample in Phase II, the plotting statistic of the density control chart under normality is essentially the likelihood function, while the GLR test based G, on the other hand, calculates the generalised likelihood ratio for testing H0 : ¼ 0
and ¼ 0 . In both plotting statistics, the mean and standard deviation are either assumed known or can be
estimated based on Phase I data.
We compare the performance of the density control chart with that of the GLR chart. We use the same data set as in Example 1 and treat the in-control process as having a normal distribution with 0¼33.2133 and 20 ¼3:3595.
The out-of-control process is considered to be normally distributed with ¼ 0þ 0 and ¼ 0 ( 40).
Numerous combinations of (, ) are considered. The performance measure is based on the average run length (ARL), where the run length is equal to the number of samples needed before the first out-of-control signal shows up on a control chart. The in-control ARL is approximately equal to 370.
Note that for the density chart, the ARL can be calculated as ARL ¼1 p, 30 31 32 33 34 35 36 37 0 2 4 6 8 10 12 14 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Semi−circle chart ¯ X S 2
Figure 3. The semi-circle chart of the vane operating example for 20 new generated samples.
0 2 4 6 8 10 12 14 16 18 20 −20 −18 −16 −14 −12 −10 −8 LCL = −16.7267 3 10 18 Density chart Sample number log L
Figure 2. The density control chart of the vane operating example for 20 new generated samples.
where p ¼ P 2 n,n2 2 2 n,1 2 ! , ð8Þ where 2
k,a denotes a non-central chi-square distribution with k degrees of freedom and non-centrality parameter a
and 2
n,1 is the 1 quantile of a central chi-square distribution with n degrees of freedom. The results of the
comparison are summarised in Table 1 and Table 2 for n ¼ 5 and 10, respectively.
As can be seen from Tables 1 and 2, the density control chart is not effective in detecting a decrease in variance (i.e. 51). It is also less effective than the GLR chart in detecting small shifts in mean only (i.e. ¼ 0, 15 1.25). On the other hand, when there is a small increase in variance only (15 2, ¼ 0) or a small increase in variance and a small shift in mean (15 2, 05 1), the proposed density chart is more effective than the GLR chart. It should be noted that in most industrial quality control applications, it is more important that a control chart can effectively detect quality deterioration as characterised by a shift in the mean (40) or an increase in variance ( 41) or both. Therefore, as far as the performance under normal process is concerned, the density control chart represents a viable alternative to the existing control charts for simultaneously monitoring process mean and variance. Furthermore, while the existing charts are limited to normal processes, the density control chart can be easily extended to non-normal processes (see Sections 4 and 5 for detailed discussion).
As mentioned earlier, replacing the parameters in the likelihood function by their estimates obtained from Phase I data is likely to affect the performance of the density control chart. Under normal processes, here we examine via simulations the effect of Phase I estimates on the performance of the density chart for n ¼ 5, and various total numbers of Phase I samples (m), assuming that 0¼0 and 0¼1. Given m, m Phase I samples of size 5 each were first
generated, andb0andb0were calculated. These estimates were then used to evaluate the ARL. Such a simulation was
then repeated 50,000 times, resulting in 50,000 ARLs, and the average of these 50,000 ARLs was calculated and tabulated.
Table 1. The ARLs of the density control chart (DCC) and the GLR chart when n ¼ 5 ( ¼ 0þ0, ¼ 0).
n ¼5 0.87 1.00 1.15 1.52 2.00 2.15 2.30 2.55 2.75 3.00 0.00 DCC 4713.61 370.37 58.25 6.14 2.11 1.79 1.58 1.37 1.27 1.18 GLR 302.65 370.38 242.13 20.80 3.95 2.99 2.41 1.86 1.61 1.42 0.25 DCC 2222.05 239.33 45.54 5.71 2.07 1.77 1.57 1.36 1.26 1.18 GLR 225.47 236.30 142.30 17.41 3.79 2.90 2.36 1.84 1.60 1.41 0.50 DCC 495.44 91.75 25.35 4.71 1.96 1.70 1.52 1.34 1.25 1.17 GLR 101.06 87.09 52.34 11.43 3.37 2.67 2.23 1.78 1.56 1.39 0.75 DCC 104.93 31.92 12.69 3.63 1.81 1.60 1.46 1.31 1.23 1.16 GLR 35.04 28.91 19.61 7.02 2.85 2.37 2.04 1.69 1.51 1.36 1.00 DCC 26.41 12.10 6.54 2.76 1.65 1.50 1.39 1.27 1.20 1.14 GLR 12.12 10.67 8.40 4.42 2.38 2.07 1.84 1.58 1.44 1.32 1.25 DCC 8.51 5.36 3.69 2.13 1.49 1.39 1.31 1.22 1.17 1.13 GLR 4.90 4.71 4.21 2.95 1.99 1.80 1.66 1.48 1.37 1.28 1.50 DCC 3.59 2.84 2.33 1.71 1.36 1.30 1.25 1.18 1.14 1.11 GLR 2.46 2.53 2.48 2.13 1.69 1.58 1.50 1.38 1.30 1.23 1.75 DCC 1.97 1.80 1.66 1.43 1.25 1.22 1.19 1.14 1.12 1.09 GLR 1.56 1.65 1.69 1.64 1.47 1.41 1.36 1.29 1.24 1.19 2.00 DCC 1.36 1.34 1.31 1.25 1.17 1.15 1.14 1.11 1.09 1.07 GLR 1.20 1.26 1.32 1.36 1.31 1.29 1.26 1.22 1.19 1.15 2.25 DCC 1.12 1.13 1.14 1.14 1.11 1.10 1.10 1.08 1.07 1.06 GLR 1.06 1.10 1.13 1.20 1.20 1.19 1.18 1.16 1.14 1.12 2.50 DCC 1.03 1.04 1.05 1.07 1.07 1.07 1.06 1.06 1.05 1.04 GLR 1.01 1.03 1.05 1.10 1.12 1.12 1.12 1.11 1.11 1.09 2.75 DCC 1.01 1.01 1.02 1.03 1.04 1.04 1.04 1.04 1.04 1.03 GLR 1.00 1.01 1.02 1.05 1.07 1.08 1.08 1.08 1.08 1.07 3.00 DCC 1.00 1.00 1.01 1.01 1.02 1.03 1.03 1.03 1.03 1.02 GLR 1.00 1.00 1.00 1.02 1.04 1.05 1.05 1.05 1.05 1.05
Table 3 summarises the effects of Phase I estimates on ARL under the true parameters. The results reported in Table 3 indicate that using Phase I parameter estimates will increase the false alarm rate when the process is in control, as evidenced by the decrease in the in-control ARL. Take m ¼ 20 and n ¼ 5 as an example. The in-control ARL assuming that 0¼0 and 0¼1 are known is set at approximately 370. On the
other hand, if the parameters are replaced by estimates obtained from 20 Phase I samples of size 5 each, the in-control ARL is only approximately 175.68. As the total number of Phase I sample increases, thus increasing the total number of Phase I observations, the ARL becomes closer to the value obtained when assuming that 0¼0 and 0¼1 are known. Note that these observations are consistent with what has been reported in the
literature. For example, Bischak and Trietsch (2007) argue that not only is the in-control ARL affected but that the ARL itself may be a confusing performance measure when used with control charts with estimated in-control parameters.
4. The density control chart under a negative exponential distribution
Suppose that the quality characteristic of interest of a process follows a negative exponential distribution having the following density function
f ðx; 1, 2Þ ¼ 1 1 exp x 2 1 , x 2,
where 140 and 22Rare unknown parameters. Given m samples of size n, let xij, i ¼ 1, 2, . . . , n, j ¼ 1, 2, . . . , m,
denote the ith observation of the jth sample. One can estimate 1and 2by
b 1¼ 1 m Xm j¼1 b 1j ð9Þ
Table 2. The ARLs of the density control chart (DCC) and the GLR chart when n ¼ 10 ( ¼ 0þ0, ¼ 0).
n ¼10 0.87 1.00 1.15 1.52 2.00 2.15 2.30 2.55 2.75 3.00 0.00 DCC 9909.26 370.37 38.19 3.23 1.33 1.20 1.13 1.06 1.04 1.02 GLR 222.01 370.59 141.64 6.85 1.73 1.45 1.29 1.15 1.09 1.05 0.25 DCC 3644.81 213.03 28.51 3.01 1.31 1.20 1.12 1.06 1.04 1.02 GLR 114.84 131.64 59.49 5.78 1.69 1.43 1.28 1.14 1.08 1.05 0.50 DCC 485.09 63.13 14.30 2.51 1.27 1.17 1.11 1.05 1.03 1.02 GLR 27.18 25.64 15.80 3.91 1.56 1.36 1.24 1.13 1.08 1.04 0.75 DCC 63.26 17.14 6.50 1.99 1.21 1.14 1.09 1.05 1.03 1.01 GLR 6.64 6.58 5.30 2.55 1.41 1.28 1.19 1.10 1.06 1.04 1.00 DCC 11.68 5.62 3.22 1.59 1.15 1.10 1.07 1.04 1.02 1.01 GLR 2.40 2.54 2.42 1.78 1.28 1.20 1.14 1.08 1.05 1.03 1.25 DCC 3.45 2.47 1.89 1.32 1.10 1.07 1.05 1.03 1.02 1.01 GLR 1.35 1.45 1.49 1.37 1.17 1.13 1.09 1.06 1.04 1.02 1.50 DCC 1.64 1.47 1.34 1.16 1.06 1.04 1.03 1.02 1.01 1.01 GLR 1.07 1.11 1.15 1.16 1.09 1.08 1.06 1.04 1.03 1.02 1.75 DCC 1.15 1.13 1.11 1.07 1.03 1.02 1.02 1.01 1.01 1.00 GLR 1.01 1.02 1.04 1.06 1.05 1.04 1.03 1.02 1.02 1.01 2.00 DCC 1.02 1.03 1.03 1.03 1.02 1.01 1.01 1.01 1.01 1.00 GLR 1.00 1.00 1.01 1.02 1.02 1.02 1.02 1.01 1.01 1.01 2.25 DCC 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.00 1.00 1.00 GLR 1.00 1.00 1.00 1.01 1.01 1.01 1.01 1.01 1.01 1.00 2.50 DCC 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 GLR 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 2.75 DCC 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 GLR 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
and
b
2¼minfb21, b22, . . . , b2mg, ð10Þ
where b2j¼minfx1j, x2j, . . . , xnjg and b1j¼1=n
Pn
i¼1ðxij b2jÞ, j = 1, 2, . . . , m, are the estimates of 2 and 1,
respectively, obtained from the jth sample.
When we consider b1 and b2 as the true 1 and 2, respectively, it is easy to see that for any given sample,
ðX1 b2Þ=b1, ðX2 b2Þ=b1, . . . , ðXn b2Þ=b1 are i.i.d. as exp(1) and therefore Pni¼1ðXi b2Þ=b1ðn, 1Þ, which is a
Gamma distribution. The inequality
LðX1, X2, . . . , Xn; b1, b2Þ l ðb1, b2Þ
Table 3. The simulated ARLs of the density control chart (DCC) when the estimates obtained from Phase I data are used for n ¼5, m ¼ 20, 25, 30 ( ¼ 0þ0, ¼ 0). 0.87 1.00 1.15 1.52 2.00 2.15 2.30 2.55 2.75 3.00 m ¼20 n ¼5 0.00 1292.97 175.68 38.16 5.39 2.03 1.74 1.55 1.35 1.25 1.18 0.25 689.79 121.28 30.84 5.05 2.00 1.72 1.54 1.34 1.25 1.17 0.50 196.34 53.28 18.48 4.24 1.90 1.66 1.50 1.32 1.24 1.17 0.75 53.66 21.43 10.03 3.35 1.76 1.57 1.44 1.29 1.22 1.15 1.00 16.93 9.25 5.57 2.60 1.61 1.47 1.37 1.26 1.19 1.14 1.25 6.54 4.56 3.34 2.05 1.47 1.37 1.30 1.22 1.17 1.12 1.50 3.15 2.61 2.21 1.67 1.35 1.29 1.24 1.18 1.14 1.10 1.75 1.88 1.74 1.62 1.41 1.25 1.21 1.18 1.14 1.11 1.09 2.00 1.35 1.33 1.31 1.24 1.17 1.15 1.13 1.11 1.09 1.07 2.25 1.13 1.14 1.14 1.14 1.11 1.10 1.09 1.08 1.07 1.06 2.50 1.04 1.05 1.06 1.07 1.07 1.07 1.06 1.06 1.05 1.04 2.75 1.01 1.02 1.02 1.04 1.04 1.04 1.04 1.04 1.04 1.03 m ¼25 n ¼5 0.00 1626.95 201.05 41.24 5.52 2.05 1.75 1.56 1.35 1.26 1.18 0.25 840.78 136.50 33.06 5.16 2.01 1.73 1.54 1.35 1.25 1.17 0.50 228.62 58.41 19.51 4.32 1.91 1.67 1.50 1.33 1.24 1.17 0.75 59.80 22.89 10.44 3.40 1.77 1.58 1.44 1.30 1.22 1.15 1.00 18.17 9.67 5.72 2.63 1.62 1.48 1.37 1.26 1.19 1.14 1.25 6.82 4.68 3.39 2.06 1.47 1.38 1.30 1.22 1.17 1.12 1.50 3.21 2.65 2.23 1.67 1.35 1.29 1.24 1.18 1.14 1.11 1.75 1.89 1.75 1.63 1.41 1.25 1.21 1.18 1.14 1.11 1.09 2.00 1.35 1.33 1.31 1.24 1.17 1.15 1.13 1.11 1.09 1.07 2.25 1.13 1.14 1.14 1.14 1.11 1.10 1.09 1.08 1.07 1.06 2.50 1.04 1.05 1.06 1.07 1.07 1.07 1.06 1.06 1.05 1.04 2.75 1.01 1.01 1.02 1.04 1.04 1.04 1.04 1.04 1.04 1.03 m ¼30 n ¼5 0.00 1890.74 219.66 43.39 5.61 2.06 1.76 1.56 1.36 1.26 1.18 0.25 967.30 148.40 34.71 5.24 2.02 1.74 1.55 1.35 1.25 1.18 0.50 255.97 62.55 20.33 4.38 1.92 1.67 1.51 1.33 1.24 1.17 0.75 65.01 24.10 10.77 3.43 1.78 1.58 1.45 1.30 1.22 1.16 1.00 19.23 10.02 5.85 2.65 1.62 1.48 1.38 1.26 1.20 1.14 1.25 7.06 4.78 3.44 2.07 1.48 1.38 1.31 1.22 1.17 1.12 1.50 3.27 2.68 2.25 1.68 1.35 1.29 1.24 1.18 1.14 1.11 1.75 1.90 1.76 1.63 1.42 1.25 1.21 1.18 1.14 1.11 1.09 2.00 1.35 1.33 1.31 1.24 1.17 1.15 1.13 1.11 1.09 1.07 2.25 1.12 1.13 1.14 1.14 1.11 1.10 1.09 1.08 1.07 1.06 2.50 1.04 1.05 1.06 1.07 1.07 1.07 1.06 1.06 1.05 1.04 2.75 1.01 1.01 1.02 1.04 1.04 1.04 1.04 1.04 1.04 1.03
subject to Pb1,b2 ðLðX1, X2, . . . , Xn; b1, b2Þ l ðb1, b2ÞÞ ¼1 yields l ðb1, b2Þ ¼ 1 bn 1 exp 2 2n,1 2 ( ) : ð11Þ
Therefore, for the negative exponential distribution, the density control chart plots, for any given sample of observations x1, x2, . . . , xn, Lðx1, x2, . . . , xn; b1, b2Þ ¼ 1 b n 1 exp Pn i¼1ðxi b2Þ b 1 ( ) ð12Þ and an out-of-control signal is detected as soon as the plotting statistic is below
LCL ¼ 1 b n 1 exp 2 2n,1 2 ( ) : ð13Þ
Alternatively, one may choose to use 1=nPni¼1ðxi b2Þ as the plotting statistics. Note that given b1 and b2,
2 0¼2
Pn
i¼1ðxi b2Þ=b1is distributed as 22n, and that Lðx1, x2, . . . , xn; b1, b2Þ LCL if and only if 2022n,1. That
is, the density control chart in this case is equivalent to the chi-square control chart where the plotting statistic is 1=nPni¼1ðxi b2Þand
UCL ¼b1 2n
2
2n,1: ð14Þ
Given b1 and b2, and specific out-of-control parameters 1and 2, it is also quite straightforward to calculate the
ARL of the density chart as ARL ¼ 1/p, where the probability p is defined as P 22n b1 1 22n,12nð2 b2Þ 1 ! : ð15Þ
Here we look at simple cases when n = 5 and 10. When the process is in control, we assume b1¼1:0 and b2¼0, and
the in-control ARL is set at 370. The ARLs of different out-of-control scenarios are summarised in Table 4. The ARL decreases when either the location parameter (2) shifts or the scale parameter (1) changes, or both.
Not surprisingly, the ARL decreases as the sample size increases. Moreover, the density chart seems to be more effective in detecting changes in the scale parameter than shifts in the location parameter.
Example 2: A particle counts example
The semiconductor manufacturing processes often involve quality characteristics whose distributions are non-normal. Levinson and Polny (1999) studied a set of particle counts data generated from an Applied Materials etcher. See Levinson and Polny (1999) for a detailed account of the data set and its generation. A three-parameter Gamma distribution with the following density function
f ðx; 1, 2, 3Þ ¼ 1 ð3Þ13 ðx 2Þ31exp x 2 1 , x 2
was fit and shown to be appropriate for the data set. The maximum likelihood estimates of the three parameters are b
1¼222:32, b2¼34, and b3¼1:172. Levinson and Polny (1999) also constructed a Shewhart chart for the mean
¼ 13þ2with n ¼ 1, and the resulting control limits are LCL ¼ 362.63 and UCL ¼ 951.79. They mentioned
that this classical X-chart has a serious problem since negative particle counts do not exist in practice.
Since the estimated b3 is close to 1, the negative exponential distribution may provide a simpler but also
appropriate distribution to model the particle counts data. Here we use this example to demonstrate how the density
control chart can be readily applied to a negative exponential distribution. We carried out both the chi-square goodness-of-fit test and the Kolmogorov–Smirnov test for testing the hypothesis that the particle counts data follow a negative exponential distribution, and the p-values were 0.2466 and 0.7889, respectively. Both tests did not provide strong evidence to reject the hypothesis that the particle counts follow a negative exponential distribution.
A total of 12 samples each with size 4 were collected and the resulting estimates for 1and 2were b1¼160:25
and b2¼35:5, respectively. Based on the density chart described in (12) and (13), the plotting statistic in (12), for
each of the 12 samples with size 4, was calculated by 1 ð160:25Þ4exp P4 i¼1ðxi35:5Þ 160:25 ( ) , and the chart signals if the plotting statistic falls below
LCL ¼ 160:254exp 2 8,0:9973 2 ! ¼0:0001153 1010:
Since the 12 plotting statistics obtained from (12) and LCL are all very large (in the magnitude of 1010), for the chart to be visible to the reader, these values are divided by 1010and then transformed on log-scale. The density control chart of the 12 samples (with the transformed LCL ¼ lnð0:00011531010
1010 Þ ¼ 9:0683) is shown in Figure 4. It is evident from Figure 4 that samples 2 and 6 are out of control.
In this case, we can use the plotting statistic described in (14) to help us diagnose the possible causes of any out-of-control signal. Specifically, for a given sample x1, x2, . . . , xn, we denote v ¼ 1=nPi¼1n ðxixð1ÞÞand u ¼ x(1), where
x(1)¼min {x1, x2, . . . , xn}. Given b1 and b2, the diagnostics can be carried out by looking at the equation
v þ ðu b2Þ ¼
b1
2n
2
2n,1: ð16Þ
Here the v measures the variation of the sample and u b2measures possible location shift. The diagonal line with
axes u and v from the above equation represents the control limit. Any sample producing a point lying above the diagonal line is considered as an out-of-control sample. If the point is away from the line u ¼ b2, it is an indication
Table 4. The ARL of the density control chart for the negative exponential distribution (1is the scale parameter and 2is the location parameter).
2 1 0.0 0.2 0.5 1.0 2.0 3.0 1.00 370.37 180.62 64.03 13.06 1.36 1.00 1.25 56.37 33.32 15.72 5.10 1.17 1.00 1.50 17.83 11.95 6.79 2.97 1.09 1.00 1.75 8.40 6.14 3.97 2.12 1.05 1.00 n ¼5 2.00 5.01 3.91 2.77 1.71 1.03 1.00 2.25 3.47 2.84 2.15 1.48 1.02 1.00 2.50 2.66 2.25 1.80 1.34 1.01 1.00 2.75 2.17 1.90 1.58 1.25 1.01 1.00 3.00 1.87 1.67 1.44 1.18 1.01 1.00 1.00 370.37 115.49 23.55 2.97 1.00 1.00 1.25 35.10 15.95 5.64 1.64 1.00 1.00 1.50 9.25 5.35 2.67 1.26 1.00 1.00 1.75 4.16 2.83 1.77 1.12 1.00 1.00 n ¼10 2.00 2.53 1.93 1.40 1.06 1.00 1.00 2.25 1.85 1.52 1.22 1.03 1.00 1.00 2.50 1.51 1.31 1.13 1.02 1.00 1.00 2.75 1.32 1.19 1.08 1.01 1.00 1.00 3.00 1.21 1.12 1.05 1.00 1.00 1.00
that there is a possible mean shift. On the other hand, if the point is away from the line v ¼ 0, it is an indication that there is a possible change in the variance.
For the particle counts data, the equation turned out to be v þ ðu 35:35Þ ¼ 401:83:
The diagonal line with axes u and v, along with all 12 samples are shown in Figure 5. The two out-of-control samples, sample 2 and sample 6 produced (u, v) ¼ (207.0, 383.5) and (334.5, 175.9), respectively. Figure 5 indicates that both samples 2 and 6 are due to mean shift and variance change. On the other hand, the level of mean shift seems to be larger in sample 6 than in sample 2, while the degree of change in variance seems to be greater in sample 2 than in sample 6.
5. Approximate density control chart
In previous sections, we discussed situations in which the distribution of the plotting statistic of the density control chart can be readily derived. However, in other applications it may be too complicated to derive such a distribution. In general, given b1, b2, . . . , bp, if the distribution of LðX1, X2, . . . , Xn; b1, b2, . . . , bpÞ is too complicated, one can
repeatedly generate samples from f ðx; b1, b2, . . . , bpÞ and calculate the likelihood of each of the, say, k generated
samples, which are denoted by l1, l2, . . . , lk. Then the LCL can be estimated as LCL ¼ ld ð½kþ1Þ, where
l(1)l(2) l(k) are the ordered statistics and [a] is the largest integer smaller than or equal to a.
Here we look at another distribution to illustrate the use of simulations to approximate the LCL. Suppose that a quality characteristic follows a Gamma distribution having the following density function
f ðx; 1, 2Þ ¼ 1 ð2Þ12 x21exp x 1 , x4 0:
The Gamma distribution has been used to model a variety of lifetime data, including some situations involving the negative exponential distributions. For the Gamma distribution, it is rather difficult to derive the distribution of the likelihood function. Suppose that, based on the Phase I data, both of two parameters, 1and 2, are estimated to
be equal to 2, i.e. b1¼ b2¼2. Based on these estimates and assume that the subgroup size is 5, we then generated
10 million samples of size 5 each and calculated, for each sample, the likelihood function f ðx1, x2, . . . , x5; b1¼ b2Þ ¼ 1 ðð2ÞÞ5252 Y5 i¼1 xi !21 exp P5 i¼1xi 2 ( ) : 0 50 100 150 200 250 300 350 400 450 0 50 100 150 200 250 300 350 400 450 U V U=35.5 U+V=437.18 2 (207.0,383.5) 6 (334.5,175.9)
Figure 5. The chi-square control chart of the particle counts example. 0 2 4 6 8 10 12 −10 −8 −6 −4 −2 0 LCL = −9.0683 2 6 Density chart Sample number log L 1
Figure 4. The density control chart of the particle counts example with the plotting statistics.
From the 10 million likelihoods, we obtained dLCL ¼ lð27,000Þ. Using this dLCL, we then simulated the ARLs for
a variety of combinations of 1and 2, including the in-control case when 1¼2¼2. The results are summarised
in Table 5. From Table 5, the results seem to indicate that the approximate density control chart is more effective in detecting changes in the shape parameter (2) than in the scale parameter (1) in the case of the Gamma distribution.
As an additional example of using simulation to obtain the control limit for the approximate density control chart, we assume that the estimates of 1and 2from Phase I data are both equal to 1, i.e. b1¼ b2¼1. Note that if
we treat these estimates as the true parameter values in Phase II, the corresponding Gamma distribution is equivalent to a negative exponential distribution with scale parameter equal to 1 and location parameter equal to 0. The simulated ARL values for this example are summarised in Table 6. The dLCL for this case was also obtained based on 10 million simulated likelihood values. The approximate density control chart seems to be more effective when the in-control parameter 1¼2¼1 than when the in-control parameters 1¼2¼2, in terms of producing
smaller ARL value for the same level of change in each parameter, in the case of the negative exponential distribution than in the case of the Gamma distribution.
Table 6. The ARLs of the approximate density control chart for the negative exponential distribution (n ¼ 5, 1is the scale
parameter, and 2is the shape parameter).
2 1 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 1.00 370.70 56.46 17.82 8.38 5.01 3.47 2.66 2.17 1.87 1.66 1.52 1.41 1.33 1.25 97.83 18.98 7.24 3.96 2.67 2.04 1.69 1.48 1.35 1.26 1.19 1.15 1.11 1.50 33.74 8.26 3.75 2.35 1.76 1.47 1.30 1.20 1.14 1.10 1.07 1.05 1.04 1.75 14.33 4.38 2.34 1.66 1.36 1.21 1.13 1.08 1.05 1.03 1.02 1.01 1.01 2.00 7.22 2.72 1.69 1.32 1.17 1.09 1.05 1.03 1.02 1.01 1.01 1.00 1.00 2.25 4.21 1.93 1.36 1.16 1.07 1.04 1.02 1.01 1.01 1.00 1.00 1.00 1.00 2.50 2.77 1.51 1.18 1.07 1.03 1.01 1.01 1.00 1.00 1.00 1.00 1.00 1.00 2.75 2.01 1.28 1.09 1.03 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.00 1.59 1.15 1.04 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.25 1.34 1.08 1.02 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.50 1.20 1.04 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 3.75 1.11 1.02 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 4.00 1.06 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 Table 5. The ARLs of the approximate density control chart for the Gamma distribution (n ¼ 5, 1is the scale parameter, and 2
is the shape parameter).
2 1 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 2.00 370.59 100.81 37.52 17.80 9.98 6.35 4.45 3.35 2.68 2.23 1.93 1.71 1.56 2.25 160.95 47.19 19.10 9.74 5.84 3.96 2.94 2.33 1.94 1.69 1.51 1.39 1.30 2.50 74.22 23.89 10.55 5.83 3.75 2.71 2.12 1.77 1.54 1.39 1.28 1.21 1.16 2.75 36.90 13.13 6.36 3.80 2.63 2.01 1.66 1.44 1.31 1.21 1.15 1.11 1.08 3.00 19.83 7.85 4.16 2.69 1.98 1.61 1.39 1.25 1.17 1.11 1.08 1.05 1.04 3.25 11.48 5.05 2.93 2.04 1.60 1.36 1.23 1.14 1.09 1.06 1.04 1.02 1.02 3.50 7.18 3.50 2.21 1.65 1.37 1.21 1.13 1.08 1.05 1.03 1.02 1.01 1.01 3.75 4.79 2.58 1.77 1.40 1.22 1.12 1.07 1.04 1.02 1.01 1.01 1.00 1.00 4.00 3.40 2.02 1.49 1.25 1.13 1.07 1.04 1.02 1.01 1.01 1.00 1.00 1.00 4.25 2.57 1.66 1.31 1.15 1.07 1.04 1.02 1.01 1.00 1.00 1.00 1.00 1.00 4.50 2.03 1.43 1.19 1.09 1.04 1.02 1.01 1.00 1.00 1.00 1.00 1.00 1.00 4.75 1.69 1.28 1.12 1.05 1.02 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 5.00 1.46 1.18 1.07 1.03 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
6. Conclusion and discussion
We have proposed and studied the density control chart which can be used to monitor multiple parameters simultaneously on a single chart for any given type-I error probability. The approach of the density chart is quite general which allows easy extensions to non-normal processes. We gave examples of non-normal processes and demonstrated how the density chart can be constructed and applied in those cases.
Under the normality assumption, we compared the performance of the density chart with that of the GLR chart of Hawkins and Deng (2009). The density chart performs better than the GLR chart when there is a small increase in the variance. On the other hand, when there is a decrease in the variance and a small shift in the mean, the density chart is ineffective. When only mean shifts but variance remains unchanged, the two charts have comparable performance.
The density chart studied in the current article is a Shewhart type control chart. It is rather straightforward to calculate the EWMA of the likelihoods in very much the same manner as those discussed in Yeh et al. (2005). It would be worthwhile to study to what extent the EWMA density chart improves the performance of the Shewhart type density chart.
It should be noted that, unlike most of the simultaneous charts that exist in the literature which assume n41, the proposed density chart is also applicable when n ¼ 1. It would be interesting to develop the density chart for individual observations and compare it with some of the existing control charts such as the exponentially weighted moving squared deviations chart studied in MacGregor and Harris (1993).
The generality of the density control charting mechanism lends itself nicely for multivariate processes, provided that the density function is known. There have been some recent studies on simultaneously monitoring multiple parameters for multivariate normal processes (see, for example, Cheng and Thaga 2005, Chen et al. 2005, Reynolds and Chao 2006, and Machado et al. 2009). It would be worthwhile to extend the density chart to multivariate normal and non-normal processes.
Acknowledgments
The authors would like to thank Professor Alan H. Welsh for his valuable comments. They would also like to thank two referees for their comments and suggestions which improved the presentation of the article. This research work of H-CC and L-AC was partially supported by a grant from the National Science Council of the Republic of China (NSC 95-2118-M-009-007).
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