3.3.1.1 Shewhart Control Charts for Monitoring Process Mean

For monitoring the mean of a process, we typically use the sample mean (). An example follows.

3.3.1.2 Shewhart Control Charts for Monitoring Process Variation

Variation is an important aspect of any analysis and thus it is necessary to monitor the process variation or spread and ensure that it is IC. Moreover, as we see in Equation 3.1 , the Shewhart control limits for the process mean depend on the process standard deviation. Thus, unless the standard deviation remains IC, the control chart for the mean will not be very informative. So, we need to monitor the variance or the standard deviation using a control chart.

There are several possible statistics that can be used to monitor variation. The most popular choices are the sample range (), the sample standard deviation (), and the sample variance ().

Typically, we use a control chart to monitor the process mean together with a control chart to monitor the process variation. If the variation is IC, we go ahead and examine the control chart for the mean. For example, a Shewhart chart for the mean is often used together with a Shewhart chart for the spread. Note that, for illustration, we consider the Shewhart chart even though recent literature recommends using a different spread chart, such as the Shewhart chart; see, for instance, Mahmoud et al. (2010). We do this because the Shewhart chart is simple and continues to be used in the industry.

In Case K, the values of and are known or are specified so that they can be used to construct the respective control charts. We illustrate the Shewhart R and S charts for the known standard deviation .

3.5.2.1 Signaling Indicators

Let the charting statistic for the subgroup be denoted by for 1, 2, 3, … Let the indicator functions for the one‐sided control charts be denoted by

(3.17)

and

(3.18)

respectively. Thus, equals 1 and indicates a signal for an upper one‐sided chart when plots on or above the UCL. Similarly, takes a value 1 and indicates a signal for a lower one‐sided chart when plots on or below the LCL. Thus, these binary variables are called signaling indicators. For a two‐sided control chart, the signaling indicator may be denoted by

(3.19)

so that indicates whether the charting statistic of plots on or above the UCL (), between the two limits (), or on or below the LCL (). The values 1 and 2 indicate that there is a signal, on the high or the low side, respectively, whereas the value 0 indicates there is not a signal and that the process is IC.

3.5.2.1.1 The 1‐of‐1 Runs‐type Signaling Rule

The most commonly used and least complex charts are the usual Shewhart charts, called 1‐of‐1 charts, which are not supplemented by any runs‐rules. However, they are known to be insensitive in detecting small process shifts. These charts signal when the event or , or occurs, respectively, where

1. The 1‐of‐1 runs‐type signaling rule for an upper one‐sided chart: .
2. The 1‐of‐1 runs‐type signaling rule for a lower one‐sided chart: .
3. The 1‐of‐1 runs‐type signaling rule for a two‐sided chart: .

Figure 3.12 illustrates points (i) and (ii) of the 1‐of‐1 runs‐type signaling rule for an upper one‐sided and a lower one‐sided chart, respectively. Panel (a) detects an upward shift at time i = 7 when plots above the UCL, whereas Panel (b) detects a downward shift at time i = 7 when plots below the LCL. For each chart the process is declared OOC and a search for assignable causes can be started.

Figure 3.13 illustrates point (3) of the 1‐of‐1 runs‐type signaling rule for a two‐sided chart. Both charts signal at time i = 7; at first an upward shift is detected when plots above the UCL, and then a downward shift is detected when plots below the LCL.

3.5.2.1.2 The k‐of‐k and k‐of‐w Runs‐type Signaling Rules

Several runs‐type signaling rules, such as the 2‐of‐2 and 2‐of‐3 runs‐rules, have been investigated by many authors and proved that their “runs‐rules enhanced” charts outperform the 1‐of‐1 chart.

One‐sided k‐of‐k and k‐of‐w Runs‐type Signaling Rules

To illustrate, we start with the 2‐of‐2 runs‐type signaling rule. This rule uses the last two charting statistics, and , to determine whether the process is IC or OOC. For the 2‐of‐2 runs‐type signaling rule, an upper one‐sided chart signals when event occurs and a lower one‐sided chart signals when event occurs where

1. an upper one‐sided chart supplemented by a 2‐of‐2 runs‐type signaling rule:
2. a lower one‐sided chart supplemented by a 2‐of‐2 runs‐type signaling rule: .

Panels (a) and (b) in Figure 3.14 illustrate these signaling events and , and both signal at time i = 7 where two consecutive points plots above (below) the UCL (LCL). For each chart, the process is declared OOC and a search for assignable causes can be started.

Next, we illustrate the 2‐of‐3 runs‐type signaling rule for an upper one‐sided chart and a lower one‐sided chart, respectively. These charts use two of the last three charting statistics , , and to determine whether the process is IC or OOC. Although there are three ways that two of the three charting statistics can result in a signal, only on two of these possibilities does the last charting statistic plot on or above (below) the upper (lower) control limits. The one‐sided charts supplemented by the 2‐of‐3 runs‐type signaling rule signal when event or = occurs where:

An upper one‐sided chart supplemented by the 2‐of‐3 runs‐type signaling rule

(1) (2)

A lower one‐sided chart supplemented by the 2‐of‐3 runs‐type signaling rule

(3) (4)

Figure 3.15 illustrates these signaling events , , , and of an upper one‐sided chart and a lower one‐sided chart supplemented by the 2‐of‐3 runs‐type signaling rule, respectively. The upper one‐sided charts in Panel (a) signal at time i = 7 where two of the last three points plot above the UCL. For each chart, the process is declared OOC and a search for assignable causes can be started. Likewise, the lower one‐sided charts in Panel (b) signal at time i = 7 where two of the last three points plot below the LCL. For each chart, the process is declared OOC and a search for assignable causes can be started.

Some of the patterns of the 2‐of‐3 runs‐type signaling rules are excluded. If the first and the second charting statistics plot outside the control limits, but the third (or last) charting statistic plots between the control limits, the process cannot be declared OOC, because the process has shifted back to being IC and, accordingly, these patterns are excluded from the 2‐of‐3 runs‐type signaling rules. Refer to Figure 3.16, where the upper one‐sided chart supplemented by the 2‐of‐3 runs‐type signaling rule and the lower one‐sided chart supplemented by the 2‐of‐3 runs‐type signaling rule are excluded as signaling events.

Two‐sided Charts Supplemented by k‐of‐k and k‐of‐w Runs‐type Signaling Rules

A two‐sided chart has both an upper and a lower control limit, thus it can detect either an upward or a downward shift. To illustrate, we start with the 2‐of‐2 runs‐type signaling rule. A two‐sided chart supplemented by a 2‐of‐2 runs‐type signaling rule is much the same as that of the one‐sided charts in the sense that it uses the last two charting statistics, and , to determine whether the process is IC or OOC, but it is able to detect both an upward or a downward shift. A two‐sided chart supplemented by the 2‐of‐2 runs‐type signaling rule signals when event or occurs where

1. both charting statistics plot on or above the UCL:
2. both charting statistics plot on or below the LCL:
3. the first charting statistic plots on or above the UCL and the second plots on or below the LCL:
4. the first charting statistic plots on or below the LCL and the second plots on or above the UCL: .

Figure 3.17 illustrates these signaling events , , , and for the two‐sided chart supplemented by 2‐of‐2 runs‐type signaling rules, respectively. Panels (a) and (b) illustrate events and ; both charts signal at time i = 7, where two consecutive points plots above (below) the UCL (LCL). Panels (c) and (d) illustrate events and when an upward shift is immediately followed by a downward shift, or vice versa.

Next, we illustrate a two‐sided chart with 2‐of‐3 runs‐type signaling rules. Similar to one‐sided chart, a two‐sided chart uses two of the last three charting statistics, , , and , to determine whether the process is IC or OOC. There are up to 12 scenarios where exactly two of the three charting statistics can plot outside the control limits; however, we will only focus on four signaling events or that may occur where:

Two of the three charting statistics plot on or above the UCL

(1) (2)

Two of the three charting statistics plot on or below the LCL

(3) (4)

Figure 3.18 illustrates these signaling events , , , and for a two‐sided chart supplemented by 2‐of‐3 runs‐type signaling rules. Panel (a) illustrates events and , where exactly two out of the three charting statistics plot on or above the UCL, whereas Panel (b) illustrates events and , where exactly two out of the three charting statistics plot on or below the LCL.

Figure 3.19 illustrates eight of the 12 scenarios that are excluded as signaling events. Panel (a) shows a possible swing detected, whereas Panel (b) illustrates a downward or upward trend in the process. As mentioned in Figure 3.16 , the four events illustrated in Panel (c) will be excluded as signaling events due to the fact that the last point plots between the control limits; therefore, the process cannot be declared OOC, because the process has shifted back to being IC.

3.6.1.1 The Exact Approach (for Shewhart and some Shewhart‐type Charts)

For the Shewhart and some Shewhart‐type control charts, it is easy to calculate the characteristics of the run‐length distribution exactly in Case K. For the two‐sided 3‐sigma Shewhart chart, the run‐length, which is the number of samples until the first signal, follows a geometric distribution with probability , which equals the FAR, denoted , when the process is IC. We provide more details on this later. Thus, in Case K, all IC properties of the Shewhart control chart can be characterized in terms of the FAR, , and obtained from the properties of the geometric distribution. Thus, for example, the IC ARL equals , the IC SDRL equals , and so on. Figure 3.20 has a display of the run‐length distribution for . The right‐skewed nature of the distribution is obvious. For smaller values of , such as 0.0027, which is fairly typical in the industry, the run‐length distribution is even more right‐skewed with an even longer right tail.

For some well‐known control charts like the CUSUM, the EWMA, and the Shewhart charts with runs‐rules, a more common approach to determining the run‐length distribution has been the clever application of the idea of a Markov chain (MC). The MC theory is more commonly used in the area of stochastic processes and the control charting literature has borrowed some important results from this literature. We describe some of the concepts and the key results below.

3.6.1.2 The Markov Chain Approach

Some theory and concepts in the area of what is called a finite homogeneous MC have been applied to derive the run‐length distribution and various characteristics of the run‐length distribution of some time‐weighted charts, such as the CUSUM and the EWMA. The reader can look up, for example, Fu and Lou (2003) and Balakrishnan and Koutras (2002) for a more detailed treatment of the topics; however, most existing SPC books don't seem to treat this important topic. In this section, some of the basic terminologies, definitions, results, and theorems are provided that give the necessary background for calculating the run‐length distribution of some control charts via the MC approach. These theorems and results are critical to the following chapters; illustrations are provided.

The MC approach entails that the charting statistic is viewed as following a MC (as a stochastic process), with a state space S and a transition probability matrix . The state space consists of two types of states:

1. one absorbing state (i.e., this state is entered when the chart signals, i.e., when the charting statistic is greater than or equal to the UCL, or less than or equal to the LCL for a two‐sided chart); and
2. transient or non‐absorbing states, so that there are + 1 states in total.

The ( + 1) ( + 1) transition probability matrix, , is often written in a partitioned form

(3.20)

where the sub‐matrix contains all the probabilities of going from one transient state to another and is called the essential transition probability matrix. The column vector contains all the probabilities of going from each transient state to the absorbing state; is a row vector of zeros, consisting of the probabilities of going from the absorbing state to each transient state (which are all zero), and the scalar value 1 is the probability of staying in the absorbing state once it has been entered. Note that the key component in using the MC approach is to obtain the essential transition probability matrix .

In the context of MCs, the run‐length random variable N of a control chart is viewed as the waiting time for the MC to enter the absorbing state for the first time. Using this analogy, and assuming that the process starts IC when the chart is implemented, the probability mass function (pmf), the expected value (ARL), the standard deviation (SDRL), and the cdf of N are given by (see Fu and Lou, 2003, Chapter 5)

(3.21)

(3.22)

(3.23)

(3.24)

respectively, where is the identity matrix, is the essential transition probability matrix, is a column vector with all elements equal to one, and is a row vector called the initial probability vector, which contains the probabilities that the MC starts in a given state. The vector is typically chosen such that .

The zero‐state and steady‐state modes of analysis are used to characterize the short‐ and long‐term run‐length properties of a control chart, respectively. That is, the zero‐state run‐length is the number of charting statistics plotted before a signal is given when the process begins in some initial state; however, the steady‐state run‐length is the number of charting statistics plotted before a signal is given when the process begins and stays IC for a very long time. Then, at some random time, an OOC signal is observed; see, for example, Champ (1992) and Machado and Costa (2014a, b). Let denote the row vector of initial probabilities associated with the zero‐state mode, that is

(3.25)

so that the initial state element on the transition probability matrix corresponds to the value of 1 in Equation 3.25. It should be noted that the 1 is in the middle of the row vector so that the process starts IC. This is done by forcing the initial state to be as close to the CL as possible, that is, the selection of the middle value corresponding to the middle discretized subinterval which, in turn, is the closest one can start to the CL. In the steady‐state, the vector is replaced by a vector , that is, the steady‐state initial probability vector. This is given by

(3.26)

where the elements in Equation 3.26 sum to unity. For example, for , the may be equal to . This would indicate that one would start closer to the CL with the highest probability of 0.3, or one would start further away from the CL with smaller probability 0.2, and, finally, one could start close to the control limits with the smallest probability of 0.15. The smallest probability is assigned to starting close to the UCL and close to the LCL, respectively, because this is not ideal. Another example could be, for we have . In this example, we have some elements in that equal zero and other elements that are non‐zero. Here, it indicates that one would start closer to the CL with the highest probability of 0.8, or one would start further away from the CL with smaller probability 0.1, and, finally, one could start close to the control limits with zero probability.

From Equation 3.24, it is clear that in order to apply the MC approach to find the run‐length distribution for a given control chart one needs to specify the essential transition probability matrix . Once the matrix is found, the run‐length distribution and associated characteristics can be obtained easily using the formulas given above for a given initial probability vector. Now, in order to calculate the elements of , the state space first needs to be identified. For an attributes control chart with a discrete charting statistic this is easy, since the number of states will be fixed. However, for a variables control chart, which uses a charting statistic with a continuous distribution, the states of the process thus need to be defined carefully. The steps of how this is typically done are provided below.

It should be noted that, with runs‐type signaling rules, the number of states is defined by the type of rule used, for example, whether one uses a 2‐of‐2 or a 2‐of‐3 runs‐rules. There is no need to discretize the interval between the LCL and the UCL into = 2 + 1 subintervals (as explained in Step 1 below), since, as stated before, for runs‐type signaling rules the number of states depends on the type of rule. Also, it should be noted that, with CUSUM charts, the number of states is a fixed number (see Example 3.6). There is no need to discretize the interval between the LCL and the UCL into = 2 + 1 subintervals (as explained in Step 1 below), since, as stated before, for CUSUM charts the number of states will be a fixed number. The steps below apply to the EWMA charts.

• Step 1: Divide the interval between the LCL and the UCL into = 2 + 1 subintervals of width 2 = (UCL – LCL)/ (see Figure 3.21), where each subinterval corresponds to a transient (or an accessible) state of the MC. State is said to be accessible from state if, and only if, starting in state , it is possible that the process will at some stage enter state . Note that 2 = (UCL – LCL)/ will equal 2 = 2UCL/ so that = UCL/ if the control limits are symmetrically placed around zero (i.e., – LCL = UCL). It should be noted that taking larger values of would result in more accurate answers. A detailed discussion on the choice of is given in Note 3.2.
• Step 2: Choose the number of discretized subintervals to be an odd positive integer so that there is a unique middle entry.
• Step 3: Declare the charting statistic, , to be in the transient state at time if for and for , where denotes the midpoint of the th interval.
• Step 4: Calculate the one‐step transition probabilities ('s) where denotes the probability of moving from state to state in one step at any point in time.
• Step 5: Construct the transition probability matrix, consisting of the one‐step transition probabilities, to find the run‐length distribution.

Note that the midpoints (defined in Step 3) can be found using the following general calculation formula

since we assume (for simplicity) that – LCL = UCL. The charting statistic is said to be in the absorbing or OOC state (i.e., ) if falls on or outside the control limits. This region is considered absorbing since the process is stopped when a signal is given by the chart. Hence, the process is declared to be OOC whenever is in the absorbing state, whereas the process is considered IC whenever is in a transient state (also referred to as an IC state). Therefore, the IC region consists of v non‐absorbing states, whereas the OOC region is treated as a single absorbing state.

As explained before, in order to apply the MC approach, we need expressions for the signaling probabilities so that we can set up the transition probability matrix. The elements of the essential transition probability matrix are called the one‐step transition probabilities; for . In order to calculate these probabilities once the states have been defined or identified, we assume that the charting statistic is equal to the midpoint whenever it is in state . This is an approximation that becomes more accurate when the numbers of subintervals increase, and is commented on later. For each control chart under consideration, the one‐step transition probabilities are calculated differently. Next, we give another example of using the MC approach to calculate the run‐length distribution of a CUSUM chart.

3.6.1.3 The Integral Equation Approach

The integral equation approach utilizes mathematics and combinatorics to find a closed form expression of the run‐length distribution. This approach is sometimes challenging in that the expression obtained is typically complex or difficult to evaluate numerically. Often, the exact expression of the run‐length distribution can be found after a considerable amount of work, but here a simulation is done instead, since it is much easier.

3.6.1.4 The Computer Simulations (the Monte Carlo) Approach

Monte Carlo simulations can be used to calculate the characteristics of the run‐length distribution. The popularity of this method stems from the fact that, no matter how complicated the run‐length distribution is, theoretically, computer simulations can almost always be used with relative ease to calculate the run‐length distribution and its associated characteristics fairly accurately, provided that the simulation size is big enough. In this body of work, we use 100 000 simulations since it is well known that the error of a run‐length characteristic can be bounded by increasing the simulation size sufficiently.

The generic steps of a computer simulation procedure to calculate the run‐length distribution for a two‐sided control chart, where the charting statistic is calculated from a random sample, is given as follows. This can be implemented in any software package such as R, SAS, or Matlab.

• Step 1: Specify the necessary parameters, such as the subgroup size, nominal IC average run‐length (or FAR); calculate the control limits.
• Step 2: Assume some process distribution, such as the normal distribution and specify parameters. Generate (simulate) subgroups of random observations from the specified distribution.
• Step 3: Calculate the charting statistic for each subgroup and compare it to the control limits calculated in Step 1.
• Step 4: The number of subgroups needed until the first charting statistic plots on or outside the control limits is recorded as an observation from the run‐length distribution.
• Step 5: Repeat Steps 1 to 4 a total of 100 000 times.
• Step 6: Once we obtain a “data set” with 100 000 observations from the run‐length distribution, the run‐length distribution characteristics of interest, which are descriptive statistics, such as the average, standard deviation, median, percentiles, etc., can be calculated. For example, one can use proc univariate of SAS to find these values. Minitab or R can be used to do the same and graph the boxplots or histograms of the run‐length values for a better understanding of the run‐length distribution.

It may be noted that simulating the run‐length distribution and the associated characteristics may be particularly time consuming in the IC case since the run‐length distribution is heavily right‐skewed, which means that there may be some very large run‐length values, which can take a very long time to be realized. In some cases, one has to be careful that the run‐length characteristic of interest, say, the ARL, is finite, which may not necessarily be the case. A possible shortcut to calculating the run‐length distribution may be to use the explicit expressions to calculate the probability of a signal (and its reciprocal) within the simulations, which may be available for some charts under the assumed process distribution.

3.8.1.1 Shewhart Control Charts for the Mean in Case U

In our discussion of parametric variables control charts in Case U, we assume that the process follows a normal distribution, but at least one (or both) of the process parameters, the mean and the standard deviation, is (are) unknown. Although other process distributions can be assumed in practice, we discuss the case of the normal distribution to be consistent with the earlier developments in the chapter for the known parameter case (see Section 3.2). Before continuing, we make some finer classifications within Case U. When two parameters are in play, as is the case here, we can have four cases, namely, Case KK (mean known and standard deviation known), which is often referred to as just Case K, Case KU (mean known and standard deviation unknown), Case UK (mean unknown and standard deviation known), and Case UU (both mean and standard deviation unknown). In this section, we consider perhaps the most practical situation, Case UU, when both of these parameters are unknown. So Case U refers to one of the three cases, Case KU, UK or UU. Other situations can be treated in a similar way (see for example, Chakraborti, 2000). In addition, we assume that a reference (Phase I) sample comprising samples, each of size , is available after a successful Phase I analysis to estimate these unknown parameters.

Suppose that , , and denote the Phase I sample means and standard deviations, respectively. Also, let denote the Phase I sample ranges. The process mean is typically estimated by the average of the m sample means or the grand mean, . Conversely, several estimators have been proposed to estimate the process standard deviation . We use some of these estimators to illustrate various points. Letting denote the chosen Phase I estimator of , the control limits for the two‐sided Phase II k‐sigma Shewhart control chart for the mean in Case UU are given by

(3.31)

The control limits in Equation 3.13 may be referred to as the plug‐in limits as they are of the same form as in Case K, with estimators of the mean and the standard deviations used in place of the known IC values of the parameters. The choice of the charting constant is an interesting question. For many years, the value 3 was used in analogy with a 3‐sigma chart in Case K. It is now understood that, with plug‐in limits, is not a good choice in that it increases the number of false alarms, unless there is a very large number of Phase I data available, which may be in the hundreds or even in the thousands. In fact, in Case U, the choice of the charting constant depends on , and the given nominal IC average run‐length. In Case K, alternatively, the charting constant depends only on and the nominal IC average run‐length. More details about this are provided later.

3.8.1.2 Shewhart Control Charts for the Standard Deviation in Case U

As in Case K, the variation of a process must be monitored to ensure that it remains IC. The Shewhart control limits for the process mean in Case U depend on the process standard deviation. Thus, unless the standard deviation remains IC, the control chart for the mean will not be informative and useful. So, we need to monitor the variance or the standard deviation using a control chart.

As noted earlier, there are several possible estimators that can be used to monitor the standard deviation . The popular choices include functions of the sample range (), or the sample standard deviation (), or the sample variance (). Typically, we use a control chart to monitor the process mean together with a control chart to monitor the process variation. If the variation is IC, we go ahead and examine the control chart for the mean.

A Shewhart chart for the mean, for example, is often used together with a Shewhart chart for the spread. Note that, for illustration purposes, we use the Shewhart chart for , even though recent literature recommends using a different chart, such as the one based on the standard deviation (see, for instance, Mahmoud et al., 2010 and Epprecht, Loureiro, and Chakraborti, 2016). We do this because the Shewhart chart is simple and continues to be used in the industry.

Let denote the average range. The CL and the control limits for the Shewhart R control chart for in the unknown parameter case are given by (see Montgomery, 2009, p. 229)

(3.32)

where

and and are, respectively, the mean and the standard deviation of a range in a random sample of size n from the standard normal distribution. We provide the formulas here for completeness as these are not available in many textbooks.

where and

denotes the pdf of the range of the sample of size n from the standard normal distribution with pdf and cdf (see, for example, Gibbons and Chakraborti, 2010). The integrals involved in these expressions can be calculated numerically or estimated in a simulation study in order to evaluate the constants and . They both depend on the sample size and are tabulated in Table C of Appendix A. Note that is an unbiased estimator of so that is called an unbiasing constant.

The control limits given in Equation 3.31 are of the same form as in Case K but the values of the charting constants and are different. However, the control limits in Equations 3.32 are the 3‐sigma limits and have some serious deficiencies, see, for example, Diko et al. (2016), particularly when is small. We consider some alternatives to the R chart control limits based on probability limits later in this section.

Using a similar idea, but employing the standard deviation of the sample and not the range, the CL and the control limits for the Shewhart S control chart for in the unknown parameter case are given by (see Montgomery, 2009, p. 253)

(3.33)

where is the average standard deviation and the constants are given by

where denotes the gamma function with ! when is a positive integer. Note that is also an unbiased estimator of . The constants and can be easily calculated from the formulas and are tabulated in Table H in Appendix A.

Again, note that these control limits are the 3‐sigma limits and, as in the case of the R charts, also have some deficiencies, particularly when is small. We also consider some alternatives to the chart control limits.

It is well known that the probability limits are more appropriate for the standard deviation since the distribution of the estimator (used in the charting statistic) is right‐skewed. The skewness is more pronounced when the Phase II sample size is smaller, which is typically the case in SPC. Intuitively, the lower control limit on a chart for should be positive (and not zero) so that the control chart is able to detect both increases and decreases in . The increase in indicates process degradation and is perhaps more important, but a decrease in can also be meaningful as it indicates possible improvements in the process. However, the standard 3‐sigma limits used in Case K, which are based on the normal approximation to the distribution of can lead to a negative LCL, which is then typically reset to 0. This is, indeed, the case for , which includes the most common subgroup used in practice, namely, . Although the 3‐sigma limits are perhaps more popular due to their simplicity, note that many authors have advocated their use with charting statistics with a skewed distribution. Montgomery (2009, p. 242) mentions the probability limits and cites Grant and Leavenworth (1980) for the constants needed to set up such control limits. However, note that these limits are correct and appropriate in Case K, the known parameter case. These constants need to be adjusted or adapted for the unknown parameter case, which was recently considered in Diko et al. (2016). We present a brief discussion on this important practical problem next.

To this end, note that the Phase II probability limits for the chart (with estimator ) and the chart (with estimator ) in Case U are given by

and

respectively, where the charting statistics are the range and the standard deviation of the ith Phase II sample and the constants and , are obtained for given values of , and a nominal IC average run‐length value, such as 370. These constants are provided in Table H in Appendix A. Some details about the derivations of these constants are provided in Appendix 3.3. The value shown in these tables represents the total tail probability beyond the lower and the upper control limits, under the IC distribution of the charting statistic. This is needed while calculating the probability limits.

Finally, consider , as the “pooled” estimator of based on averaging the Phase I sample variances. This is the estimator recommended (see, for example, Mahmoud et al., 2010) in the literature because of its superior statistical properties. The CL and the control limits for the Phase II chart with used as an estimator of in the unknown parameter case can be written as

(3.34)

where the charting statistic is and the constants and are given in Table H in Appendix A for various values of and 5, and 10, and for nominal IC average run‐length values of 370 and 500, respectively. Note that one can also use the unbiased version of in the control charts, but it can be seen that the unbiasing constant is approximately equal to one for typical and values. Again, some details about the derivations of these constants are provided in Appendix 3.3.

3.9.2.1 Using the Estimator S_p

Next, we illustrate the probability limits based Shewhart S chart with as the estimator of . Note that, for this data set, the Phase I estimator 0.1391. From Table H in Appendix A, the needed charting constants are found to be and for a nominal ARL_IC = 370. Thus, the probability limits‐based Shewhart S chart is given by and with . Note that this chart is slightly narrower than the probability limits‐based Shewhart S chart using the estimator shown in Figure 3.26 . This is expected since is a more efficient estimator than .

In Figure 3.27, all of the points plot between the control limits and there are no anomalous patterns in the data. Hence, we conclude that the process is functioning in statistical control with respect to variation.

	(a)	(b)
Period, i	xi
1	9.45
2	7.99	1.46
3	9.29	1.30
4	11.66	2.37
5	12.16	0.50
6	10.18	1.98
7	8.04	2.14
8	11.46	3.42
9	9.20	2.26
10	10.34	1.14
11	9.03	1.31
12	11.47	2.44
13	10.51	0.96
14	9.40	1.11
15	10.08	0.68
16	9.37	0.71
17	10.62	1.25
18	10.31	0.31
19	8.52	1.79
20	10.84	2.32

Suppose now that at the end of the Phase I analysis that 10 prospective independent measurements are made available.

In his Table 3.1 , Hawkins (1993) recommends that, for a shift of about in the process mean, taking and so that and , gives an ARL_IC = 370. In doing so, we find the design parameters and , respectively.

We monitor the process mean prospectively in Phase II based on the 10 samples given in Table 3.17. It will be seen from Table 3.18 that the charting constant for m = 20, n = 1, = 0.5, and a nominal ARL₀ = 370 is equal to 4.1 and not 4.77, as in Case K. At this point, note that the values in Table 3.18 were obtained using simulation, and show the impact of the estimation of parameters on the Phase II control limits. The correct value of is thus . For larger values of , around 100, the values seem to converge to their values in Case K.

The CUSUM charting statistics, and , are shown in Columns (b) and (d) of Table 3.19, respectively, whereas the corresponding quantities, and , are given in Columns (c) and (e), respectively. To illustrate the calculations, consider period 1. The charting statistics are calculated as in the parameter known case, except that we use for . Thus,

and

	(a)	(b)	(c)	(d)	(e)
Period, i	xi
		0		0
1	10.90	0.217	1	0	0
2	9.33	0.000	2	0	0
3	12.29	1.607	3	0	0
4	11.50	2.424	4	0	0
5	10.60	2.341	5	0	0
6	11.08	2.738	6	0	0
7	10.38	2.435	7	0	0
8	11.62	3.372	8	0	0
9	11.31	3.999	9	0	0
10	10.52	3.836	10	0	0

Figure 3.28 displays the CUSUM statistics, and , of Columns (a) and (c), respectively, which are plotted on a parametric CUSUM control chart together with both sets of control limits, one with the 3‐sigma limits and the other with the corrected for parameter estimation control limits.

As always, once the control chart is drawn, a decision is to be made about the status of the process. In Figure 3.28 , none of the charting statistics plot outside the control limits and the points do not seem to exhibit any non‐random pattern. Hence, we conclude that the process is functioning in a state of statistical control with respect to the mean.

In this example, we've illustrated the parametric CUSUM for individual measurements. However, if rational subgroups of size n > 1 are taken, then we would simply replace with and with in the previous

equations. In this case, the optimal k and h pair may differ from the pair used in this example for individual data. In Table 3.18 , the charting constant was given for = 1. In Table 3.20 , we provide the corresponding values for = 5.

The EWMA charting statistic is given in Column (b) of Table 3.22, whereas the time‐varying lower and upper control limits are given in Columns (c) and (d), respectively. We take starting value . To illustrate the calculations, consider observation number 1. The first charting statistic is calculated as follows

Substituting , , 0.1, L = 2.65, and = 1 into Equation 3.11 , the CL and the exact control limits for the data (for = 1) in Table 3.22 are given by

The steady‐state control limits can be calculated using Equation 3.12 and are given by and , respectively.

Figure 3.29 displays the EWMA statistics, , given in Column (b) of Table 3.22 , plotted on a parametric EWMA control chart together with two sets of time‐varying control limits, one with the 3‐sigma limits (solid line) and the other with the corrected for parameter estimation control limits (dotted line).

As always, once the control chart is drawn, a decision is to be made about the status of the process. In Figure 3.29 , none of the charting statistics plots outside the control limits and the points do not seem to exhibit any non‐random pattern. Hence, we conclude that the process is functioning in a state of statistical control with respect to the mean.

In this example, we've illustrated the parametric EWMA chart for individual measurements. However, if rational subgroups of size n > 1 are taken, then simply replace with and with in the previous equations. In Table 3.21 , the charting constant was given for = 1. Here we provide a table for = 5.

As discussed in Case K, an important point to keep in mind is that, while monitoring the process mean, the process mean and the standard deviation are both monitored, since the standard deviation appears in the control limits for the mean and thus must be IC when estimated and used in the calculation of the control limits. This is a well‐recognized matter in practice, but the impact of this “joint” monitoring is not always explicitly discussed or understood.

3.10.1.1 The Shewhart Chart for the Mean in Case U

The conditioning‐unconditioning (CUC) method, which was first explicitly coined and used in Chakraborti (2000), is explained here for the parametric Shewhart control chart for the mean, assuming the normal distribution. This development is important since the “standards unknown” case, that is, Case U, is the situation often encountered in practice, and the Shewhart control chart for the mean is one of the most popular charts used in practice. A brief background is given before going into detail. Recall that in the “standards known” case, that is, when process parameters are known (or Case K), the signaling events are mutually independent so that the run‐length distribution is geometric with the probability of a success (which, in SPC, is a signal) equal to, say, some . This result completely characterizes the performance of the Shewhart control chart in Case K, so that all performance properties of the chart can be obtained from the properties of the GEO(θ) distribution. Thus, the expected value of the run‐length distribution is the reciprocal of θ. For the IC run‐length distribution, the ARL_IC, simply equals the reciprocal of the FAR, which is the probability of a signal θ when the process is IC. The IC signal probability is denoted by α and therefore, ARL_IC = so that specifying the FAR specifies the ARL_IC and vice versa. This simple relationship makes understanding the performance of the Shewhart chart easier in Case K. Conversely, when the process parameters are unknown and need to be estimated to set up the control limits before Phase II process monitoring can begin, the signaling events are no longer independent so that the run‐length distribution is no longer geometric. As a consequence, for example, the ARL is no longer the reciprocal of the probability of a signal. This is the fundamental conceptual difference between Case K and Case U, which has important implications since some may find it tempting to use the results of Case K to design control charts even when the underlying process parameters are unknown. Practitioners are cautioned against this (see, for example, Quesenberry, 1993) practice and we thus discuss how to handle the situation properly. To this end, we use an important tool called the “conditioning‐unconditioning method” proposed in Chakraborti (2000) and used extensively in the literature by many researchers. Note that Chen (1997), among others, has also used similar ideas.

Four cases can arise with the monitoring of the normal mean. First, the simplest case is when both the mean and the standard deviation are known or specified. This is called the standards known case and was referred to as Case K earlier. Henceforth, this is referred to as Case KK, each letter denoting the status/assumption about each parameter, first for the mean and the second for the standard deviation, respectively. Next, perhaps the most common case is when the process mean is specified or known (K) but the process standard deviation is unknown (U). This will be referred to as Case KU. The other two cases are, respectively, Case UK, where the mean is unknown but the standard deviation is known, and finally, the important case, Case UU, where both the process mean and the process standard deviation are unknown. Admittedly, Case UK is more academic and is perhaps less practical, thus we leave the analysis of this case as an exercise for the interested reader.

In each case, the key quantities to be dealt with are the signaling event, the probability of a signaling event, and the run‐length distribution. Once the run‐length distribution is obtained, chart performance characteristics associated with the run‐length distribution, such as the average (expected value), the standard deviation, etc., are obtained easily.

We begin with Case KK for simplicity and to set the stage. In this case, the two‐sided Shewhart control limits are given by

where and denote the IC known process mean and standard deviation, respectively. The process is declared OOC or, equivalently, the control chart signals when a charting statistic falls outside of the control limits. Thus, in Case KK, a signaling event is

or, equivalently, if

Hence, the probability of a signal is

Thus, in Case KK, the run‐length distribution of the Shewhart control chart is geometric with success probability , that is, GEO(). This follows from the fact that, for Case KK, the four conditions for the geometric distribution are satisfied, namely, the experiments/trials must be independent Bernoulli trials with a constant probability of success, where a trial/experiment here refers to a comparison of the sample mean to the control limits. Further, at each stage, we observe a Bernoulli trial, which is either a “signal,” when or , which may be classified as a success (), or a “no signal”; when , which may be classified as a failure (). Moreover, the probability of success is and the trials at different points in time are independent since the samples are drawn independently at each time point. Finally, the run‐length refers to the random variable, the number of trials until the first success, among this sequence of independent Bernoulli trials. Thus, the distribution of run‐length follows a geometric distribution with success probability .

Since the run‐length distribution is geometric, it is completely characterized by the success probability , and hence all properties of the distribution, such as the mean (ARL), the standard deviation (SDRL), and the median run‐length (MRL), etc., can all be found in terms of , for example, the ARL and the SDRL are given by

respectively. As noted earlier in Chapter 2, the 100pth percentile can be calculated from

where inf denotes the infimum and denotes the order of the percentile. Thus, is the smallest integer that is at least equal to . For example, for the median run‐length MRL, we have , and thus

Next, consider the case when the mean is known or specified and the variance is unknown, that is, Case KU. This situation can arise in problems where the mean is specified by some production or regulatory constraint but the standard deviation of the process is unknown. There is more than one way to start the monitoring process in this situation. Typically, the unknown parameter (here, the variance) is first estimated from a retrospective (Phase I) analysis of data that are already available from the process when it was deemed to be IC. This analysis is called a Phase I analysis and such data are called reference data, which consist of a fixed size sample. Other options could include estimating parameters “on the go” with what are known as self‐starting and sequential sampling schemes, but we focus here on the more traditional approach of using a fixed size reference sample. Control charts are used in Phase I to get a process under control, but the construction of control limits is not the same as in Phase II. We discuss control charts for the Phase I analysis later in Chapter 5.

In Phase II, the control limits obtained from Phase I are applied to monitor the status of the process on an ongoing basis. Assume that in Phase I the process follows a normal distribution so that for and . In addition, assume that we estimate the unknown IC process standard deviation with the point estimator . The two‐sided control limits in Case KU are given by substituting (plugging‐in) the point estimator in the Case KK control limits

In this case, note that the signals and the non‐signals refer to events in Phase II, that is, for sample ( + 1) and onward. Assume that the Phase II samples follow a distribution. Thus, the signaling event in Phase II can be written as

which can be re‐expressed as

The probability of a signaling event in Case KU is calculated in two steps. First, the probability of a signal is calculated conditionally given the parameter estimator (or ) from Phase I, which is a random variable. Hence, this probability is called the conditional probability of a signal, and plays a key role in the subsequent developments. Note that when substituting the estimate of the standard deviation in the Case KK control limits to construct the Case UU limits, one could use a charting constant from a t distribution with n − 1 degrees of freedom, particularly since n is usually small. Such an idea has been considered in the literature but we don't pursue these details here.

Let , where the quantity characterizes whether there is a shift in the mean and hence this formulation allows us to handle both the IC () and the OOC () cases. Thus, the conditional probability of a signal given , in Phase II, is

This probability can be rewritten as

Now, assuming that, along with the underlying normal distributions, the variances in Phase I and II remain unchanged (i.e., the shift is expected to be only in the mean assuming ), this last probability can rewritten as

Finally, writing , which follows a chi‐square distribution with degrees of freedom, we can further rewrite the last expression as

Hence, since follows a N(0,1) distribution, the conditional probability of a signal can be conveniently expressed as

The conditional probability clearly shows that the chart performance will be affected by the value of the random variable Y, which corresponds to the estimator of used and obtained from the Phase I data. As noted earlier, the conditional probability of a signal plays an important role in the analysis of the performance of the control chart under estimated parameters. We comment more on this later. However, since this probability is a function of the random variable , it is a random variable itself with its own probability distribution. This is the main effect of parameter estimation on the performance of control charts and is crucial to the conceptual understanding. Thus, when the control limits utilize parameter estimate(s), in Phase II, the signaling events are dependent due to the fact that the same Phase I parameter estimate of the standard deviation is used in the control limits and each Phase II sample is compared with the same control limits involving this estimate. Thus, the run‐length distribution of the Phase II control chart no longer follows a geometric distribution, since the Bernoulli trials (signal or no signal at each sample or time) involved in this case are no longer independent and hence, typically, one does not use the FAR to describe chart performance in this case since the FAR depends on the type and the value of the estimates (obtained, for example, from the Phase I analysis). In the IC case , and the quantity denotes the conditional IC probability of a signal, or the conditional FAR in Phase II.

Thus, to summarize, in Case KU, that is, when the mean is known or specified but the variance is unknown and hence estimated, for the Shewhart chart,

1. Conditionally, the Phase II run‐length distribution is geometric with success probability , where is the nominal FAR, and is the realization of a chi‐square random variable Y with degrees of freedom. In this context, note that the corresponding random variable Y is a scaled version of the estimator . Other estimators of (such as the one based on the average of the sample ranges) could be considered, but the theory for such estimators, although similar in principle, is somewhat more involved and is thus omitted here.
2. Conditionally, the properties of the Phase II Shewhart chart in Case KU can be obtained from the properties of the geometric distribution.
3. So, for example, the conditional run‐length distribution is given by its probability mass function (pmf) where we write, for brevity, .

Hence, using properties of the geometric distribution, we can completely characterize the conditional run‐length distribution in Case KU and calculate various attributes of it, for example, the conditional average run‐length () of the Phase II chart is given by . Note that, as a function of , is a random variable given by , with a value , for a value of . Similarly, the variance and the standard deviation of the conditional run‐length distribution are both random variables, given by

and

respectively.

The conditional run‐length distribution and its associated characteristics, such as the average, the standard deviation, and the percentiles, etc., can help us better understand the effects of parameter estimation on the performance of the Phase II control chart. This is an important point and is common to all Phase II control charts when parameters are estimated, including the Shewhart charts we discuss here. It is because the parameter estimator(s), which is a (are) random variable(s), when simply plugged into the Case KK control limits, significantly alter the performance of the resulting control chart relative to that in Case KK. Moreover, since the estimator is a random variable, two independent reference samples, both from an IC process, would likely produce two different parameter estimates, which in turn will produce two different control charts (limits) and hence would yield different Phase II performance. Thus, there will be variation in the performance of the control charts even though they may use estimates from independent reference samples from the same IC process, and even when the charts have all been calibrated to achieve the same nominal ARL_IC. This performance variation has been called user‐to‐user variation. Admittedly, this can be unsettling, and various issues come to the surface, particularly from a practitioner's point of view. To this end, several recent articles in the literature have suggested studying the conditional run‐length distribution in more detail and hence better understanding the impact of user‐to‐user variation on Phase II control charts. We cover some of these aspects in later chapters.

While the conditional run‐length distribution can help us better understand the effects of parameter estimation, one can further average the conditional distribution over the distributions of the parameter estimator, a process known as “unconditioning,” and study the “average” chart performance. For example, the unconditional probability of a signal in Case KU is given by

where denotes the pdf of a chi‐square distribution with degrees of freedom. Similarly, the unconditional or the marginal run‐length distribution is given by taking the conditional run‐length distribution and averaging over the distributions of the random variable, , which are independent for the normal distribution. Hence, we have the unconditional run‐length pmf given by

which is not the pmf of a geometric distribution.

This type of derivation of the marginal or the unconditional run‐length distribution and its associated characteristics is referred to as the conditioning‐unconditioning (CUC) method. The derivation proceeds by first conditioning over the parameter estimator and then by taking the expectation of the result with respect to the probability distribution of the estimator.

Thus, in Case KU, using the fact that the conditional run‐length distribution is geometric with success probability , and applying the CUC method, the average of the unconditional run‐length distribution is given by

This is the unconditional ARL of the Shewhart control chart denoted by ARL_U. Note that the same result could be obtained from the unconditional run‐length pmf given earlier.

In addition to the ARL, the variance (or the standard deviation) of the unconditional run‐length distribution is an important performance measure. The derivation of the unconditional variance is interesting and, as such, is explained further. The derivation is facilitated by using the result mentioned in Chapter 1, that for any random variable N, the unconditional variance can be obtained using the conditional mean, the conditional variance, and the formula

(see also, for example, Hogg, McKean, and Craig, 2005) where the expectation is taken over the distribution of the random variable . For the and the terms, recall that, given , the run‐length distribution is geometric with probability and hence from the properties of the geometric distribution

and

respectively. Thus,

and by substitution we get

Rewriting this expression in terms of integrals, we obtain the unconditional variance of the run‐length distribution in Case KU

where the reader should recall that , as defined earlier. Note that, although the run‐length distribution is always discrete, the monitored variable, and therefore the corresponding parameter estimator(s), can be continuous (as in the present case) as well as discrete, as in the case of attributes charts (see Montgomery, 2009, Chapter 7). In the latter case, the estimator Y will be discrete and thus the unconditional run‐length distribution, its average, and the variance can be obtained in a similar manner using the CUC method, except that now the integral will be replaced by summation over the mass points of the estimator Y, and the chi‐square pdf in the integration will be replaced by the appropriate pmf of Y. The reader is referred to Human and Chakraborti (2006) for examples.

Other moments and characteristics of the run‐length distribution can also be obtained similarly using the CUC method, which takes advantage of the facts that (i) the conditional run‐length given the parameter estimate is the geometric distribution and (ii) the unconditioning involves averaging (taking expectation) over the distribution of the parameter estimator, which follows some known probability distribution.

Typically, one refers to the unconditional run‐length distribution as the run‐length distribution of a control chart, but the conditional run‐length distribution is gaining more prominence as it captures and explains the impact of the inherent variation in the parameter estimators on the performance of the control chart. The reader is referred to the recent literature for more details.

In summary, while the unconditional run‐length properties, such as the mean and the standard deviation, are fixed, and describe the average performance of the control chart over many reference samples, the conditional run‐length properties vary based on the values of the parameter estimators for a given reference sample and show the impact of parameter estimation based on the value of the estimator that the users have for their particular situation. Both the conditional and the unconditional performance may have a role to play in the practical implementation of control charts with estimated parameters. The Case KK run‐length distribution and its associated characteristics have been the benchmarking norm in the design of a control chart for over 50 years, until recently, when the unconditional run‐length and its associated characteristics gained prominence in light of the effects of parameter estimation, but, as noted earlier, it is recognized that, although the averaging process may produce some useful overall guidance, the conditional run‐length and its associated characteristics may be very useful while implementing and understanding chart performance in practice when estimated parameters are used in control charts. The same summary applies to all types of control charts, univariate or multivariate, variables or attributes, parametric or nonparametric, where unknown parameters are estimated from the data.

Next, consider the case when both parameters are unknown, that is, Case UU. The analysis is similar to that in Case KU, except that now two parameters and their corresponding estimators are involved, which raises the complexity. We will try to point out the similarities with Case KU and highlight the main differences. As we noted earlier, in Phase II, the parameter estimators and hence the control limits are obtained from Phase I and are applied to monitor the status of the process in Phase II on an ongoing basis. Again, as in Case KU, assume that in Phase I the process follows a normal distribution so that for and . In addition, we assume that we estimate the unknown IC process mean and the unknown IC process standard deviation with the overall mean which is the grand mean of the means of the reference samples

and the pooled estimator

Thus, unlike in Case KU, two estimators, namely and , are involved in the analysis in Case UU. Note also that, unlike in Case KU, the estimator of , namely , has a scaled chi‐square distribution but with , degrees of freedom. Thus, one loses degrees of freedom (recall that in Case KU the degree of freedom was ) while estimating the mean using the Phase I samples.

Again, the two‐sided control limits are given by substituting the point estimators in the control limits for Case KK

As before, assume that the Phase II samples follow an distribution. Thus, the signaling events in Phase II can be written as

which can be re‐expressed as

Compare this signaling event with that in Case KU. As in Case KU, under the CUC method, the probability of a signaling event in Case UU is calculated in two steps. First, the probability of a signal is calculated given (conditioned on) the two parameter estimators, and , which are both random variables. This probability is called the conditional probability of a signal and it plays a key role in the subsequent developments.

To show this, we note that the probability of a signal in Case UU is given by for This can be written as

Now, writing , , and , we can rewrite the above probability as

Finally, writing , which follows a chi‐square distribution with degrees of freedom and denoting , we can further rewrite the last expression as

When the quantity . Finally, since follows a N(0,1) distribution, the conditional probability of a signal in Phase II for the mean chart in Case UU is given by

where and are independent random variables. The variable follows a standard normal distribution and follows a chi‐square distribution with degrees of freedom. Again, other estimators of could be considered, and one would have to work with its distribution in the sequel. We leave this up to the reader.

In any case, also in Case UU, the conditional probability of a signal plays an important role in the analysis of the performance of the control chart under estimated parameters. We comment more on this later. In the IC case, we have and under the typical assumption that , that is, , the quantity denotes the IC conditional probability of a signal (the FAR) in Phase II. However, since this probability is a function of random variables and , it is a random variable with its own probability distribution. This is why, as we noted earlier, when parameters are estimated in Phase II, typically, one does not use the FAR to describe chart performance since the FAR depends on the type of estimates obtained from the Phase I analysis and is itself a random variable with a probability distribution.

Moreover, as in Case KU, the Phase II signaling events are dependent in Case UU due to the fact that the same Phase I parameter estimates in the control limits and each Phase II sample is compared with the same control limits involving these estimates. Thus, in Case UU, as in Case KU, the run‐length distribution of the Phase II control chart no longer follows a geometric distribution since the Bernoulli trials involved in this case are no longer independent. However, again, as in Case KU, given the Phase I parameter estimators (or conditioning over the corresponding random variables), the signaling events are independent. This observation facilitates a lot of the statistical calculations of the Phase II chart properties in Case UU, as it did in Case KU. Thus, to summarize, in Case UU, for the Shewhart chart,

1. The conditional Phase II run‐length distribution is geometric with success probability .
2. Hence, conditionally, properties of the Phase II Shewhart chart follow from the properties of the geometric distribution, for example, the conditional run‐length distribution is given by its probability mass function (pmf)
   
   where we write, for brevity, . Now, using properties of the geometric distribution, we can easily obtain various attributes of the conditional run‐length distribution. The conditional average run‐length () of the Phase II chart, for example, is given by . Similarly, the variance and the standard deviation are given by and , respectively.

As noted earlier, parameter estimates plugged into the control limits can significantly alter/affect the performance of the control chart and the conditional run‐length distribution. Its associated characteristics, such as the average, the standard deviation, etc., can help to describe the effects of parameter estimation on the performance of the Phase II control chart. The estimators are random variables, and two independent reference samples from the same IC process would most likely produce two different sets of parameter estimates, which in turn will produce different control charts (limits), and hence different Phase II chart performance. The bottom line is that there will be variation in the performance of the control charts among users, when they use estimates from independent reference samples from the same IC process, and even though the control charts may have been calibrated to achieve the same nominal ARL_IC. This variation may be termed as user‐to‐user variation, which is not easy to control unless there are lots of data, which may not be practical in all situations. To this end, as noted before, several recent articles have suggested studying the conditional run‐length distribution and the impact of user‐to‐user variation on Phase II control charts. We cover some of these aspects in later chapters.

Averaging over the distributions of the parameter estimates, a process known as “unconditioning,” we can study the “average” chart performance, for example, the unconditional probability of a signal in Case UU is given by

where denotes the pdf of a chi‐square distribution with degrees of freedom. Note that for the normal distribution the estimators of the mean and the variance are statistically independent. Therefore, the joint distribution factors into the product of the individual distributions, one a standard normal and one a chi‐square. This may not be the situation in all cases, however. Similarly, using the CUC method, the unconditional run‐length pmf can be obtained by integrating the conditional run‐length pmf over the joint pdf of and , which is the product of the marginal pdf's of and , respectively since and are independent. Hence

Similarly, using the fact that the conditional run‐length distribution is geometric with a success probability , and applying the CUC method, the expected value of the unconditional run‐length distribution is given by

This is the unconditional ARL of the Shewhart control chart in Case UU. In general, for higher‐order moments, say, the non‐central moment of the run‐length distribution can be obtained from the conditional non‐central moment, that is, using the fact that follows a geometric distribution. In particular, the variance (or the standard deviation) of the unconditional run‐length distribution is an important performance measure. This can be shown to be equal to

The derivation of the unconditional variance (and standard deviation) is interesting and is explained further. This follows from the result that for any random variable N, (see, for example, Hogg, McKean, and Craig, 2005) where the expectation is taken over the distributions of the two independent random variables and . Furthermore, under the assumption of a normal distribution, and are independent. For the and the terms, we note that, given and , the run‐length distribution is geometric with probability and hence from the properties of the geometric distribution, and . Thus

and by substitution we get

Rewriting this expression in terms of integrals, we obtain the unconditional variance

shown above. Hence, the unconditional run‐length standard deviation is given by .

Typically, one refers to the unconditional run‐length distribution as the run‐length distribution of a control chart, which has been used in the traditional evaluation of a control chart performance. However, the conditional run‐length distribution has gained more prominence recently as it captures and explains the impact of the inherent variation in the parameter estimators on the performance of the control chart. Other moments and characteristics of the run‐length distribution can be obtained similarly using the CUC method, which takes advantage of the facts that (i) the conditional run‐length given the parameter estimates is the geometric distribution, and (ii) the unconditioning involves averaging (taking expectation) over the distributions of the two parameter estimators, which are independent, and follows some known probability distributions. In summary, while the unconditional run‐length properties, such as the mean and the standard deviation, are fixed and describe the average performance of the control chart over many samples, the conditional run‐length properties are not and show the impact of parameter estimation based on the estimators that the users have for their situation. Both conditional and unconditional performance may have a role to play in the practical implementation of control charts with estimated parameters.

For Case UK, the probability of a signal, the conditional run‐length distribution, its various moments, the unconditional run‐length distribution, and its various moments can be obtained similarly and are left for the reader as exercises. More details can be found in Chakraborti (2000).

Having considered the Shewhart chart for the mean in detail, we are now in a better position to study some of the more advanced or sophisticated charts. These include the CUSUM and the EWMA charts, and Shewhart charts with supplementary runs‐rules, which are expected to be more effective in detecting small, persistent changes in the underlying parameter. The calculation of the run‐length distribution and associated properties for these charts are more involved in Case U, but one can again apply the CUC method. This is briefly outlined as follows, first in general terms. Recall that, for these charts, expressions for the run‐length distribution and some of the associated characteristics in Case K were given in Equations 3.21 ,3.22,3.23,3.24 using the MC approach. This is the starting point in Case U. As we noted earlier, a key to using these expressions is the construction of the essential transition probability matrix for each chart. In Case U, we first use these formulas conditionally given the parameter estimates, and then apply the CUC method to find the unconditional run‐length distribution and its associated properties. Suppose, for example, that there is a single unknown parameter to be estimated with the corresponding Phase I estimator . Now we calculate the elements of the essential probability matrix , the so‐called transition probabilities, by using the conditional distribution of the charting statistic in Phase II, given the estimator . Thus, each element of the transition probability matrix is calculated conditional on , and let denote the resulting conditional essential probability matrix. Next, we can apply the expressions given in Equations 3.22 ,3.23,3.24 conditionally on , and get

(3.36)

(3.37)

(3.38)

(3.39)

Finally, using the CUC method, the unconditional run‐length distribution is obtained by taking the expectation (averaging or integrating) over the distribution of . So, for example, the unconditional of the chart is given by

(3.40)

where is assumed to be continuous with a pdf . Other moments and associated run‐length distribution characteristics can be found in a similar manner. Note that a similar analysis is possible if case is discrete, in which case one would obtain these expressions for the unconditional run‐length distribution by summing over the pmf of . Interested readers are asked to try out some examples. Some examples can be found in Chakraborti, Eryilmaz and Human (2009).

3.10.1.2 The Shewhart Chart for the Variance in Case U

The run‐length distribution of the Shewhart charts for the variance can be obtained similarly, as in the case of the mean, using the CUC method. We sketch a brief example here for the chart with as the estimator of and derive the IC run‐length distribution.

Note that, in this case, the probability limits of the Phase II chart are given by , , and , where , , and is the unconditional probability that the Phase II charting statistic, the standard deviation, plots outside either the lower or the upper control limit when the process is IC. Note that, in the parameter known case, this is the probability of a false alarm or the FAR and it only depends on the sample size n, but in the unknown parameter case, the probability depends on both m and n. Once the quantity is determined, the probability limits are easily obtained using the corresponding percentiles of the chi‐square distribution with degrees of freedom. Although we may refer to as the FAR (probability of a signal when the process is IC) in Case U, for convenience, note that it does not have the same clear interpretation as in Case K since the signaling events are dependent in Case U. Some details about the derivation of are given in Appendix 3.3. The values are calculated and shown in Table H in Appendix A along with the corresponding control limits and for n = 5 and 10 and some values of m from 5 to 100, with nominal ARL_IC values of 370 and 500. Note that the values depend both on m and n, and for a given value of n, they increase for increasing values of m (increasing the number of Phase I samples).

Other moments of the conditional and the unconditional IC run‐length distribution can be found in a similar way. Note that when other estimators of , such as the one based on the or the are used in the Phase I control limits, or a different chart such as the R chart is used in Phase II, a similar approach can be used to derive the run‐length distribution and its various moments by applying the CUC method. We basically need the probability distributions of the charting statistic and of the Phase I estimator of . Note that the same approach can be also used to derive the OOC run‐length distribution, which is useful in studying the performance of these charts in shift detection and comparisons. We leave the details to the reader.

3.10.1.3 The CUSUM Chart for the Mean in Case U

The reader is referred back to Example 3.6 in Section 3.6.1.2. In this example, the underlying process distribution is assumed to be N(0,1). Let us consider, for example, the one‐step transition probability, . It was shown that . However, recall that the was obtained using the cdf of a N(0,1) distribution, which requires not only the knowledge of the normality of the process distribution but also that the mean and the standard deviation, assumed to be 0 and 1, respectively. Suppose now that we are in Case KU, that is, the process mean is known (say, is equal to zero), but the process variance is unknown. In this case, we have a single unknown parameter to be estimated by some estimator (which is typically obtained from a Phase I analysis) and we calculate the elements of the essential probability matrix , that is, the transition probabilities, conditionally, given the estimator . Now is calculated as where is some estimator of . The same applies to the other one‐step transition probabilities. Thus, , the estimated conditional essential probability matrix, is obtained and substituted into Equations 3.36,3.37,3.38,3.39. Finally, using the CUC method, the unconditional run‐length distribution is obtained by taking expectation over the distribution of , as explained above and illustrated in Equation 3.40 for the unconditional ARL. Case UK follows similarly. However, this time the process mean needs to be estimated and dealt with and, for Case UU, where both the process mean and the process variance need to be estimated, we condition on, say, and , where denotes the estimator for the process mean and denotes the estimator for the process variance. The reader is also referred to Jones, Champ, and Rigdon (2004) for an alternative derivation of the run‐length distribution of the CUSUM chart with estimated parameters.

3.10.1.4 The EWMA Chart for the Mean in Case U

The reader is referred back to Example 3.7 in Section 3.6.1.2 . In this example, the underlying process distribution is assumed to be N(0,1). Let us consider the one‐step transition probability, , in that example. It was shown that is equal to but is obtained using the cdf of a N(0,1) distribution. However, even when we assume normality, one or more of the parameters can be unknown and would need to be estimated. In such a case, we proceed as in the case of the CUSUM chart discussed above, and calculate the elements of the transition probability matrix conditioned on the parameter estimators. Then, the resulting conditional essential probability matrix is obtained and substituted into Equations 3.36 ,3.37,3.38,3.39. Finally, using the CUC method, the unconditional run‐length distribution is obtained by taking expectation over the distribution of the estimator(s), as explained above and illustrated in Equation 3.40 for the unconditional ARL. The reader is referred to Jones, Champ, and Rigdon (2001) for a detailed discussion on the run‐length distribution of the EWMA chart with estimated parameters.

In the unknown parameter case, the CUC method has been successfully used not only for various parametric charts but also for nonparametric charts, as will be seen in the next chapter.