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Scientia Iranica E (2017) 24(4), 2152{2163

Sharif University of Technology Scientia Iranica

Transactions E: Industrial Engineering www.scientiairanica.com

Research Note

Performance evaluation of joint monitoring control charts T. Mahmooda; , H.Z. Nazirb , N. Abbasa , M. Riaza and A. Alib;c a. Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, KSA. b. Department of Statistics, University of Sargodha, Sargodha, Pakistan. c. SAA Center for Improvement of Working Conditions and Environment Labour and Human Resource Department, Government of Punjab, Lahore, Pakistan. Received 8 May 2015; received in revised form 25 February 2016; accepted 25 July 2016

KEYWORDS

Average run length; Contaminations; Cucconi; Lepage; Robustness; Shewhart charts.

Abstract. Shewhart-Cucconi and Shewhart-Lepage are two nonparametric control charts used for monitoring joint shifts in the process location and scale parameters. This study investigates impact of the light and heavy-tailed distributions on the performances of these charts. The eect of reference and test samples is also a part of this study. ©

2017 Sharif University of Technology. All rights reserved.

1. Introduction Statistical Process Control (SPC) is a set of wellknown tools used to monitor the performance of a process. Control chart is a major tool of SPC that consists of a Lower Control Limit (LCL), Central Line (CL), and an Upper Control Limit (UCL). It helps us dierentiate between natural and unnatural variations that refer to In-Control (IC) and Out-OfControl (OOC) states, respectively, in a process. Normality is a typical assumption needed for parametric charts, while non-parametric charts are free from any such constraints. Reference can be made for further check out to Chakraborti et al. [1,2], Chowdhury et al. [3], and Mukherjee and Sen [4] in the literature. Moreover, a traditional approach used in SPC is to monitor each parameter separately; however, simultaneous monitoring of more than one parameter is *. Corresponding author. E-mail addresses: [email protected] (T. Mahmood); ha [email protected] (H.Z. Nazir); [email protected] (N. Abbas); [email protected] (M. Riaz); [email protected] (A. Ali)

also becoming popular in industry. Chowdhury et al. [5], McCracken and Chakraborti [6], Mukherjee and Chakraborti [7], and Mukherjee et al. [8] and the references therein may be seen in literature on simultaneous charts. Recently, Mukherjee and Chakraborti [7] have proposed a Shewhart-type distribution-free chart for joint monitoring of the process parameters. It is based on the Lepage test, a combination of Wilcoxon rank sum test for location and Ansari Bradley test for scale (cf. Lepage [9]), and this chart is hereafter named as Shewhart-Lepage (SL) chart. On the same lines, Chowdhury et al. [5] developed a distribution-free Shewhart chart for joint monitoring that utilizes Cucconi test proposed by Cucconi [10], hereafter referred to as Shewhart Cucconi (SC) chart. Marozzi [11] provided a comparative analysis of Cucconi test versus Lepage test under some distributional setups and favored Cucconi test over Lepage test. This study intends to investigate the impact of light and heavy-tailed distributions on the performance of SL and SC charts. In addition, the eect of reference/test samples is included in this study. The rest of the article is organized as follows: Section 2

T. Mahmood et al./Scientia Iranica, Transactions E: Industrial Engineering 24 (2017) 2152{2163

provides the description of SL and SC charts. Section 3 explores the properties of these charts under dierent distributional environments and also examines the eects of reference/tests samples. Section 4 deals with a real application related to these charts. Section 5 concludes the study with the main ndings.

Var(T1 jIC) =

2. Description of SC and SL charts

Var(T2 jIC) =

Let U1 ; U2 ; ; Um and V1 ; V2 ; ; Vn be independent random samples from their respective populations with continuous cumulative distribution functions: F (U ) = Q( U ) and G(V ) = Q( V ); 2 R; > 0; this is where Q refers to some unknown continuous functions. Constants and represent unknown location and scale parameters, respectively. Let us introduce indicator variable, Ik = 0, or 1 depending on whether or not kth order statistic of the combined sample of N = m + n observations belongs to U or V . It is to be mentioned that m is reference sample (phase I) and n is the test sample (phase II). Further, we assume that R is the linear ranks assigned to the values of the combined sample. The popular nonparametric Wilcoxon Rank Sum (WRS) test statistic, T1 , is de ned as follows: T1 =

N X k=1

For the equality of two scale parameters, Ansari Bradley (AB) is an ecient nonparametric test whose statistic, T2 , is de ned as follows: T2 =

k=1

1 (N + 1) Ik : 2

Consider S1 as the sum of the square of the ranks of Vi 's in the combined sample, i.e.: S1 =

N X k=1

R2 Ik :

Further, note that quantities (N + 1 R)Ik , for k = 1; 2; ; N , may be considered as the contrary ranks of Vi 's. The sum of squares of the contrary ranks of Vi 's in the combined sample, say S2 , is given by: S2 =

N X

(N + 1 R)2 Ik = n(N + 1)2

k=1

2(N + 1)T1 + S1 : Assuming that = 0 and = 1 refer to IC state (F = G), we have the following properties: 1 E (T1 jIC) = n(N + 1); 2

1 mn(N + 1); 12

8 nN > < 4

when N is even

> : n(N 2 1)

when N is odd

4N

8 (N 2 4) 1 > < 48 mn N 1 > : 1 mn(N +1)(N 2 +3)

N2

48

E (S1 jIC) = E (S2 jIC) =

when N is even when N is odd

n(N + 1)(2N + 1) ; 6

Var(S1 jIC) = Var(S2 jIC) mn (N + 1)(2N + 1)(8N + 11): 180 The combination of AB and WRS is known as Lepage statistic (cf. [9]) and is given as follows:

=

L=

(T1 E (T1 jIC))2 (T2 E (T2 jIC))2 + ; VAR(T1 jIC) VAR(T2 jIC)

(1)

and Cucconi [10] statistic for testing both location and scale is de ned by: C=

RIk :

N X R

E (T2 jIC) =

2153

W 2 + Z 2 2W Z ; 2(1 2 )

(2)

where W and Z are the standardized statistics given as follows: S E (S1 jIC) W = p1 VAR(S1 jIC) =

6S1 n(N + 1)(2N + 1) ; 5 (N + 1)(2N + 1)(8N + 11)

p mn

S E (S2 jIC) Z = p2 VAR(S2 jIC)

=

6S2 n(N + 1)(2N + 1) ; 5 (N + 1)(2N + 1)(8N + 11)

p mn

when > 0 and = 1, E (W ) > 0 and E (Z ) < 0; when = 0 and > 1, E (W ) > 0 and E (Z ) > 0; in general, when 6= 0 and 6= 1, E (W ) 6= 0 and E (Z ) 6= 0. Similar inequalities may be observed in other possible cases, when either diers from 0 or diers from 1 in any direction. Also, note that E (W jIC) = E (Z jIC) = 0 and V (W jIC) = V (Z jIC ) = 1. Moreover, when F = G, the correlation coecient between W and Z is given as (cf. [11]): = Corr(W; Z jIC) =

2(N 2 4) (2N + 1)(8N + 11)

1:

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2.1. Design of control charting constants of distribution-free charts

Construction and design of both SC and SL charts depend on the distributions of the statistics given in Eqs. (1) and (2). The lower control limits of both charts is zero as both statistics can never be negative (cf. [5,7] and the upper control limits of both charts, say H , are used to make decision. The values of H are provided in [5,7] for some selective values of n and m. We have covered more combinations of n and m to nd the upper control limit, say H , for both charts, using a simulation study with 100,000 replicates (in R 3.1.1). We have taken the retrospective samples, i.e. m = 30; 50; 100; 150; 500, and 1000 and prospective samples, i.e. n = 5; 8; 11; 16, and 25, for this study, xing ARL0 = 500. The results are reported in Table 1 for SL and SC charts. The decision procedure for the two charts is given as follows: - SL chart: Statistic L is used for plotting in SL chart. If L is greater than H , then the process is declared OOC. For the follow-up analysis, we compute p-values of the WRS test for location and AB test for scale with phase I sample and ith test sample, and they are denoted as p1 and p2 , respectively. If p1 is very low except p2 , a shift of location is detected, or if p2 is very low and p1 is relatively high, a shift in scale parameter is detected. When both WRS and AB p-values are very low, a joint shift in the location and scale is considered; - SC chart: Statistic C is used for plotting in SL chart. If C exceeds H , the process is declared OOC. For the follow-up analysis, we compute the p-values for Wilcoxon test (p3 ) and Mood test (p4 ) based on two samples (reference and test samples), (cf. [5]). The shift in location is noted when p3 is very low except p4 ; if p3 is relatively high except p4 , then there is the indication of a shift in scale. If both p3 and p4 are very low, shift is noted in both location

n=5 n=8 n = 11 n = 16 n = 25 n=5 n=8 n = 11 n = 16 n = 25

and scale. Sometimes, neither p3 nor p4 is very low, though the plotting statistic C is high; in this situation, the eect is due to the relation between location and scale changes or due to false alarm. So, to overcome this problem, combine ith and (i 1)th prospective samples and recalculate (p3 ) and (p4 ) for further decision.

3. Performance analyses of SL and SC charts In this section, we will investigate the performances of SL and SC charts under dierent distributional environments. We will also examine the eects of reference and test samples on the performance of these charts. We will use Average Run Length (ARL) and Standard Deviation Run Length (SDRL) as performance measures. The ARL value is denoted by ARL0 for incontrol situation and ARL1 for out-of-control situation. The distributional p covered in this study include: p setups Uniform: U ( 3; 3), Student's t: t4 , Lognormal: LN(1; 1), Gamma: G(1; 1), and contaminated normal (C 1: with 10% contaminations, and C 2: with 30% contaminations). The rst two are symmetric and light tailed, the next two are skewed and heavy tailed, and the last two are contaminated distributions. Abbasi and Miller [12], Alfaro and Ortega [13], Ali et al. [14], Human et al. [15], and Marozzi [11] are some useful references on the said distributional environments. The graphical displays of these distributions are given in Figure 1.

3.1. OOC performance

In order to examine the OOC performance of SL and SC charts, we have considered shifts in location and scale for these choices: = 0, 0.25, 0.50, 0.75, 1.00, 1.50, 2.00, and = 0:50, 0.75, 1, 1.25, 1.5, 1.75, 2. We have chosen m = 30, 50, 100, 150, 500, and 1000 and n = 5, 8, 11, 16, and 25. It makes a total of 30 pairs (m; n). The properties of SL and SC charts, in terms of ARL and SDRL, are evaluated for dierent

Table 1. Constant H for SC and SL charts at ARL0 = 500. m = 30 m = 50 m = 100 m = 150 m = 500 m = 150 SC 4.48 4.31 4.45 4.47 4.18

5.25 4.77 4.8 4.85 4.7

5.98 5.56 5.34 5.31 5.25

9.4 9.28 9.24 9.11 8.4

10.32 10.22 10.1 9.95 9.5

11.25 11.15 11.07 10.9 10.74

SL

6.25 5.91 5.67 5.56 5.49

6.65 6.42 6.29 6.11 6

6.73 6.54 6.42 6.28 6.16

11.5 11.53 11.45 11.32 11.17

12.02 12.1 12.06 12.04 12.02

12.14 12.24 12.22 12.21 12.2


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Figure 1. Probability density plots of dierent distributions.

Figure 2. ARL1 curve with varying location shifts and xed = 1:25. combinations, and . These results are provided in Tables 2 and 3 under dierent distributions. For the sake of brevity, we only discuss the results of the pair (100, 5). Moreover, some useful ARL curves are also produced and provided in Figures 2 and 3. The useful ndings about the two charts are listed as follows: 1. In general, the run length follows right skewed distribution; the run length distributions of both charts decrease with the increase in the location and scale shifts; shift in the scale parameter is detected faster than the shift in the location parameter; both

charts are sensitive to shifts in location and scale, but both charts react more quickly to detect a shift in standard deviation rather than mean; 2. For the case of uniform distribution, SC chart performs slightly better than SL chart. For instance, when = 0:25 and = 1:25, ARL1 values of SC and SL charts are 19.97 and 31, respectively; when = 0:0 and = 1:25, ARL1 values of SC and SL charts are 22.81 and 39.65, respectively; and when = 0:25 and = 1:00, ARL1 values of SC and SL charts are 111.64 and 133.09; 3. SC chart performs slightly better than SL chart

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0.5

0.75

1

1.25

1.5

1.75

2


0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00

Table 2. ARLs of SC and SL charts under dierent distributions using m = 100 and n = 5. p p U ( 3; 3) t(4) LN(1; 1) G(1; 1) C1 C2 SC SL SC SL SC SL SC SL SC SL SC SL

133.60 153.60 4596.41 2185.38 16.18 1.59 1.00 3639.70 10453.54 201.87 25.75 8.05 2.07 1.11 506.23 111.64 31.67 13.19 6.75 2.52 1.36 22.81 19.97 14.46 9.47 6.06 2.90 1.66 8.09 7.83 7.01 6.17 5.18 3.14 1.97 4.74 4.64 4.42 4.20 3.80 3.12 2.20 3.38 3.35 3.24 3.17 2.99 2.68 2.32

1171.98 1184.99 990.82 182.76 11.66 1.39 1.00 5250.33 1930.49 122.63 21.42 7.39 1.96 1.07 499.45 133.09 37.89 15.55 7.89 2.85 1.41 39.65 31.65 19.18 11.07 6.96 3.25 1.81 14.67 13.64 11.14 8.41 6.29 3.49 2.14 8.46 8.15 7.33 6.29 5.29 3.60 2.39 5.81 5.68 5.41 4.97 4.40 3.41 2.55

15598.55 15053.16 3651.45 271.02 13.11 1.14 1.01 5776.59 2348.37 285.72 34.85 6.00 1.32 1.03 511.47 267.71 66.51 16.09 5.17 1.51 1.07 112.22 73.29 28.97 10.80 4.71 1.68 1.14 40.09 30.78 16.53 8.05 4.32 1.82 1.23 19.39 16.38 10.88 6.43 3.96 1.92 1.31 11.54 10.24 7.75 5.25 3.63 1.98 1.38

4741.50 4721.17 6725.76 3925.29 2888.23 1607.68 795.95 9287.89 8441.62 6249.00 4173.70 3025.55 1435.98 589.88 503.09 245.33 55.74 13.54 4.64 1.49 1.09 136.30 84.49 29.93 10.74 4.68 1.72 1.17 56.25 40.54 19.25 9.00 4.63 1.91 1.27 28.97 23.38 13.94 7.71 4.50 2.08 1.37 18.11 15.29 10.65 6.69 4.32 2.20 1.47

1958.81 4446.38 9842.98 12483.72 9251.59 1044.82 47.68 12139.71 6892.77 1657.41 269.82 55.96 5.97 1.80 503.04 261.97 71.47 20.79 7.64 2.10 1.20 76.10 45.91 17.54 7.31 3.62 1.50 1.09 24.25 17.07 8.28 4.20 2.45 1.31 1.05 11.72 9.06 5.19 3.03 1.97 1.21 1.03 7.12 5.86 3.77 2.42 1.70 1.16 1.03

1351.42 1405.95 1701.95 1703.05 1183.41 164.16 11.86 1784.44 1146.05 405.53 104.72 28.35 4.12 1.57 505.16 259.18 69.17 20.23 7.64 2.16 1.23 102.55 57.89 20.25 7.97 3.85 1.57 1.11 37.13 23.69 10.24 4.81 2.71 1.36 1.07 18.94 13.34 6.62 3.51 2.18 1.26 1.04 11.84 8.79 4.89 2.88 1.89 1.20 1.03

14130.31 12805.36 5629.85 1238.17 218.56 3.66 1.05 10500.39 3579.71 866.22 183.01 38.16 2.86 1.08 503.96 782.65 240.77 72.60 22.60 3.02 1.14 18.35 177.22 121.88 48.95 19.29 3.62 1.26 7.06 18.87 61.19 38.18 18.40 4.38 1.49 4.31 7.47 15.58 27.12 17.83 5.24 1.84 3.20 4.55 7.11 11.64 14.87 6.14 2.29

332.10 2288.34 1595.17 296.32 35.65 1.32 1.00 1406.33 290.75 150.12 55.13 11.77 1.43 1.01 506.55 534.14 162.23 53.58 16.62 2.23 1.06 28.47 199.10 110.31 43.15 16.75 2.85 1.14 11.07 31.48 69.09 37.91 17.78 3.83 1.32 6.69 12.63 25.51 30.59 18.30 5.07 1.65 4.87 7.44 12.18 17.69 16.88 6.36 2.15

4244.22 3179.54 1644.35 625.48 215.57 22.81 3.79 1497.88 1198.41 645.93 287.55 105.60 17.23 3.91 511.82 423.98 266.94 133.49 63.85 13.94 4.40 203.03 180.53 123.96 73.42 41.02 12.83 4.65 100.81 88.99 68.48 46.47 29.86 11.48 5.08 59.37 50.90 43.14 32.22 22.62 10.84 5.39 40.46 33.66 29.54 24.08 18.16 9.79 5.54

771.56 652.55 479.31 226.86 71.55 5.46 1.49 692.08 535.60 270.15 107.38 36.96 5.48 1.71 500.85 404.52 216.13 96.20 39.75 8.47 2.86 198.85 172.20 109.05 54.95 28.50 8.47 3.24 97.56 89.52 63.88 38.58 22.70 8.05 3.72 55.99 54.32 42.68 29.04 19.19 8.22 4.10 35.55 37.23 31.10 23.29 16.44 8.20 4.51

753.65 717.98 655.09 560.32 453.31 288.71 185.64 631.99 612.85 552.64 483.19 412.44 273.54 185.71 506.78 494.57 455.29 411.23 359.26 260.57 190.03 410.70 401.98 383.55 353.64 315.83 244.72 186.30 340.70 342.21 324.87 301.81 281.01 229.81 186.46 288.24 283.87 271.38 264.18 247.84 214.13 180.64 248.38 250.30 242.79 234.72 225.19 195.52 170.17

374.45 342.99 314.40 281.93 238.19 133.39 68.50 360.24 345.37 314.30 257.58 207.27 118.28 71.33 507.24 490.28 450.29 387.42 321.20 207.84 139.47 389.44 380.56 359.37 317.61 276.75 197.99 135.86 307.79 301.97 286.24 268.04 238.85 181.89 135.55 250.96 246.25 237.93 227.34 208.32 168.13 132.26 209.49 207.66 200.31 193.50 184.62 153.90 132.57


0.5

0.75

1

1.25

1.5

1.75

2

0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00 0.00 0.25 0.50 0.75 1.00 1.50 2.00

2157

Table 3. SDRLs of SC and SL charts under dierent distributions using m = 100 and n = 5. p p U ( 3; 3) t(4) LN(1; 1) G(1; 1) C1 C2 SC SL SC SL SC SL SC SL SC SL SC SL

1523.32 3023.32 14639.87 7716.73 22.29 1.07 0.07 13916.04 18810.79 327.14 30.13 8.37 1.58 0.36 851.89 134.50 34.76 13.56 6.61 2.04 0.73 23.81 20.59 14.58 9.29 5.77 2.42 1.08 7.77 7.45 6.64 5.79 4.76 2.62 1.39 4.26 4.16 3.90 3.71 3.31 2.61 1.64 2.87 2.82 2.70 2.65 2.46 2.13 1.76

3438.77 3506.40 2769.44 499.28 15.20 0.83 0.05 9698.16 4380.18 185.02 24.54 7.76 1.49 0.29 702.26 162.99 42.01 16.23 7.78 2.42 0.79 42.05 33.27 19.67 11.07 6.69 2.78 1.25 14.68 13.42 10.96 8.12 5.90 3.00 1.59 8.03 7.76 6.98 5.85 4.83 3.10 1.82 5.39 5.22 4.92 4.52 3.88 2.89 2.02

26154.47 22552.20 10064.97 2384.35 106.34 0.71 0.07 9485.65 5963.97 1039.44 221.31 17.96 0.79 0.17 853.95 535.68 146.08 33.98 8.02 1.04 0.29 163.10 109.80 44.99 15.30 5.48 1.21 0.42 51.18 41.65 21.10 9.63 4.59 1.35 0.55 22.75 19.15 12.77 7.11 3.93 1.42 0.65 12.63 11.26 8.42 5.36 3.42 1.47 0.73

15223.50 14967.63 14773.53 13315.89 11305.25 8864.32 6302.91 19606.95 19060.68 16231.99 13388.02 11673.86 8030.86 4801.92 712.30 417.24 130.01 21.43 5.66 0.95 0.32 174.78 123.10 44.47 14.19 5.09 1.20 0.46 67.60 51.14 23.69 10.33 4.73 1.43 0.60 32.97 26.84 16.22 8.44 4.47 1.60 0.74 19.82 16.66 11.68 6.95 4.17 1.72 0.85

10820.01 15555.80 21235.22 21560.04 17742.19 4978.54 408.96 17472.98 12867.26 4985.29 942.35 136.19 8.67 1.43 818.90 461.83 124.03 31.48 9.10 1.70 0.51 97.38 59.77 21.03 7.96 3.41 0.91 0.31 27.76 19.44 8.71 4.00 1.99 0.65 0.23 12.28 9.33 5.03 2.59 1.43 0.51 0.19 7.01 5.68 3.37 1.91 1.11 0.43 0.16

3922.19 3377.95 4046.90 3839.40 2831.89 777.76 41.36 2883.92 1896.86 859.17 262.07 54.03 4.79 1.10 720.95 424.46 111.20 27.53 8.94 1.76 0.55 126.44 72.91 24.47 8.59 3.63 0.99 0.35 41.45 26.69 10.88 4.58 2.29 0.72 0.27 19.95 13.94 6.52 3.11 1.65 0.57 0.21 12.18 8.62 4.58 2.42 1.33 0.49 0.18

24104.39 20772.47 13380.19 5596.81 1910.50 35.83 0.64 16745.08 8592.19 3075.42 908.14 186.15 9.81 0.60 836.01 1915.06 662.72 188.94 52.61 5.51 0.69 18.82 281.84 222.75 101.57 33.07 5.50 0.85 6.73 20.10 84.64 55.69 26.45 5.75 1.19 3.85 7.27 16.98 34.93 22.75 6.26 1.62 2.70 4.14 6.94 12.64 17.57 6.84 2.11

1212.42 7520.43 5877.11 2573.27 475.51 2.47 0.08 2724.92 523.63 329.09 165.33 34.21 1.87 0.14 711.49 1037.74 313.50 137.30 36.38 3.76 0.39 29.69 283.74 182.89 69.38 27.11 3.74 0.57 10.82 33.57 96.83 53.46 25.01 4.69 0.91 6.29 12.66 28.03 38.86 22.82 5.95 1.36 4.41 7.13 12.39 19.66 19.59 7.10 1.96

5762.79 4863.72 3319.82 1796.22 818.40 149.53 30.15 2298.94 2021.06 1381.87 907.08 427.66 134.33 27.68 723.14 760.01 617.50 386.99 256.72 99.17 35.49 286.95 327.52 271.87 200.18 157.51 60.55 24.00 138.08 160.81 167.35 111.48 97.94 55.60 27.12 80.30 95.33 93.15 75.73 62.09 41.53 26.37 58.03 57.65 53.91 50.35 45.03 27.69 31.29

1152.54 973.78 767.23 496.24 281.56 22.06 2.89 865.22 744.40 467.73 363.98 123.80 17.72 2.10 804.96 646.67 437.40 268.98 147.65 54.36 27.53 335.96 271.34 214.53 126.31 85.21 43.06 15.72 168.01 137.43 113.32 75.83 53.64 21.25 8.49 97.01 76.66 69.49 48.14 38.74 19.02 6.29 64.65 51.27 46.07 39.54 26.37 15.50 7.96

1165.85 1135.92 1063.13 1013.83 823.35 655.10 494.31 1005.81 989.55 918.59 859.89 785.62 615.42 452.18 827.01 871.64 765.56 738.65 723.87 590.70 495.19 728.38 733.74 701.67 674.40 611.01 548.87 470.69 686.31 601.55 668.38 576.60 547.22 511.73 493.54 568.63 506.45 487.01 481.79 472.90 442.68 443.42 436.40 478.36 449.94 457.56 439.34 430.07 387.64

462.47 419.56 417.88 394.96 410.21 319.56 185.81 461.84 432.95 438.71 395.57 370.67 264.77 203.34 729.03 711.38 703.94 669.15 629.51 493.13 424.78 570.31 587.17 579.38 563.25 529.24 476.71 373.64 478.92 485.66 463.00 472.61 470.79 434.71 391.42 394.23 404.74 398.63 418.78 405.47 385.83 341.43 347.44 342.76 332.65 345.47 356.90 348.88 351.30

2158


Figure 3. (ln ARL1 ) pro le with respect to scale shift () on xed = 0:5. under t4 . For example, when = 0:25 and = 1:25, ARL1 values of SC and SL charts are 73.29 and 84.49, respectively; when = 0:0 and = 1:25, ARL1 values of SC and SL charts are 112.36 and 136.30, respectively, while when = 0:25 and = 1:00, ARL1 values of SC and SL charts are 267.71 and 245.33; 4. For the case of lognormal distribution, SC chart performs slightly better than SL chart. Due to an upward shift in = 0:25 and = 1:25, ARL1 decreasing status concerning 45.91 of SC and 57.89 of SL charts is clear. When = 0:0 and = 1:25, ARL1 values of SC and SL charts are 76.10 and 102.55, respectively, while when = 0:25 and = 1:00, ARL1 values of both SC and SL charts decrease as approximately 48%; 5. Gamma (1,1) provides substantial results when = 0:0 and = 1:0. When = 0:25 and = 1:00, ARL1 values of both charts are greater than the intended ARL0 values, making both charts less eective and ARL biased for such a shift. By varying , we observe the same eect on the results of the said charts. Moreover, having = 1:5 and 2 with = 1:25 shows an increasing trend as compared to the results when remains IC. Similar type of the nding of the exponential distribution was also noted by Riaz and Does [16]; 6. In contaminated environment (C 1 and C 2), eectiveness of detecting the shift in location and scale is aected for both SC and SL charts as compared to other environments. SL chart performs slightly

better than SC chart. In C 1, reduction in ARL1 values of SC and SL charts is reported as: 64% and 66% on = 0:25 and = 1:25; 59% and 60% on = 0:0 and = 1:25; and approximately 15% and 19% on = 0:25 and = 1:00. On the other hand, in C 2, reduction in ARL1 values of SC and SL charts is as: 20% and 24% on = 0:25 and = 1:25; 18% and 22% on = 0:0 and = 1:25; and approximately 1.08% and 1.9% on = 0:25 and = 1:00; 7. Consider the eect of speci c shift in = 1:25 on the charts with respect to dierent environments. The shifts in (on horizontal axis) and ARL1 (on vertical axis) are portrayed in Figure 2. The results revealed better performances of SC and SL charts with the increase of . Further, results from Figure 3 show better performances of SC and SL charts with an increase in on xed = 0:5. Moreover, in light-tailed distributions, SC chart performs well, while SL chart is superior in heavy-tailed environments; both chartslose their performance in the case of C 2.

3.2. Eect of reference sample and test sample on the performance of charts

Control limits of nonparametric charts are estimated from reference sample (m), and this may have signi cant eect on the performance of phase-II chart, which is reported in Table 4. In general, increasing m produces a decreasing trend in ARL1 of both charts under all environments. Speci cally, at xed = 1:5, ARL1 of SC chart under G(1; 1) decreases about 44.5%


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2160


due to an increase in m from 30 to 50, while at xed = 0:75, it decreases about 64.1%, 68.4%, 70.7%, and 71.8% from 30 to 100, 150, 500, and 1000 samples, respectively. On the other hand, in SL chart, 23.5%, 32.9%, 36.3%, 40.1%, and 40.7% fall out is reported in ARL1 from m = 30 to 1000, respectively, on the xed location parameter = 0:25. Moreover, the same ndings are examined for dierent at xed = 1. The test sample (n) also exhibits signi cant effects on the performance of the phase-II chart and its pro le study is given in Table 5. At xed = 1:25, ARL1 of SC chart under t4 environment decreases about 53.9% due to increase in n from 5 to 8 at xed = 0:75, while it decreases 66.5%, 76.6%, and 84.7% from 5 to 11, 16, and 25 samples, respectively. On the other hand, a decrease of 39.6%, 66.4%, 74.5%, and 84.7% in ARL1 of SL chart is reported with n = 5 to 25, respectively, on the xed location parameter = 0:75. The same ndings are also observed at xed = 1 and varying values of . In general, increasing the test sample size produces decreasing trend in ARL1 of both charts under all environments.

Distributions Chart G(1; 1)

p p U ( 3; 3)

t(4)

LN(1; 1)

C1

C2

SC SL SC SL SC SL SC SL SC SL SC SL

4. Illustrative example In this section, we apply our SC and SL charts to a dataset containing duration of contract strikes in US manufacturing industries (cf. [17]). A strike is a refusal of employees to perform their required work as a form of protest. In industries, strikes may cause losses in manufacturing and production departments. So, administration and human resource management always try to avoid it. In case of a strike, they monitor the strike duration to minimize loss. From the said data, we have considered the data from January 1968 to October 1976. Further (following Mukherjee and Sen [4]), we have considered 100 observations between January 1968 and February 1969 as a reference sample and remaining 460 data points as test samples (each of size 10). The control limits for SC and SL charts are obtained by the same simulation procedure, as mentioned in Table 1, and they are given as: 5.37 for SC chart and 11.1 for SL chart at ARL0 = 500. The values of the plotting statistics for SC and SL charts, along with test samples, are reported in Table 6 and their corresponding control charts are given in Figure 4.

Table 5. Pro le of ARL and SDRL using m = 50 and = 1:25. 5 8 11 16 0.75 2 0.75 2 0.75 2 0.75 2

ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL ARL SDRL

66.23 206.14 59.59 171.20 5.81 5.55 8.28 8.18 13.29 33.41 13.29 33.41 7.57 9.77 8.72 11.01 121.24 541.81 101.05 370.64 371.56 1008.69 388.05 943.40

1.53 2.53 1.35 1.68 1.94 1.39 2.16 1.64 1.17 0.49 1.17 0.49 1.09 0.32 1.12 0.38 21.19 317.18 13.09 225.37 246.12 934.20 242.54 854.49

30.14 111.94 30.20 63.90 5.45 5.37 8.12 8.05 6.12 13.75 8.03 23.69 4.06 5.19 5.38 7.40 61.43 369.53 71.54 347.72 305.37 1021.67 374.93 994.99

1.03 0.30 1.03 0.31 1.17 0.46 1.29 0.66 1.02 0.14 1.03 0.18 1.01 0.09 1.01 0.12 2.81 46.72 3.88 28.89 137.23 962.15 182.64 976.19

15.81 37.51 16.91 36.08 4.52 4.46 6.19 6.53 4.46 9.94 4.46 9.94 3.14 3.86 3.82 4.93 43.57 197.72 53.65 235.46 257.64 662.90 325.04 786.37

1.00 0.09 1.00 0.10 1.14 0.41 1.22 0.54 1.00 0.05 1.00 0.05 1.00 0.02 1.00 0.04 1.38 1.59 1.66 22.57 21.62 65.37 75.86 409.76

6.20 17.84 5.94 17.68 3.65 3.60 4.42 4.68 3.11 5.49 3.39 6.81 2.33 2.79 2.58 3.15 27.96 110.00 28.68 121.34 145.77 311.41 178.23 444.82

1.00 0.00 1.00 0.00 1.01 0.10 1.01 0.13 1.00 0.02 1.00 0.02 1.00 0.00 1.00 0.01 1.13 0.54 1.14 0.59 6.97 11.49 8.41 15.83

25 0.75 2

1.88 7.67 1.86 5.49 1.68 1.18 2.10 1.72 2.03 4.60 2.03 3.15 1.61 1.62 1.69 1.68 16.04 62.73 17.18 77.65 105.54 269.41 121.87 318.78

1.00 0.00 1.00 0.00 1.00 0.06 1.00 0.07 1.00 0.00 1.00 0.00 1.00 0.00 1.00 0.00 1.03 0.21 1.03 0.22 3.54 5.15 3.79 6.16


Table 6. Contract strikes, test samples, and corresponding SC and SL statistics. Test samples (n) SC SL Serial no. 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

5 77 46 2 3 46 6 32 62

19

21 168 3 40 64 11 3 15 70 20 92 6 38 39 10 5 9 16 16 15 15 16 101 23 19 94 55 101 45 62 49 139 14 29 42 107

18 1 47 13 4 47 16 71 64

28

52 2 5 47 148 12 4 61 70 24 153 139 2 41 4 44 13 21 24 18 17 22 130 27 20 116 160 8 50 38 8 2 14 99 117 5

44 2 50 25 5 77 18 7 68

72

109 11 7 57 4 29 10 98 1 84 17 2 6 2 5 56 14 37 59 20 22 24 1 32 20 1 5 11 61 51 13 14 106 118 2 5

44 7 77 31 6 2 31 27 82

99

3 19 17 1 6 50 12 22 11 117 226 25 61 4 22 6 15 41 115 26 24 31 2 33 20 8 8 15 106 98 2 15 127 2 12 29

59 10 9 31 7 11 42 14 3

60 18 37 35 9 16 6 26 13

7 23 41 44 14 147 7 4 30

14 25 49 45 23 2 32 4 154

31 36 52 53 26 2 44 43 3

32 42 119 111 37 4 70 60 17

0.942 0.466 2.47 0.784 2.407 1.745 0.92 0.186 1.502

5 26 23 5 12 235 34 24 18 1 13 85 18 5 27 21 28 2 123 34 39 31 2 35 23 15 13 22 142 133 6 33 131 12 19 151

9 30 30 10 12 10 88 38 19 23 23 13 64 7 28 33 50 2 141 84 53 34 3 43 24 15 20 58 36 9 37 143 140 12 22 9

26 36 104 15 28 19 101 64 50 25 2 125 122 9 36 109 60 20 146 122 107 38 4 43 33 22 42 60 52 86 28 42 141 21 75 16

52 47 108 19 105 41 102 84 90 59 38 4 11 13 39 125 135 24 146 174 5 42 8 44 33 23 53 108 99 141 36 8 163 21 126 29

61 50 192 28 112 52 104 5 9 63 3 54 16 38 85 127 5 57 3 4 9 65 11 100 63 26 59 31 38 9 48 122 22 27 8 35

148 87 18 42 163 100 124 6 15 179 3 91 31 3 191 8 7 8 15 14 10 74 22 2 67 27 83 42 47 5 136 56 23 38 36 65

0.674 0.416 0.326 0.594 2.12 0.769 2.629 0.445 0.084 1.582 1.417 2.193 0.046 1.911 0.119 0.762 0.184 0.651 3.952 0.347 0.782 2.706 4.704 1.378 2.719 0.176 0.782 0.832 4.435 2.534 0.07 0.886 4.387 0.247 0.216 0.137

104 114 152 153 216

15

6.95

1.557 1.361 4.58 2.149 4.86 3.181 1.628 0.639 2.632

12.23 1.258 1.124 0.129 1.346 3.774 1.534 5.101 0.886 0.149 2.399 1.311 3.807 0.089 3.883 0.7 1.574 0.253 2.062 6.63 1.274 1.978 6.862 7.843 4.147 8.275 1.844 1.755 1.656 8.895 5.763 0.118 1.503 7.672 1.458 0.36 0.274

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Figure 4. Control chart displays: (a) SC chart, and (b) SL chart. It is evident that both SC and SL charts indicate an OOC signal at 10th point. For the follow-up diagnosis of shift by SL chart, we have computed the p-values for Wilcoxon test (p1 = 0:001684) and Ansari Bradley test (p2 = 0:1267), indicating a shift in location parameter. Similarly, for SC chart, we got the p-values for Wilcoxon test (p3 = 0:001684) and mood test (p4 = 0:04445), referring to locational shift. The results of this example are also in line with those of Mukherjee and Sen [4] which concluded that there is no scale shift in the process.

5. Concluding remarks This study investigated two nonparametric, SC and SL, charts for the joint monitoring of location and scale parameters. The performance analysis revealed that SC takes an edge over SL under light-tailed distributions, while SL is a good alternative under heavy-tailed distributions. Moreover, a reasonably larger reference and test samples produce better ARL performance of these charts. Some interesting future research directions might include studying the performance of these charts: EWMA and CUSUM setups; under multiple structural breaks; when a shift occurs at steady-state.

Acknowledgements The authors are grateful to the editor(s) and anonymous reviewers for their constructive comments that led to a substantial improvement of the article. Authors Tahir Mahmood and Nasir Abbas would like to acknowledge the support provided by the Deanship of Scienti c Research (DSR) at KFUPM for funding this work through project # SR151010.

References 1. Chakraborti, S., Qiu, P. and Mukherjee, A. \Editorial to the special issue: Nonparametric statistical process control charts", Quality and Reliability International, 31(1), pp. 1-152 (2015). 2. Chakraborti, S., van der Laan, P. and Bakir, S.T. \Nonparametric control charts: An overview and some results", Journal of Quality Technology, 33(3), pp. 304-315 (2001). 3. Chowdhury, S., Mukherjee, A. and Chakraborti, S. \Distribution-free phase II CUSUM control chart for joint monitoring of location and scale", Quality and Reliability Engineering International, 31(1), pp. 135151 (2015). 4. Mukherjee, A. and Sen, R. \Comparisons of Shewharttype rank based control charts for monitoring location parameters of univariate processes", International Journal of Production Research, 53(14), pp. 4414-4445 (2015). 5. Chowdhury, S., Mukherjee, A. and Chakraborti, S. \A new distribution-free control chart for joint monitoring of unknown location and scale parameters of continuous distributions", Quality and Reliability Engineering International, 30(2), pp. 191-204 (2014). 6. McCracken, A.K. and Chakraborti, S. \Control charts for simultaneous monitoring of unknown mean and variance of normally distributed processes", Journal of Quality Technology, 45(4), p. 360 (2013). 7. Mukherjee, A. and Chakraborti, S. \A distributionfree control chart for the joint monitoring of location and scale", Quality and Reliability Engineering International, 28(3), pp. 335-352 (2012). 8. Mukherjee, A., McCracken, A.K. and Chakraborti, S. \Control charts for simultaneous monitoring of parameters of Shifted exponential distribution", Journal of Quality Technology, 47(2), pp. 176-192 (2015).


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9. Lepage, Y. \A combination of Wilcoxon's and AnsariBradley's statistics", Biometrika, 58(1), pp. 213-217 (1971). 10. Cucconi, O. \Un nuovo test nonparametrico per il confronto fra due gruppi di valori campionari", Giornale degli Economisti e Annali di Economia, pp. 225-248 (1968). 11. Marozzi, M. \Some notes on the location-scale Cucconi test", Journal of Nonparametric Statistics, 21(5), pp. 629-647 (2009). 12. Abbasi, S.A. and Miller, A. \On proper choice of variability control chart for normal and non-normal processes", Quality and Reliability Engineering International, 28(3), pp. 279-296 (2012). 13. Alfaro, J.L. and Ortega, J.F. \A robust alternative to Hotelling's T2 control chart using trimmed estimators", Quality and Reliability Engineering International, 24(5), pp. 601-611 (2008). 14. Ali, A., Mahmood, T., Nazir, H. Z., Sana, I., Akhtar, N., Qamar, S. and Iqbal, M. \Control charts for process dispersion parameter under contaminated normal environments", Quality and Reliability Engineering International, 32(7), pp. 2481-2490 (2016). 15. Human, S.W., Kritzinger, P. and Chakraborti, S. \Robustness of the EWMA control chart for individual observations", Journal of Applied Statistics, 38(10), pp. 2071-2087 (2011). 16. Riaz, M. and Does, R.J. \An alternative to the bivariate control chart for process dispersion", Quality Engineering, 21(1), pp. 63-71 (2008). 17. Kennan, J. \The duration of contract strikes in US manufacturing", Journal of Econometrics, 28(1), pp. 5-28 (1985).

Minerals (KFUPM), Dhahran, Saudi Arabia. His current research interests include statistical process control, nonparametric techniques, and linear pro le monitoring.

Biographies Tahir Mahmood is a student of MS in Applied

Amjid Ali is a student of MPhil Statistics at the

Statistics at the Department of Mathematics and Statistics, King Fahd University of Petroleum and

Ha z Zafar Nazir obtained his PhD in Statistics from

the Institute for Business and Industrial Statistics, University of Amsterdam, the Netherlands. He is serving as an Assistant Professor in the Department of Statistics, University of Sargodha, Pakistan. His current research interests include statistical process control, nonparametric techniques, and robust methods.

Nasir Abbas is serving as an Assistant Professor at

Department of Mathematics and Statistics, KFUPM. He obtained his PhD in Industrial Statistics from the Institute for Business and Industrial Statistics, University of Amsterdam, the Netherlands in 2012. His current research interests include mathematical statistics and quality control, particularly control charting methodologies under parametric and nonparametric environments.

Muhammad Riaz obtained PhD degree in statistics

from the Institute for Business and Industrial Statistics, University of Amsterdam, the Netherlands in 2008. He holds the position of Associate Professor in the Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. His current research interests include statistical process control, non-parametric techniques, mathematical statistics, and experimental design. Department of Statistics, University of Sargodha, Sargodha, Pakistan. His current research interests include statistical process control, multivariate techniques, biostatistics, and development of new robust methods.