A Class of Gauged Inter Quantile Deviation Control Charts

A Class of Gauged Inter Quantile Deviation Control Charts

Sharada V. Bhat, Shradha Patil

Department of Statistics, Karnatak University, Dharwad-580003, India

Abstract – Control charts are widely used in statistical quality control to monitor and detect shifts in the process variation. A class of control charts based on gauged inter quantile deviation (GIQD) is proposed for monitoring process variability. This class allows for the flexibility in selecting control charts based on the deviation of a pair of quantiles such as percentiles, deciles or quartiles. The proposed class of GIQD control charts are evaluated under various distributions. Its performance is analyzed and optimality of the class is discussed. An illustrative example is provided to demonstrate its practical applicability.

Keywords – average run length, control limits, inter quantile deviation, process variation, robust, scale parameter

1. INTRODUCTION

   Control charts play a vital role in statistical process control (SPC). They detect the unfavorable changes in an ongoing process with regard to quality characteristics. The process parameters such as process location, process variance, process standard deviation, etc represent various entities of the production process. As consistency of the process is ensured by process variability, the control chart for process variation gain prominence in SPC. The details of Shewhart’s range (R), sample standard deviation and sample variance () control charts for assessing process variability are discussed in Montgomery (2019). These control charts rely on the assumption that, the process variables follow normal distribution. However, in reality this assumption need not hold. For instance, in manufacturing industries, tool wear often exhibits non-normal distributions. As tools deteriorate over time, the measurements of dimensional accuracy such as the diameter or surface finish of a machined part may follow a skewed distribution or a symmetric distribution other than normal distribution. Similarly, quality characteristic such as the tensile strength of paper do not follow normal distribution. According to Korteoja et al. (1998), this quality characteristic is adequately captured by Laplace distribution. Hence, the control charts for process variation are inevitable under non normality.

    

   Numerous control charts under the assumption of normality have been extensively developed in the literature to monitor process variation. Control charts based on one sided cumulative sum (CUSUM) and the Shewhart control chart with

    

   warning limits based on was discussed by Page (1963). Run length distribution of , and control charts were derived by Chen (1998). Shewhart’s and control charts based on runs rules was discussed by Lowry et al. (1995). CUSUM control chart based on to detect shift in process standard

   deviation () was studied by Acosta-Mejia (1998). Modified

   Shewhart control chart to improve the detection of small shifts in process was suggested by Klein (2000). The probability limits for control chart was provided by Ryan

    

   (2011). The modified control chart for monitoring the process variance was proposed by Khoo and Lim (2005). Control charts based on prior information for process variation due to Menzefricke (2007, 2010), Bhat and Gokhale (2014, 2016, 2017), Saghir et al. (2020) and Bhat and Malagavi (2021) are given under normal model.

   Unlike parametric control charts, nonparametric control charts are distribution free and do not rely on specific assumptions about the underlying distribution. Some of the nonparametric control charts for process are due to Amin et al. (1995) based on the sign statistic, Abu and Abdullah (2000) on Downton estimator due to Downton (1966), Riaz and Saghir (2007) on Gini’s mean difference and Zombade and

   Ghute (2014) on Sukhatme and Mood test statistics. Also, the control charts due to Das (2008) based on Ansari-Bradley two sample rank sum test, Abu (2008) on median absolute deviation (MAD), Riaz and Saghir (2009) on mean absolute

   deviation from the median exist for process . Abbasi and Miller (2012) compared control charts based on eight different estimators. Rajmanya and Ghute (2014) developed a robust control chart by proposing a synthetic D control chart.

   In this paper, we propose a class of control charts to monitor process variability based on gauged inter quantile deviation () scaled by a positive constant. As quantile based control charts are robust, they are useful in situations when process variables deviate from normality. These control charts

   have advantages when data contains outliers and process has heavy tailed or skewed distributions. This motivates the current study to introduce a class of control charts utilizing the difference of two equidistant quantiles from the centre of the data gauged by a constant to provide a robust and effective control chart for monitoring process scale.

    

   The paper is organized in 6 sections. In section 2 and section 3, we introduce respectively, the concept of gauged ( ), its properties and class of control charts based on

   . Section 4 deals with performance evaluation of control charts and its optimality. Section 5 provides an illustrative example and section 6 contains conclusions based on our findings.

2. GIQD AND ITS PROPERTIES

    

   Suppose is a random sample of size from a continuous distribution with location parameter and scale parameter . When , are arranged in ascending order, the is known as order statistic.

    

   Let and denote respectively the population and sample quantiles of , then according to Gibbons and Chakraborti (2020), is given by

    

   (1)

   and is given by ,

   where (2)

   is the largest integer and .

   According to Cramer (1946), suppose and are respectively the and sample quantiles, then

   (3)

   where represents normal distribution and ,

   is the probability density function ( ) of the distribution evaluated at .

    

   The GIQD based on the deviation of two equidistant quantiles with as gauged constant is given by

   , , (4)

   is an estimator of the scale parameter. The selection of various values of and results in the flexibility of in its adjustability as it contains various members of this class of

   estimators.

    

   For , we have IQD given by

   gauged inter percentile deviation . When and , we respectively have and semi

    

    

    

   . Also, and are respectively known as inter percentile range ( ) and semi inter percentile range ().

   As IQD estimator is a robust measure of scale parameter,

    

    

   ,

    

   the class of GIQD estimators contain the members that are robust to outliers on either side of the partition values where is the index of quantile deviation. That is, if , then is resistant to outliers. If

    

    

   , then is robust to

   outliers where as is robust to outliers and so on. Similarly, if , then

   and are respectively robust to and

   outliers. Likewise, various members of are resistant to various numbers of outliers. Hence, the class of estimators provide a wide range of adaptable robust measures

   which are useful in distinct situations.

    

   From (2) and (3), asymptotically follows normal distribution with mean and variance .

    

   That is,

   (7)

   where

    

   .

    

   (8)

    

   (5)

   and for , we have semi IQD () given by

   . (6)

    

   Similarly, for various values of , we get various estimators.

    

   For a specified and , simplifies to the gauged inter quartile range ( ). Specifically, when

   , it reduces to inter quartile range ( ) and when

   it becomes semi inter quartile range ( ).

3. CLASS OF CONTROL CHARTS BASED ON GIQD

    

   ,

    

   and

    

   (9)

    

   In this section, we propose a class of Shewhart type control charts based on GIQD to monitor process scale parameter. The proposed class of control charts is given in terms of its control limits, viz.

   where , and respectively being

    

    

    

    

   the upper control limit, center line and lower control limit of control chart. The and are

   respectively mean and of control chart. The width of the control chart is given by

    

    

    

    

   Similarly, for fixed and

    

   with

    

   we get gauged inter decile deviation (

   , we have representing

    

   ,

   ). When

   . Similarly,

    

   for we get ,

    

   respectively given by

   , we have

   . Also, the

   ( ) and

    

   ,

   and when

    

   and

   and

    

   . When

    

   , we get semi

    

   is known as inter decile range

   is known as semi inter decile range

    

   ( ).

   Further, on similar grounds, for specific and with , the corresponding estimator is

    

   . (10)

   The control charts are developed for light tailed distribution viz. Uniform (U) distribution, skewed

   distribution viz. Exponential (E) distribution, medium tailed distributions such as Normal (N), Logistic (LG) distributions and heavy tailed distributions like Laplace

   (L), Cauchy (C) distributions. These distributions are modified such that, the scale parameter of the distributions is their .

    

    

   Here, the control limits of the control chart is derived under normal distribution, viz. with

    

    

   Taking square root of (18),

   The is

   . (11)

   (12)

   (19)

   where is the error function defined as

   .

    

   Taking and substituting for in (11), we get

   Hence, . (13)

   Similarly, . (14)

    

   Also, by (11), (13) and (14)

   Similarly, , are obtained for other distributions and furnished in Exhibit 1.

   D
   U
   E
   LG
   L
   C

    

   Exhibit 1: and under various distributions

   (15)

   and

    

    

   (16)

   Hence from (7), the

    

   (17)

    

   (18)

    

   and is obtained by substituting (15) and (16) in (8). Hence,

   * ,

   Since it is not feasible to obtain control charts based on all members of the proposed class, considering , for various values of , we obtain the control limits based on some members of this class.

   The and under normal distribution is obtained as

    

    

    

   (20)

   (21)

    

   .

   On similar grounds, the mean and

   ,

    

   (22)

   of the

   and

    

   , … , estimators are derived under

    

   and

    

    

   .

    

   The control limits of control chart which is control chart under normal distribution is given by

   various distributions. These values which are useful in obtaining control limits for some members of the class along with their widths are provided in Exhibit 2.

    

   Exhibit 2: Mean, , of for various distributions and values of

   in terms of		U	E	N	LG	L	C
   			2.2976	2.3263	2.5334	2.7662	31.8205
   				1.6449	1.6234	1.6282	6.3138
   			1.0986	1.2816	1.2114	1.138	3.0777
   			0.8673	1.0364	0.9563	0.8513	1.9626
   			0.6931	0.8416	0.7643	0.6479	1.3764
   				0.6745	0.6057	0.4901	1
   in terms of				2.6264	3.8983	4.9497	222.8902
   				1.4544	1.7410	2.1213	19.2565
   			1.4907	1.1396	1.2252	1.4142	6.5798
   			1.1716	0.9827	0.9908	1.0801	3.4925
   				0.8749	0.8440	0.866	2.2273
   			0.8165	0.7867	0.7351	0.7071	1.5708
   in terms of		1.4549	29.8481	15.7586	23.3897	29.6985	1337.3414
   		3.1177	13.0586	8.7264	10.4462	12.7279	115.5387
   		4.1569	8.9443	6.8377	7.3511	8.4853	39.4790
   		4.7624	7.0294	5.8963	5.9447	6.4807	20.9550
   		5.0912	5.8095	5.2496	5.0643	5.1962	13.3641
   		5.1962	4.8990	4.7203	4.4106	4.2426	9.4248

4. PERFORMANCE OF SIQD CONTROL CHARTS

    

    

   In this section, we evaluate the performance of proposed control charts in terms of power , average run length

   , median run length and of run length .

   The Power measures ability of a control chart to detect shift in the process scale parameter. The ARL being average number of samples required before a shift is signalled by the control chart, a higher ARL is desired when there is no shift. Conversely, when a shift occurs, a smaller ARL is preferred. The MRL is the median number of samples and behaves like ARL. The SDRL indicating the consistency captures spread in the run length distribution and smaller SDRL is beneficial when process is out of control.

   These performance measures of control chart are given by

   , (23)

   , (24)

   , (25)

   (26)

   and .(27)

   The values of , , and for the distribution under consideration are

   determined by setting . The values of and for various values of and are given in Table 1.

   Also, and are respectively given in Figure 1 and Figure 2 for .

    

   Figure 1: under various distributions

    

    

    

    

    

    

    

    

   Figure 2: under various distributions

    

    

    

   From Table 1, Figures 1 and 2, we observe that, for different distributions, when there is no shift, is 0.0027 while

    

   , and respectively are approximately 370, 256 and 369. For fixed , as increases, increases while , and

   decrease. Also, the exhibits least value for uniform

    

   distribution followed by normal, logistic, exponential, Laplace and Cauchy distributions.

   From Figures 1 and 2, we observe that, for and different values of , the and of control chart is accordingly increasing from uniform distribution to

   normal, logistic, exponential, Laplace and Cauchy

   distributions.

   For uniform distribution,

    

   for exponential distribution,

    

   and for normal distribution,

    

   . Similarly for Logistic and Laplace distributions,

    

    

   Also, the

    

   follows the same pattern as

    

   .

    

   and for Cauchy distribution

    

    

    

   .

   and axis gives

    

   As it is observed that, the optimality of the control charts under each distribution corresponds to a specific value of , considering , , the ARL of control chart is plotted in Figure 3 for numerous values of , viz.

   Here, the axis represents various

    

   . A line is dropped on axis from the line representing minimum of .

    

    

   Figure 3: Optimality of control charts under various distributions

    

   From Figure 3, we observe that, the is minimum for specific quantiles of control charts. Under uniform distribution, the is minimum at ,

    

   indicating control chart is optimal. Under, exponential distribution, the decreases upto and then increases, highlighting control chart as optimal. On similar lines, the optimal control chart is control chart under normal distribution, control chart under logistic,

   Laplace distributions and control chart under Cauchy distribution.

   Considering the computations due to Abu (2008), we compare

   Shewhart’s control chart and MAD control chart as competitors to the proposed class of control charts. It is observed that, for , and , all the members of class of control charts perform better than control chart under normal, logistic and Laplace distributions and better than MAD control chart under normal and Laplace distributions.

   and

    

   , where

   , and is the of

   sample. Here and .

   Using Exhibit 3(a), 3(b), ,

    

   control charts are plotted in Figure 4 and using Exhibit 3(a), 3(c) the optimal control charts under various distributions along with control

   chart are plotted in Figure 5. The represents optimal control chart as under uniform distribution along with similar interpretation to other optimal

   control charts shown in this figure.

    

   	Values of
   1	2.0628	2.0142	1.9535	1.8192	1.6640	1.4575	1.6751
   2	1.8653	1.5862	1.2375	1.0445	0.8960	0.7688	1.2569
   3	2.6958	2.3790	1.9830	1.6150	1.2550	0.8700	1.7713
   4	3.3870	2.7552	1.9655	1.4977	1.1220	0.8213	2.1541
   5	2.6811	2.4057	2.0615	1.7802	1.5170	1.2350	1.8407
   6	2.7994	2.1567	1.3535	1.0647	0.9230	0.7750	1.6465
   7	2.0017	1.8882	1.7465	1.4455	1.0990	0.8075	1.3902
   8	1.1060	1.0502	0.9805	0.7620	0.5010	0.3750	0.7734
   9	3.1138	2.8690	2.5630	1.9630	1.2790	0.9150	2.1074
   10	2.2968	1.9440	1.5030	1.3280	1.2290	1.1162	1.6059
   	2.4010	2.1049	1.7348	1.4320	1.1485	0.9141	1.6222

    

   Exhibit 3(a): Computed values of , and their averages to data given by Yang and Arnold (2016)

5. ILLUSTRATION

   In this section, we illustrate the application of the class of control charts using an example due to Yang and Arnold (2016). The data represents service times (in minutes)

    

    

    

    

   recorded at a bank branch in Taiwan. The dataset comprises of samples collected at times. In Exhibit 3(a) the computations of , for and their averages are carried out and in Exhibit 3(b), , , and are computed. Also, in Exhibit 3(c), the

   , and of various optimal control charts are obtained.

    

    

    

    

   Since is to be estimated from the sample observations for different values of , we calculate by , where

   is the mean of and is a varying constant for various values of given in Exhibit 2 under various distributions. The control limits and width of control chart is

   given by , ,

   The control limits and width of control chart is given by

    

    

    

   , and .

    

    

   Exhibit 3(b): and control limits of control charts for various distributions

   	D						D
   	U	0.1085	2.7264	2.0756	0.6508		U	0.2965	2.3214	0.5426	1.7787
   	E	1.6439	7.3327	0.0000	7.3327		E	0.6117	3.2672	0.0000	3.2672
   	N	0.8572	4.9726	0.0000	4.9726		N	0.4294	2.7201	0.1439	2.5762
   	LG	1.1683	5.9059	0.0000	5.9059		LG	0.4692	2.8395	0.0245	2.8150
   	L	1.3586	6.4767	0.0000	6.4767		L	0.5745	3.1556	0.0000	3.1556
   	C	5.3183	18.3558	0.0000	18.3558		C	0.8058	3.8495	0.0000	3.8495

   http://www.ijert.org Published by :

   International Journal of Engineering Research & Technology (IJERT)

   ISSN: 2278-0181

   Vol. 14 Issue 02, February-2025

   	U	0.2219	2.7705	1.4393	1.3312	0	U	0.2965	2.0381	0.2589	1.7792
   	E	0.9840	5.0569	0.0000	5.0569		E	0.5073	2.6705	0.0000	2.6705
   	N	0.5885	3.8705	0.3393	3.5312		N	0.3776	2.2812	0.0158	2.2653
   	LG	0.7138	4.2464	0.0000	4.2464		LG	0.4015	2.3531	0.0000	2.3531
   	L	0.8672	4.7065	0.0000	4.7065		L	0.4854	2.6048	0.0000	2.6048
   	C	2.0301	8.1951	0.0000	8.1951		C	0.5877	2.9116	0.0000	2.9116
   	U	0.2743	2.5576	0.9119	1.6457		U	0.2891	1.7813	0.0469	1.7344
   	E	0.7444	3.9679	0.0000	3.9679		E	0.4297	2.2032	0.0000	2.2032
   	N	0.4878	3.1981	0.2714	2.9268		N	0.3372	1.9256	0.0000	1.9256
   	LG	0.5548	3.3992	0.0703	3.3290		LG	0.3508	1.9666	0.0000	1.9666
   	L	0.6817	3.7799	0.0000	3.7799		L	0.4171	2.1653	0.0000	2.1653
   	C	1.1728	5.2531	0.0000	5.2531		C	0.4541	2.2763	0.0000	2.2763

   Exhibit 3(c): Control limits and width of optimal control charts

   	0.01	0.13	0.07	0.10	0.10	0.25
   	U	E	N	LG	L	C
   	2.7264	3.5182	3.5822	3.3992	3.7799	2.2763
   	2.0756	0.0000	0.3315	0.0703	0.0000	0.0000
   	0.6508	3.5182	3.2507	3.3290	3.7799	2.2763

    

    

   Figure 4: control charts for different values of p

    

    

   Figure 5: Optimal control charts and S control chart under various distributions

    

   From Exhibit 3 (a), 3(b) and Figure 4, we notice that, is minimum under uniform distribution and maximum under Cauchy distribution. The , and control

    

   charts indicate the process is in control. However, control chart shows out of control status for uniform distribution.

   From Exhibit 3(c) and Figure 5, we observe that, is maximum under Laplace distribution and is minimum under uniform distribution. The optimal control

   charts and control chart shows slight similarity in their

   patterns under exponential, normal, logistic and Laplace distributions, whereas they show slightly different pattern for uniform and Cauchy distributions. Further, using R software, it is seen that, the given data follows exponential distribution

   and both and control charts show that the process is in control.

6. CONCLUSIONS

In this section, based on our findings we record our conclusions on the proposed class of control charts.

• A class of control charts based on gauged inter quantile deviation (GIQD) for monitoring process scale parameter is proposed. It includes , and

  control charts as its subclasses.

• The class of control charts contains numerous members that are resistant to varying number of outliers.
• The GIQD control charts are developed under both symmetric and skewed distributions.
• For and , the class reduces respectively to the IQD and SIQD control charts.
• For a specified shift, as sample size increases, the power

   

  of control chart increases whereas, ARL, MRL and SDRL of control chart decrease.

• Among control charts, that is among control charts, the optimal control charts are identified as

  control chart under uniform distribution, control

  chart under exponential distribution, control chart under normal distribution, control chart under logistic and Laplace distributions and control

  chart under Cauchy distribution.

• The control charts perform better than Shewhart’s control chart under normal, logistic, Laplace distributions and outperforms MAD control charts due to Abu (2008) for normal, Laplace distributions.
• As the proposed class of control charts are robust to outliers, they are useful when process variables are taken from non-normal models, that is when underlying distributions are heavy tailed or skewed distributions.

Appendix

Table 1: and for various values of under different distributions

REFERENCES

1. Abbasi, S. A. and Miller, A. (2012). On proper choice of variability control chart for normal and non‐normal processes. Quality and Reliability Engineering International, 28(3), 279-296.

2. Abu-Shawiesh, M. O. (2008). A simple robust control chart based on MAD. Journal of Mathematics and Statistics, 4(2), 102.

3. Abu-Shawiesh, M. O. and Abdullah, M. B. (2000). Estimating the process standard deviation based on Downton’s estimator. Quality Engineering, 12(3), 357-363.

4. Acosta-Mejia, Cesar A. Monitoring reduction in variability with the range. IIE Transactions 30 (1998): 515-523.

5. Amin, R. W., Reynolds Jr, M. R. and Saad, B. (1995). Nonparametric quality control charts based on the sign statistic. Communications in Statistics-Theory and Methods, 24(6), 1597-1623.

6. Bhat, S. V. and Gokhale, K. D. (2014). Posterior Control Charts for Process Variance Based on Various Priors. Journal of the Indian Society for probability and statistics. 2014b, 15, 52-66.

7. Bhat, S. V. and Gokhale, K. D. (2016). Posterior Control Chart for standard deviation based on Conjugate Prior. Journal of the Indian Statistical Association, 54(1).

8. Bhat, S. V. and Gokhale, K. D. (2017). Bayesian S and S-2 Control Charts for Process Variation. International Journal of Agricultural and Statistical Sciences, 13(1), 33-42.

9. Bhat, S. V. and Malagavi, C. (2021). Range-Based Posterior Average Control Chart. Stochastic Modeling and Applications. 2021a, 25(1), 189-203.

10. Chen, G. (1998). The run length distributions of the R, s and s2 control charts when is estimated. Canadian Journal of Statistics, 26(2), 311-322.

11. Cramér, H. (1946). Mathematical methods of statistics.

12. Das, N. (2008). A note on the efficiency of nonparametric control chart for monitoring process variability. Economic Quality Control, 23 (1), 85 – 93.

13. Downton, F. (1966). Linear estimates with polynomial coefficients. Biometrika, 53(1/2), 129-141.

14. Gibbons, J.D. and Chakraborti, S. (2020). Nonparametric Statistical Inference (6th ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9781315110479

15. Khoo, M. B. and Lim, E. G. (2005). An improved R (range) control chart for monitoring the process variance. Quality and Reliability Engineering International, 21(1), 43-50.

16. Klein, M. (2000). Modified S-charts for controlling process variability. Communications in Statistics-Simulation and Computation, 29(3), 919-940.

17. Korteoja, M., Salminen, L. I., Niskanen, K. J. and Alava, M. J. (1998). Strength distribution in paper. Materials science and engineering: A, 248(1-2), 173-180.

18. Lowry, C. A., Champ, C. W. and Woodall, W. H. (1995). The performance of control charts for monitoring process variation. Communications in Statistics-Simulation and Computation, 24(2), 409-437.

19. Menzefricke, U. (2007). Control charts for the generalized variance based on its predictive distribution. Communications in Statistics— Theory and Methods, 36(5), 1031-1038.

20. Menzefricke, U. (2010). Control charts for the variance and coefficient of variation based on their predictive distribution. Communications in Statistics—Theory and Methods, 39(16), 2930-2941.

21. Montgomery, D. C. (2019). Introduction to statistical quality control. John wiley & sons.

22. Page, E. S. (1963). Controlling the standard deviation by CUSUMS and warning lines. Technometrics, 5(3), 307-315.

23. Rajmanya, S. V. and Ghute, V. B. (2014). A synthetic control chart for monitoring process variability. Quality and Reliability Engineering International, 30(8), 1301-1309.

24. Riaz, M. and Saghir, A. (2009). A mean deviation-based approach to monitor process variability. Journal of Statistical Computation and Simulation, 79(10), 1173-1193.

25. Riaz, M. and Saghir, A. (2007). Monitoring process variability using Gini’s mean difference. Quality Technology & Quantitative Management, 4(4), 439-454.

26. Ryan, T. P. (2011). Statistical methods for quality improvement. John Wiley & Sons.

27. Saghir, A., Aslam, M., Faraz, A., Ahmad, L. and Heuchenne, C. (2020). Monitoring process variation using modified EWMA. Quality and Reliability Engineering International, 36(1), 328-339.

28. Yang, S. F. and Arnold, B. C. (2016). Monitoring process variance using an ARL‐unbiased EWMA‐p control chart. Quality and Reliability Engineering International, 32(3), 1227-1235.

29. Zombade, D. M. and Ghute, V. B. (2014). Nonparametric Control Charts for Monitoring Process Variability. International Journal of Engineering Research, 3(5).