The Synthetic D Chart Under Non-Normality

The Synthetic D Chart Under Non-Normality

Rajmanya S V a and Ghute V B b*

aDepartment of Statistics, Sangameshwar College, Solapur, Maharashtra, India.

bDepartment of Statistics, Solapur University, Solapur-413255, Maharashtra, India.

*Correspondance to: V. B. Ghute, Department of Statistics, Solapur University, Solapur, India.

Abstract

The synthetic control chart based on Downtons estimator (Synthetic D chart) for monitoring the process variability developed by Rajmanya and Ghute (2013) is based on the assumption that the underlying process distribution is normal. In this paper, the performance of the synthetic D chart under non-normality is studied and is compared with the synthetic S chart. The comparison result shows that the synthetic D chart under non-normal distributions is more efficient than the synthetic S chart in detecting shifts in process variability.

Keywords- Downtons Estimator, Process variability, Non-normality, Average run length, Conforming run length, Control chart.

1. INTRODUCTION

   Control charts are statistical process control tools that are widely used for controlling and monitoring a process. Shewharts S chart is most widely used control chart to monitor process variability of a quality characteristic of interest. The S chart is based on the fundamental assumption that the underlying distribution of the quality characteristic is normal. It is easy to implement and is effective in the detection of large shifts in process standard deviation but become less effective for small shifts because it is based on only the most recent observation. Recently, Abbasi and Miller (2011) proposed the control chart based on Downtons estimate (D chart) of process standard deviation as an efficient alternative to S chart for monitoring process variability. It was shown that for normally distributed process, the D chart is equally efficient to the S chart for detecting shifts in the process variability.

   In the literature synthetic charting concept is used by many researchers for improving the performance of the control charts. Huang and Chen (2005) developed synthetic S chart for process dispersion by combining the sample standard deviation, S and conforming run length (CRL) charts. The CRL chart is an attribute control chart proposed by Bourke (1991) for monitoring fraction nonconforming. With the objective of improving the performance of D chart, Rajmanya and Ghute (2013) developed synthetic D chart as a combination of D chart and CRL chart. It was shown that synthetic D chart is close competitor of synthetic S chart for detecting shifts in process standard deviation. The synthetic D chart was designed and evaluated under the assumption that the underlying process distribution is normal.

   The behaviors of the commonly used control charts have been widely studied when some aspects of the assumption do not hold. One violation of the assumption is non-normality, which has been discussed by many authors. Borror et al. (1999) examined the performance of the EWMA chart for the mean in non-normal cases. Stoumbos and Reynolds (2000) studied the effect of non-normality and autocorrelation on the performance of various individual control charts for monitoring the process

   mean and/or variance. Calzada and Scariano (2001) have studied the robustness of the synthetic X chart when the normality assumption is violated and show that for sample size n = 6, the synthetic chart is robust to violations of the normality assumption. Kao and Ho (2007) examined the performance of R chart and found that R chart is robust to non-normality. Schoonhoven and Does

   (2010) studied the design schemes for the X control chart under non-normality.

   The objective of this paper is to investigate the effect of non-normality on the statistical performance of the synthetic D control chart for monitoring process variability of a continuous process. Non-normal distribution can be a symmetric distribution or a skewed distribution. In this study, we selected a t distribution as symmetric distribution and Weibull distribution as the skewed distribution to examine and compare the performance of a synthetic D chart under non-normality with synthetic S chart.

2. MATERIALS AND METHODS

   The D Control Chart

   Let X1, X2 ,…, Xn

   represents a random sample of size n from a normally distributed process with

   process mean and standard deviation and X(1) X(2) … X(n) denote the corresponding

   order statistics. Downton (1966) proposed the following estimator of process standard deviation

   (i)

   (i)

   2 n 1

   D i (n 1) X

   n (n 1) 2

   (1)

   i1

   which is unbiased estimator of for normally distributed quality characteristics. Let 0 and 0 be

   the in-control values of and respectively. When a shift in process standard deviation occurs, we

   have change from the in-control value 0

   to the out-of-control value 1 0

   ( 0 1) . When

   0

   0

   1, the process is considered to be in-control . For considered to be out-of-control and an upper control limit

   1 an increase in occurs, process is

   0

   0

   k of D chart is required, and a signal is

   issued if

   D k . For 1 decrease in occurs, process is considered to be out-of-control and

   0

   0

   0

   0

   lower limit k of D chart is required, and a signal is issued if D k . In this study we consider

   the case of increase in the process standard deviation.

   The average run length (ARL) which denotes the average number of D samples required to detect a change in of the D chart can be calculated as

   ARLD

   1

   0

   0

   Pr (D k |

   0 )

   1

   k

   Pr Z

   1

   (2)

   k

   1 F

   where, Z D

   and F(.) its cumulative distribution function.

   Synthetic D Chart

   0

   0

   The synthetic D control chart developed by Rajmanya and Ghute (2013) is a combination of the D chart and the CRL chart. The synthetic D chart consists of D sub-chart and CRL sub-chart. For detecting increase in standard deviation, the D sub-chart has upper control limit UCL k . The CRL sub-chart has a lower control limit L. These limits are called as design parameters of the synthetic D chart. According to the synthetic procedure, the signal is based on the CRL. The CRL is the number of conforming samples between two consecutive nonconforming samples including the end nonconforming sample.

   Let ARLS ( ) denote the average number of D samples required for a synthetic D chart to signal a shift of magnitude in process standard deviation . Then the ARL values of the synthetic D chart for a given shift of magnitude is given by the formula,

   ARLS ( )

   1

   k

   1

   k L1

   (3)

   1 F

   1 F

   When 1, in-control ARL of the synthetic D chart is

   ARLS

   (1)

   1

   1 F(k )

   1

   1 F (k )L1

   (4)

   To design synthetic |D| control chart means to obtain two design parameters L and k+ suitably that

   guarantee a minimum ARL for predetermined shift of magnitude

   * . The design shift *

   is the

   magnitude considered larger enough to seriously impair the quality of the products; thus

   corresponding

   ARLS *)

   should be as small as possible.

   ARLS (1)

   is decided by the requirement

   on the false alarm rate. The synthetic D control chart is properly designed by solving an optimization

   problem. The objective function

   ARLS ( *) minimum ,

   (5)

   subject to the equality constraint in Eq. (4).

   The optimal design procedure for the synthetic D chart uses the approach described in the following steps.

   1. Specify n, * and in-control ARL.

   2. Initialize L as 2.

   3. Obtain k by solving Eq.(4).

   4. Calculate ARL for * from current value of L and k using Eq.(3) (use *) .

   5. If L is not equal to 2, go to the next step; otherwise, L is increased by one and go back to step (3).

   6. If the current ARL for *

      is greater than the preceding one, go to the next step; otherwise, L is

      increased by one and go back to step (3).

   7. Take the preceding L and k as the optimal design parameters for the synthetic chart.

   To illustrate the design of the synthetic D chart, an example with n =10, * 1.4 and in-control

   S

   S

   ARL (1) 200 . Table 1 shows that each set of (L, k+) results in different

   ARLS ( *) . The ARL

   first declines and then increases. The ARLS ( *)

   reaches its minimum at 3.0926 when L = 7 and k =

   1.5180. So in this case the design parameters of the synthetic D chart are L = 7 and k = 1.5180.

   Table-1. Sets of (L, k ) and corresponding ARL for * 1.4 .

   (L, k )	ARL for *
   (2, 1.4098)	4.5127
   (3, 1.4530)	3.5726
   (4, 1.4780)	3.2783
   (5, 1.495)	3.1542
   (6, 1.5073)	3.1005
   (7, 1.5180)	3.0926
   (8, 1.5264)	3.0999
   (9, 1.5340)	3.1252
   (10, 1.5400)	3.1473

   The optimal values of L and k for sample size n = 5, 8 and 10 provided in-control ARL = 200 and 370 under * 1.2, 1.4 and 1.6 are presented in Table 2.

   Table-2. Optimal design parameters of the synthetic D chart.

   n	*	ARLS	(1) 200	ARLS	(1) 370
   		L	k	L	k
   5	1.2	17	1.843	24	1.927
   	1.4	10	1.786	11	1.856
   	1.6	7	1.750	8	1.825
   8	1.2	12	1.595	18	1.658
   	1.4	7	1.550	9	1.612
   	1.6	5	1.522	5	1.567
   10	1.2	12	1.5192	15	1.5638
   	1.4	6	1.469	7	1.518
   	1.6	4	1.438	5	1.495

   Effect of Non-normality on Synthetic D chart

   The synthetic D control chart developed for monitoring the process variability is based on the assumption that the underlying process distribution is normal. In this section, we examine the effects of non-normality on the statistical performance of the synthetic D control chart. In order to study the effect of non-normality, we considered heavy-tailed symmetric and skewed distributions. Specifically, we simulated observations in the heavy tailed symmetric case from t distribution and in the skewed case from the Weibull distribution.

   The probability density function of t distribution with degrees of freedom is

   1

   x

   x

   1/ 2

   f (x)

   2 2

   1

   , x ; 0.

   (6)

   / 2

   The probability density function of Weibull distribution with shape parameter and scale parameter is

   x

   1

   x

   f (x)

   e

   , x 0 ; 0, 0

   (7)

3. RESULTS AND DISCUSSIONS Performance comparisons for Non-normal process

The synthetic D chart is designed by using the values of design parameters (L, k) with

* 1.2 from Table 2, which are, technically, only appropriate for normally distributed process data.

The design parameter k of the charts is then adjusted so that both synthetic charts have approximately the same in-control ARL value 200. The ARL values of the synthetic S and synthetic D charts are computed using 10000 simulations when underlying process distribution is normal, heavy tailed and skewed. To study the effect of non-normality on the performance of the synthetic D control chart, we have considered the process data from t distribution with 5 and Weibull distribution with shape

parameter = 2 and scale parameter = 1.

Table 3 presents the ARL profiles of the synthetic S and synthetic D charts for increase in the process standard deviation when underlying process data follows standard normal distribution with in- control ARL = 200, * 1.2 and sample sizes n = 5, 8 and 10.

Table-3. ARL comparison of synthetic charts under normal distribution.

From Table 3, when underlying process distribution is normal, we observe that for any range of shifts, the synthetic S control chart produces slightly smaller out-of-control ARL than that of the synthetic D chart. The synthetic S chart performs slightly better than the synthetic D chart.

Table 4 and Table 5 present the ARL profiles of the synthetic S and synthetic D control charts for increase in the process standard deviation when underlying process data follows t distribution and Weibull distribution respectively each with in-ontrol ARL = 200, * 1.2 and sample sizes n = 5, 8

and 10.

Table-4. ARL comparison of synthetic charts under t distribution.

Table-5. ARL comparison of synthetic charts under Weibull distribution.

From Table 4 and Table 5, when underlying process distribution is heavy tailed and skewed we observe that, the synthetic D chart consistently produces smaller out-of-control ARL values than that of the synthetic S chart for entire range of shifts in the process standard deviation.

Conclusions

In this paper, a synthetic D chart based on Downtons estimator is studied under non-normality of process data. By comparing synthetic S and synthetic D charts when process distribution is heavy tailed or skewed, we found that the synthetic D chart detects the shifts in process standard deviation

quicker than the synthetic S chart. Hence for non-normal processes the synthetic D chart is uniformly better than the synthetic S chart.

References

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