Abstract: A nonparametric control chart based on a bivariate signed-rank test is developed for monitoring the changes in the location of a bivariate process.
Quality Technology & Quantitative Management Vol. 9, No. 4, pp. 317-328, 2012
QTQM © ICAQM 2012
A Nonparametric Signed-Rank Control Chart for Bivariate Process Location V. B. Ghute1 and D. T. Shirke2 1
Department of Statistics, Solapur University, Solapur, India Department of Statistics, Shivaji University, Kolhapur, India
2
(Received July 2010, accepted July 2011)
_____________________________________________________________________________ Abstract: A nonparametric control chart based on a bivariate signed-rank test is developed for
monitoring the changes in the location of a bivariate process. The average run length performance of the proposed nonparametric chart is investigated using a simulation study and is compared with a parametric control chart under bivariate normal and bivariate double exponential distributions. Further the performance of the proposed nonparametric chart is improved by using runs rules and the concept of a synthetic control chart.
Keywords: Average run length, conforming run length, runs rules, signed-rank test.
_____________________________________________________________________________
1. Introduction
C
ontrol charts are useful tools for monitoring/controlling a manufacturing process. Univariate control charts are used to monitor processes that manufacture products with a single quality characteristic of interest. There are many situations in which a process is characterized by more than one quality characteristics. Multivariate control charts are best suited to monitor such processes. Most of the multivariate control charts are based on the assumption that the underlying distribution of the process is multivariate normal. In reality this assumption may not hold in all the situations. In such situations, development and application of control charts that do not depend on a particular distributional assumption is desirable. Nonparametric control charts can serve this purpose.
In the literature, several nonparametric control charts are proposed for monitoring the location of a univariate process. Some of these are based on sign and/or rank statistics by assuming a known in-control target value for the process location. Amin et al. [1] developed Shewhart and cumulative sum (CUSUM) control charts based on sign test statistic. Bakir and Reynolds [6] developed a nonparametric CUSUM to monitor the process center based on within group signed-ranks. Amin and Searcy [2] used with-in group signed ranks to develop an exponentially weighted moving average (EWMA) control chart. Bakir [4] developed a distribution-free Shewhart control chart for monitoring the process center based on the signed-ranks of grouped observations. Bakir [5] proposed Shewhart, CUSUM and EWMA control charts based on signed-rank-like statistics of grouped data for monitoring a process center when the in-control target center was not specified and studied the robustness of the charts against outliers. Chakraborti et al. [9] presented an extensive overview of the literature on univariate nonparametric control charts. For monitoring multivariate process very few nonparametric control charts are available in literature. Hayter and Tusi [14] proposed a Shewhart-type multivariate nonparametric control scheme for the mean vector. Kapatou and Reynolds [15,16] proposed EWMA-type multivariate
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control charts for groups based on the sign and signed-rank statistics. Liu [17] proposed a Shewhart-type multivariate nonparametric control chart based on simplicial data depth. Qiu and Hawkins [18] suggest a nonparametric multivariate CUSUM procedure based on the antiranks of the measurement component. The purpose of this paper is to develop a nonparametric control chart for monitoring the location of a bivariate symmetric process. When the process distribution is normal, Hotelling’s T 2 is an appropriate control chart for monitoring the process location. If the underlying process distribution is non-normal, then we consider nonparametric control chart based on the appropriate nonparametric test. Many nonparametric competitors to Hotelling’s T 2 have been proposed in the literature. In this paper, the well-known bivariate signed-rank test statistic proposed by Bennett [7] is used as a charting statistic. The proposed control chart will be used for detecting shifts in the location of a bivariate process. It should be noted that the nonparametric control charts can be more efficient than their parametric counter parts when the samples are from skewed or heavy tailed symmetric distributions and less efficient for normal or normal-like distributions with light tails. To increase the sensitivity of parametric control charts various techniques are used in literature. Some of these are the use of adaptive sample sizes, double sampling, warning limits, runs rules and synthetic control chart. In order to improve the performance of Hotelling’s T 2 chart Aparisi et al. [3] studied the chart with four commonly recommended supplementary runs rules. They show that after introducing supplementary runs rules, the ARL of the chart decreases considerably when there is a shift in the process. Wu and Spedding [20] have developed univariate synthetic X control chart for detecting changes in the mean of a normally distributed process. This chart synthesizes the salient features of a Shewhart X chart and conforming run length (CRL) chart, due to Bourke [8] into a single chart. The CRL chart is described in detail in Section 4. It was observed that the synthetic X chart has greater power for detecting all range of shifts in the process mean than the Shewhart X chart. Ghute and Shirke [12] extended the synthetic X chart proposed by Wu and Spedding [20] to the multivariate case by combining Hotelling’s T 2 chart and the CRL chart. Ghute and Shirke [13] also developed a multivariate synthetic S chart for monitoring the process dispersion. It was shown that the synthetic T 2 chart detects shifts in process mean faster than the traditional Hotelling’s T 2 chart and the synthetic S chart detects shifts in covariance matrix faster than the traditional S chart. With the objective of improving the performance of the proposed nonparametric control chart, we apply the concepts of runs rules and the synthetic control chart. The rest of the paper is organized as follows. In Section 2, a bivariate nonparametric control chart based on signed-rank test statistic is discussed. The performance of the proposed nonparametric control chart is studied with runs rules in Section 3. In Section 4, the nonparametric synthetic control chart is developed by combining the nonparametric chart and CRL chart. In Section 5, the performance of proposed nonparametric control chart is compared with a parametric control chart. A robustness study of the proposed nonparametric synthetic control chart to violations of the symmetric distribution assumption is presented in Section 6. Some conclusions are given in Section 7.
2. Bivariate Nonparametric Signed-rank Control Chart Let X i ( X1i , X 2i ), i 1,2,...,n be a subgroup sample from some continuous and symmetric bivariate distribution with the location and the covariance matrix . .
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Let 0 and 0 denote the in control values of the process parameters that are either known or estimated at the end of phase I process control. Without loss of generality, we assume that 0 (0, 0) and
1 0
, 1 1. 1
We are interested in detecting shifts in 0 . When the distribution of observations is bivariate normal, Hotelling’s T 2 is an appropriate chart for this problem with charting statistic
T 2 n X 0 -1 0 X 0 ,
(1)
where X is the sample mean vector, 0 is the in-control mean vector and 0 is the covariance matrix. When the process is in-control, 0 the statistic T 2 is distributed as chi-square random variable with 2 degrees of freedom. The chart operates by plotting T 2 values on a chart with upper control limit UCL 2,2
(2)
where 2,2 is the upper 100 percentage point of chi-square distribution with 2 degrees of freedom and is the Type I error probability of the T 2 control chart. If 0 , the statistic T 2 is distributed as a non-central chi-square random variable with 2 degrees of freedom and a non-centrality parameter n 2 , where
01 .
(3)
If underlying process distribution is non-normal, then we consider a nonparametric control chart based on the popular signed-rank test statistic proposed by Bennett [7]. This test statistic is given in the quadratic form involving coordinate-wise signed-rank statistics. At each inspection point, a nonparametric signed-rank statistic is computed for each variate in X i = ( X1i , X 2i ) using n observations in a sample. Define the indicator variable 1 when X ji 0 C X ji 0 when X ji 0.
(4)
Let T1 and T2 be the two signed-rank statistics corresponding to two variables in a sample of size n. For the j th variate, define the signed-rank statistic as
T j C X ji R X ji , j 1, 2. n
(5)
i 1
where R ( X ji ), j 1, 2 is the rank of X ji among X j 1 , X j 2 , , X jn . Let E (T j 0 ) v j , j 1, 2. The combined into a quadratic form
two
signed-rank
W T v ˆ 1 T v ,
statistics
T1 and T2
are (6)
where T (T1 , T2 ) is a vector of coordinate-wise univariate signed-rank statistics, v (v1 , v2 ) and
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11
β
21
12 22
is the variance-covariance matrix of the vector T . When the process is in-control, the means, variances, and covariances of the signed-rank statistics are given by the following results (Dietz [11]):
1 2 and
n (n 1) , 4
11 22
n (n 1) (2n 1) 24
n
sgn( X1i ) sgn( X 2i ) R X1i R X 2i
i 1
4
12 21
.
(7)
These three results from Equation (7) are then used in the computation of the control statistic W, as shown in Equation (6). The resulting statistic in Equation (6) equals Bennett’s [7] bivariate signed-rank test statistic. When 0 , the statistic W in Equation (6) is asymptotically 22 . We consider W as the control statistic for the nonparametric control chart and the chart is referred as nonparametric W (denoted NP-W) chart. The chart indicates that a shift has occurred if W UCL,
(8)
where UCL is the upper control limit of the chart.
3. Nonparametric W Chart with Runs Rules In this section we study the performance of the proposed NP-W control chart using runs rules. Chakraborti and Eryilmaz [10] developed a Shewhart-type nonparametric control chart based on Wilcoxon signed-rank statistic and the following run types rules. Note that we would usually use only one of these rules at a time. Each rule would have different control limits. The process is declared to be out of control when (a). A single point plots outside the control limit (one-of-one rule). (b). k consecutive points plot outside the control limit (k-of-k rule). (c). k of the last w points plot outside the control limits (k-of-w rule). In order to improve the performance of NP-W chart we consider rule (b) of Chakraborti and Eryilmaz [10]. That is, the process is declared to be out of control when k consecutive points plot outside the control limit (k-of-k rule). The ARL performance of nonparametric W chart with runs rules (denoted NPR-W chart) is investigated for k = 2 using a simulation study in Section 5.
4. Nonparametric Synthetic W Control Chart In this section, we develop a nonparametric synthetic control chart based on signed-rank statistic (denoted NPS-W chart) for detecting shifts in the process location of a bivariate symmetric process. The proposed NPS-W control chart is a combination of the NP-W chart and the Conforming Run Length (CRL) chart proposed by Bourke [8]. The NP-W sub-chart has an upper control limit CL. The CRL sub-chart has a lower control limit L. The value of CL is used to classify W as conforming unit or nonconforming unit and the value of L is used to decide whether process is in-control or out-of-control.
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4.1. Conforming Run Length Chart
The Conforming Run Length (CRL) chart proposed by Bourke [8] was originally developed for attribute quality control to detect a shift in the fraction nonconforming , when 100% inspection is in use. The run length of conforming items between successive nonconforming items was taken as the basis for the control chart. The CRL is defined as the number of conforming units between two consecutive nonconforming units including the end nonconforming unit. To illustrate, in Figure-1, the white circle represents conforming unit and the black circle show nonconforming unit. The three values of CRL are shown CRL1 4, CRL 2 5 and CRL3 3.
CRL1 = 4
CRL2 = 5
CRL3 = 3 Nonconforming unit
Conforming unit
Figure 1. Conforming Run Length.
Because CRL is geometrically distributed, the mean value and cumulative probability function of CRL can be calculated as follows:
CRL
1
F CRL 1 1
(9) CRL
, CRL 1, 2,
When the goal is to detect an increase in , only a lower control limit L
ln (1 CRL ) ln (1 0 )
is required for the CRL chart, where CRL 1 (1 0 ) L F0 ( L ) is the Type I error of the CRL chart and 0 is the in-control fraction nonconforming. If a sample CRL is smaller than or equal to L, it suggests that the fraction nonconforming has increased and an out of control signal is given. The average number of CRL samples, ARL CRL , required to detect an out-of-control fraction nonconforming is given by (Wu and Spedding [20])
ARL CRL
1
F L
1 1 1
L
.
(10)
Let ANS CRL be the average number of the inspected units required to signal a fraction nonconforming shift then ANS CRL CRL ARL CRL
1
1 . 1 (1 ) L
(11)
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4.2. Operation of the NPS-W Chart
The operation of the NPS-W chart is as follows: 1. Decide on the upper control limit CL of the NP-W sub-chart and the lower control limit L of the CRL sub-chart. 2. Take a random sample of n bivariate observations at each time and calculate signed-rank based control statistic W. 3. If W CL, W is regarded as a conforming unit and control flow returns to Step (2). Otherwise, W is considered to be a nonconforming unit, and control flow proceeds to the next step. 4. Count the number of units between the current and previous nonconforming units. This number is taken as a sample CRL value of the CRL sub-chart. 5. If CRL L, the process is considered to be in-control and control flow goes back to Step (2). Otherwise the process is taken as out-of-control, and continues to the next step. 6. An out-of-control signal is given. 7. Action to locate and remove the assignable cause will be taken and then the control flow goes back to Step (2). Let ARL S ( ) denote the average number of W samples required for a NPS-W chart to signal a shift of magnitude in process location. Then ARL S ( ) can be calculated using the formula (Wu and Spedding [20]) ARL S ( )
1 1 , P ( ) 1 1 P ( ) L
(12)
where P ( ) P W CL | 0 4.3. Design of the NPS-W Control Chart
The NPS-W chart requires determining the two design parameters L and CL that minimize the ARL for a predetermined shift of magnitude * in the process location. The design shift * is the magnitude considered large enough to seriously impair the quality of the products; thus corresponding ARL S ( *) should be as small as possible. ARL S (0) is decided by the requirement on the false alarm rate. The NPS-W control chart is properly designed by solving an optimization problem. The objective function is Minimize ARLS ( *), (13) subject to ARL S (0) ARL0 , where ARL0 is the user specified in-control ARL. For given values of ARL S (0), n and * the solution to the optimization problem is found by increasing L incrementally starting with the value L = 1. At each step, we solve for CL in Equation (12) with 0 and then substitute CL and L in Equation (12) to compute ARL S ( *). This iteration is continued until increasing the value L no longer reduces ARL S ( *). This method gives the optimal values of L and CL for the user-specified values ARL0 , n and *.
AS an example, when the underlying process data are normally distributed and Hotelling’s T 2 is the charting statistic, we choose ARL 0 380, n 25 and * 0.5, Table 1 shows that each set of (L, CL) results in different ARLS ( *). The ARL S ( *)
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first declines and then increases. The ARL S ( *) reaches its minimum at 2.1374 when L = 4 and CL = 7.2869. So in this case with ARL 0 380, n 25 and * 0.5 the design parameters of the parametric synthetic chart are L = 4 and CL = 7.2869. Table 1. Different sets of values of L and CL for the NPS-W chart. CL L 1 5.9400 2 6.6135 3 7.0085 4 7.2869 5 7.5022 6 7.6779 7 7.8261 8 7.9542 9 8.0669 10 8.1675
ARL S ( *) 2.7141 2.2654 2.1572 2.1374 2.1506 2.1776 2.2102 2.2445 2.2788 2.3122
The optimal values of design parameters L and CL for sample sizes n = 15, 20 and 25 provided in-control ARL = 200, 370 and 500 under predetermined shift * 0.4 (0.1) 0.8 are provided in Table 2. The results given in Table 2 are for the multivariate normal case.
n
*
15
0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8
20
25
Table 2. Optimal design parameters for the NPS-W chart. ARLS (0) 200 ARLS (0) 370 ARLS (0) 500 L CL L CL L CL 8.6239 12 8.2290 11 7.3970 9 8.1086 7 7.7986 7 7.0153 6 7.7829 5 7.4749 5 6.6296 4 7.5663 4 6.9815 3 6.3549 3 7.2867 3 6.9815 3 5.9657 2 8.3503 9 7.9266 8 7.1609 7 7.7829 5 7.4749 5 6.6296 4 7.5663 4 6.9815 3 6.3549 3 7.2867 3 6.9815 3 5.9657 2 6.8902 2 6.5879 2 5.9657 2 5 6.8424 6 7.6504 7 8.1086 4 6.6296 4 7.2598 4 7.5663 3 6.3549 3 6.9815 3 7.2867 2 5.9657 2 6.5879 2 6.8902 2 5.9657 2 6.5879 2 6.8902
5. ARL Comparisons To examine the ability of proposed NP-W, NPR-W and NPS-W charts to detect location shift in a bivariate process, we consider underlying process distributions as bivariate normal and bivariate double exponential with on-target mean vector 0 (0, 0),
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0 0
0
with 0, 0.6 and the sample size n = 25. The bivariate double exponential distribution was chosen to study how a heavy tailed distribution would affect the performance of the control chart. The NPS-W chart is designed by using the values of design parameters (L, CL) from Table 1, which are, technically, only appropriate for Hotelling’s T 2 chart when process data are normally distributed. The upper control limits of all the charts are then adjusted so that all charts have approximately the same in-control ARL value 380. Except for the Hotelling’s T 2 chart under bivariate normal distribution, the ARL values of the various control charts are computed using 10000 simulations when underlying process distributions are bivariate normal and bivariate double exponential. Table 3 and Table 4 provide the ARL values of the Hotelling’s T 2 chart and the proposed nonparametric control charts when the underlying process data actually follows bivariate normal distribution with 0 and 0.6 respectively. Table 3. ARL values for the various control charts under bivariate normal distribution when 0.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T 2 Chart NP-W Chart NPR-W Chart NPS-W Chart UCL = 11.8804 UCL = 10.513 UCL = 5.750 L = 4, CL = 6.920 380 380 380 380 53.24 63.96 88.49 68.67 5.47 7.34 14.27 9.53 1.74 2.03 3.95 2.59 1.16 1.18 1.74 1.33 1.03 1.02 1.18 1.05 1.00 1.00 1.03 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
Table 4. ARL values for the various control charts under the bivariate normal distribution when 0.6.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T 2 Chart NP-W Chart NPR-W Chart NPS-W Chart UCL = 11.8804 UCL = 10.50 UCL = 5.750 L = 4, CL = 6.920 380 68.67 9.53 2.59 1.33 1.05 1.00 1.00 1.00 1.00 1.00
380 103.84 17.50 4.51 1.93 1.23 1.05 1.01 1.00 1.00 1.00
380 72.22 8.47 2.23 1.21 1.03 1.00 1.00 1.00 1.00 1.00
380 60.13 6.33 1.88 1.19 1.03 1.00 1.00 1.00 1.00 1.00
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Examination of Table 3 and Table 4 lead to the following findings: When the underlying process data follow bivariate normal distribution with 0,
The out-of-control ARL values of the NPR-W chart and NPS-W charts are smaller than that of the Hotelling’s T 2 chart for small shifts in the process location and for large shifts the performance these charts is equivalent to the Hotelling’s T 2 chart.
The out-of-control ARL values of the NPS-W chart are smaller than that of the corresponding NPR-W chart and the NP-W chart. For example, for a shift of 0.2, Table 3 shows that the NPS-W chart has ARL = 53.24 which is smaller than the ARL value 88.49 of the NP-W chart. The improvement in the ARL is about 40%. The improvement in the ARL increases as the shift size increases up to 0.4 The largest improvement of the ARL is produced when the shift is 0.4 In this case approximately 62% improvement of the ARL is obtained. While for 0.6 1.2, the improvement in the ARL declines and for 1.2 all the three control charts have the same ARL.
When the underlying process data follow bivariate normal distribution with 0.6, Table 4 shows the performance improvement similar to that of the uncorrelated variables. Table 5 and Table 6 provide the ARL values of the Hotelling’s T 2 chart and the proposed nonparametric control charts when the underlying process data actually follows bivariate double exponential distribution with 0 and 0.6 respectively. Table 5. ARL values for the various control charts under the bivariate double exponential distribution when 0.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T 2 Chart
UCL = 12.68 380 82.18 11.80 2.95 1.40 1.06 1.01 1.00 1.00 1.00 1.00
NP-W Chart NPR-W Chart NPS-W Chart UCL = 10.52 UCL = 5.762 L = 4 CL = 6.92 384 380 380 26.19 35.61 51.42 3.08 4.11 7.69 1.40 1.54 2.61 1.10 1.11 1.48 1.02 1.02 1.15 1.00 1.00 1.05 1.00 1.00 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
Examination of Table 5 and Table 6 lead to the following findings: When the underlying process data follow bivariate double exponential distribution with 0,
The NPS-W chart and NPR-W chart are uniformly better than the Hotelling’s T 2 chart, the NP-W chart is efficient only for the small shifts.
The out of control ARL values of NPS-W chart and NPR-W chart are smaller than that of the corresponding NP-W chart. For example, for a shift of 0.2,
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Table 4 shows that the NPS-W chart has ARL = 26.19 which is smaller than the ARL value 51.42 of the NP-W chart. The improvement in the ARL is about 49%. The improvement in the ARL increases as the shift size increases up to 0.4 The largest improvement of the ARL is produced when the shift is 0.4 In this case approximately 60% improvement of the ARL is obtained. While for 0.6 1.4, the improvement in the ARL declines and for 1.4 both the control charts have the same ARL. When the underlying process data follow bivariate double exponential distribution with 0.6, Table 6 shows the performance improvement similar to that of the uncorrelated variables. Table 6. ARL values for the various control charts under the bivariate double exponential distribution when 0.6. 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
T 2 Chart
NP-W Chart NPR-W Chart NPS-W Chart UCL = 12.68 UCL = 10.46 UCL = 5.762 L = 4 CL = 6.92 380 98.07 17.61 4.68 1.96 1.23 1.05 1.01 1.00 1.00 1.00
380 75.38 12.39 3.90 1.94 1.33 1.12 1.05 1.02 1.00 1.00
380 56.03 6.84 2.11 1.27 1.07 1.01 1.00 1.00 1.00 1.00
380 43.07 4.92 1.81 1.23 1.07 1.01 1.00 1.00 1.00 1.00
6. Robustness of the NPS-W Chart against Symmetry The proposed NPS-W control chart for monitoring the process location is based on the assumption that the underlying process distribution is symmetric. In this Section, we study the robustness of the chart to violations of the symmetry assumption by considering the skewed bivariate gamma distribution. Detail on the multivariate gamma distribution is provided in Stoumbos and Sullivan [19]. We denote the bivariate gamma distribution with shape parameter and scale parameter by G2 ( , ). To study the effect of non-symmetry we have considered the process data from the bivariate gamma distribution with scale parameter = 1 in combination with shape parameter = 1, 2, 4, 6, 10, 15, 20, 25, 30 respectively. Table-7 provides in-control and out-of-control ARL values for the NPS-W control chart when the underlying process data follows bivariate normal and bivariate gamma distributions. Examination of Table 7 leads to the following findings:
The in-control ARL values of the NPS-W chart are significantly smaller than the normal theory value for selected shape parameters of the bivariate gamma distributions. Accordingly, the number of false alarms would be significantly higher.
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In out-of-control cases a direct comparison of ARL values is not possible because they do not have the same in-control ARL values.
Thus one may conclude that the NPS-W chart is not very robust to violations of the symmetry assumption. Table 7. ARL values for the NPS-W chart under bivariate gamma distribution. Dist. Normal G2(30,1) G2(25,1) G2(20,1) G2(15,1) G2(10,1) G2(8,1) G2(6,1) G2(4,1) G2(2,1) G2(1,1)
0.0 0.2 0.4 380 53.24 5.47 303 79.78 6.67 290 80.14 6.67 277 83.88 6.99 259 90.45 7.40 220 99.13 7.76 192 100.82 7.78 159 108.32 8.35 119 106.10 8.66 55 88.91 9.76 20 49.04 9.39
Shift 0.6 1.74 1.80 1.82 1.80 1.82 1.80 1.80 1.81 1.81 1.73 1.56
0.8 1.16 1.14 1.13 1.14 1.13 1.13 1.12 1.11 1.09 1.05 1.01
1.0 1.03 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.00 1.00 1.00
1.2 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2.0 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
7. Conclusions In this paper, we have developed a control chart based on signed-rank test statistic to monitor the location of a bivariate symmetric process. The performance of the proposed chart is improved by using the runs rule and the synthetic chart and compared with the parametric chart under the bivariate normal and the bivariate double exponential distributions. The general conclusion of the comparison is that the NPS-W and the NPR-W charts compete very well with the Hotelling’s T 2 chart when the process distribution is normal and are more efficient than the Hotelling’s T 2 chart when underlying process distribution is heavy tailed. The NPS-W chart detects shifts in the process location faster than the NPR-W control chart. Therefore, to monitor bivariate processes operating under heavy-tailed distributions, NPS-W control chart is recommended. If one is sure that a process is operating under a bivariate normal distribution, then the Hotelling’s T 2 chart is recommended over the NPS-W control chart.
Acknowledgements The authors would like to thank the Associate Editor and the two referees for their valuable comments and helpful suggestions for the improvement of this paper.
References 1.
Amin, R. W., Reynolds, M. R. Jr. and Bakir S. T. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics: Theory and Methods, 24, 1597-1623.
2.
Amin, R. W. and Searcy, A. J. (1991). A nonparametric exponentially weighted moving average control scheme. Communications in Statistics: Simulation and Computation, 20, 1049-1072.
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3.
Aparisi, F., Champ, C. W. and Garcia-Diaz, J. C. (2004). A performance analysis of Hotelling’s 2 control chart with supplementary runs rules. Quality Engineering, 16, 359-368.
4.
Bakir, S. T. (2004). A distribution-free Shewhart quality control chart based on signed-ranks. Quality Engineering, 16, 613-623.
5.
Bakir, S. T. (2006). Distribution-free quality control charts based on signed-rank -like statistics. Communications in Statistics: Theory and Methods, 35, 743-757. Bakir, S. T. and Reynolds, M. R. Jr. (1979). A non-parametric procedure for process control based on within-group ranking. Techonometrics, 21, 175-183. Bennett, B. M. (1964). A bivariate signed rank test. Journal of the Royal Statistical Society, Series B, 26, 457-461. Bourke, P. D. (1991). Detecting a shift in fraction nonconforming using run-length control charts with 100% inspection. Journal of Quality Technology, 23, 225- 238. Chakraborti, S., Vander Der Laan, P. and Bakir, S. T. (2001). Nonparametric control charts: an overview and some results. Journal of Quality Technology, 3, 304-315. Chakraborti, S. and Eryilmaz, S. (2007). A nonparametric Shewhart-type signed -rank control chart based on runs. Communications in Statistics: Simulation and Computation, 36, 335-356. Dietz, E. J. (1982). Bivariate nonparametric tests for the one-sample location problem. Journal of the American Statistical Association, 77, 163-169. Ghute, V. B. and Shirke, D. T. (2008). A multivariate synthetic control chart for monitoring process mean vector. Communications in Statistics: Theory and Methods, 37, 2136-2148. Ghute, V. B. and Shirke, D. T. (2008). A multivariate synthetic control chart for process dispersion. Quality Technology and Quantitative Management, 5(3), 271-288. Hayter, A. J. and Tusi, K. L. (1994). Identification and quantification in multivariate quality control problems. Journal of Quality Technology, 26, 197-208. Kapatou, A. and Reynolds, M. R., Jr. (1994). Multivariate nonparametric control charts using small samples. ASA Proceedings: Section on Quality and Productivity, 241-246. Kapatou, A. and Reynolds, M. R. Jr. (1998). Multivariate nonparametric control charts for the case of known . ASA Proceedings: Section on Quality and Productivity, 77-82. Liu, R. Y. (1995). Control charts for multivariate processes, The Journal of the American Statistical Association, 90(432), 1380-1387. Qiu, P. and Hawkins, D. (2003). A nonparametric multivariate cumulative sum procedure for detecting shifts in all directions. The Statistician, 52(2), 151-164. Stoumbos, Z. G. and Sullivan, J. H. (2002). Robustness to non-normality of the multivariate EWMA control chart. Journal of Quality Technology, 34(3), 260-275. Wu, Z. and Spedding, T. A. (2000). A synthetic control chart for detecting small shifts in the process mean. Journal of Quality Technology, 32, 32-38.
6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
Authors’ Biographies: Vikas B. Ghute is Associate Professor and Head at the Department of Statistics, Solapur University, Solapur. He received his Ph.D. in Statistics from Shivaji University, Kolhapur in 2008. Digambar T. Shirke is Professor and Head at the Department of Statistics, Shivaji University, Kolhapur. He received his Ph.D. in Statistics from Shivaji University, Kolhapur in1993. His research interests are in the area of Statistical Inference, Statistical Quality Control and Statistical Computing.
