CUMULATIVE SUM STATISTICAL CONTROL CHARTS USING RANKED SET SAMPLING DATA

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1 Pak. J. Statist. 010 Vol. 6(), CUMULATIVE SUM STATISTICAL CONTROL CHARTS USING RANKED SET SAMPLING DATA Walid S. Al-Sabah Departmet of Mathematics ad Statistics Kig Fahd Uiversity of Petroleum & Mierals Dhahra 3161, Saudi Arabia ABSTRACT Differet cumulative sum (CUSUM) cotrol charts for the sample mea based o raked set samplig (RSS) data ad media raked set samplig (MRSS) data are developed ad compared to the usual CUSUM based o simple radom samplig (SRS) data usig computer simulatio. All the charts based o raked set samplig data are show to have smaller average ru legth (ARL) tha the classical CUSUM charts based o SRS if the process starts to get out of cotrol, ad have approximately the same ARL if the process is i cotrol. Real data usig the RSS ad MRSS are used to illustrate the ew developed methods ad costruct the correspodig cotrol charts. These charts are compared to usual SRS chart, it turs out that the ewly developed charts are more efficiet i.e. havig smaller ARL I this study we are assumig the uderlig distributio is ormal. KEYWORDS Average ru legth; media raked set samplig; raked set samplig ad simple radom samplig. 1. INTRODUCTION McItyre (195) was first to suggest usig raked set samplig (RSS) to estimate the populatio mea istead of the usual simple radom samplig (SRS). Takahasi ad Wakimoto (1968) supplied the ecessary mathematical theory. Dell ad Clutter (197) studied the case i which the rakig may ot be perfect i.e. there are errors i rakig the uit with respect to the variable of iterest. The Shewhart cotrol charts (see for example Motgomery, (005)) use oly the iformatio about the last poit i the process. The cumulative-sum (CUSUM) cotrol charts were proposed to overcome this problem (see for example Hawkis ad Olwell (1998)). Muttlak ad Al-Sabah (003) developed differet quality cotrol charts usig the RSS ad some of its modificatios ad compared them to the usual cotrol charts based o simple radom samplig (SRS) data. 010 Pakista Joural of Statistics 365

2 366 Cumulative sum statistical cotrol charts usig raked set samplig data This study is cocered with the idea of developig CUSUM cotrol charts usig raked set samplig (RSS) ad media raked set samplig (MRSS). These ewly developed cotrol charts are compared usig computer simulatio to the usual CUSUM cotrol charts based o the simple radom samplig (SRS) method ad show to be more efficiet. Real data sets are collected ad used to illustrate the computatios of the ewly developed cotrol charts.. DEFINITIONS AND SOME USEFUL RESULTS Let X i: j deote the i th order statistic from the i th sample of size i the j th cycle. The the ubiased estimator of the populatio mea, see Takahasi ad Wakimoto (1968) usig RSS data based o the j th cycle is defied as X rssj 1 i1 X i: j j 1,,..., r (.1) The variace of Xrssj is give by 1 rssj i : var X i1 where i: E X : i E X i:., (.) Assume that (X, Y) has a bivariate ormal distributio ad the regressio of X o Y is liear. The followig Stokes (1977) we ca write x X x Y y (.3) y where Y ad are idepedet ad has mea 0 ad variace x 1 correlatio betwee X ad Y ad x, y, x, y of the variables X ad Y., is the are the meas ad stadard deviatios Let Y i: j ad X i: j be the i th smallest value of Y ad the correspodig value of X obtaied from the i th set i the j th cycle respectively. We ca write the above equatio X x i: j x i: j y ij Y, i = 1,,,, j =1,,, r. (.4) y The ubiased estimator of the mea of the variable of iterest X with rakig based o the cocomitat variable Y, i.e. usig imperfect raked set samplig IRSS method (see Stokes (1977)), ca be writte for the j th cycle as

3 Walid S. Al-Sabah 367 X irss j 1 i1 X i: j j 1,,..., r (.5) The variace of Xirss j is give by var X 1 x irss j yi: y i1. (.6) Let Xi : m j deotes the i th media of the i th set of size i the j th cycle if the set size is odd. Also deote the (/)-th order statistic of the i th set of size (i = 1,,..., L= /) ad the ((+)/)-th order statistic of the i th set of size (i = L+1, L+,..., ) if the set size is eve. The estimator of the populatio mea (see Muttlak (1997)) usig MRSS data i the j th cycle ca the be writte as X mrss j 1 i1 X i: m j j 1,,..., r. (.7) The variace of Xmrssj is give by 1 mrss j i : m var X i1 where i: m E X : i m E X i: m., (.8) 3. CUMULATIVE SUM (CUSUM) STATISTICAL CONTROL CARTS 3.1 Cumulative sum (CUSUM) cotrol chart usig SRS Let X for i = 1,,..., ad j = 1,,..., r deote the i th uit i the j th simple ij radom sample of size ad ij, X N. The sample mea of the j th sample is 1 X srs j Xij. If the populatio mea µ is kow to be 0, the CUSUM cotrol chart i1 for the populatio mea 0, is give by C max 0, X srsj srsj 0 k C srsj1 (3.1) C max 0, 0 X srsj k srsj C srsj1 (3.) srs srs where the startig values are C 0 C 0 0.

4 368 Cumulative sum statistical cotrol charts usig raked set samplig data I (3.1) ad (3.) k is called the referece value ad usually chose about half way betwee the target value 0 ad the out of cotrol value µ 1. If the shift δ is expressed i stadard deviatio uits as 1 0, the k is oe half of the shift or 1 0 k (3.3) Note that if Csrsj or Csrsj exceed the decisio iterval h, the process is cosidered to be out of cotrol. The selectio of k ad h is every importat, as it has great effect o performace of the CUSUM, for more details see Hawkis ad Olwell (1998). We have give the developmet of the CUSUM for the case of group of size observatios, it is easy to cosider the case of a idividual observatio ( = 1), by replacig Xsrsj by X srsj ad by i ( ). 3. Cumulative sum (CUSUM) cotrol chart usig RSS Let X for i = 1,,..., ad j = 1,,..., r deote the i th uit i the j th simple ij radom sample of size ad ij, X N. Let X : i j deote the i th order statistic from the i th sample of size i the j th cycle. The RSS mea Xrssj of the j th cycle will be used to costruct the CUSUM for the RSS data if the rakig is perfect. i.e. without errors i rakig. Assumig that the populatio mea µ is kow to be 0, the CUSUM cotrol chart for the populatio mea 0 usig RSS data is proposed to be C max 0, X rssj rssj 0 k C rssj1 (3.4) C max 0, 0 X rssj k rssj C rssj1 (3.5) rss rss where the startig values are C 0 C 0 0. I (3.4) ad (3.5) k is called the referece value ad usually chose about half way betwee the target value 0 ad the out of cotrol value µ 1. If the shift δ is expressed i stadard deviatio uits as 1 0, the k is oe half of the shift or Xrssj 1 0 k (3.6) X rssj where 1 X rss j i: i1

5 Walid S. Al-Sabah 369 Note that if Crssj or Crssj exceed the decisio iterval h, the process is cosidered to be out of cotrol. The selectio of k ad h is every importat, as it has great effect o performace of the CUSUM, see Hawkis ad Olwell (1998) for more details. Rakig the variable of iterest without errors i rakig the uits is called perfect rakig. But if the uits caot be raked perfectly or the rakig is doe o a cocomitat variable we call that imperfect rakig, see previous sectio. Sice the perfect rakig ad SRS are special cases of the imperfect rakig with 1 ad 0 respectively, we will cosider the case of imperfect raked set samplig (IRSS) with differet values of. Followig the same procedure that we used i the previous sectio, we oly eed (.6) to costruct our CUSUM chart, which ca be writte as where x irss j zi: var X 1 x i1 is the variace of the variable of iterest X ad, (3.7) z ( i : ) is the variace of the i th order statistic i a sample of size from the stadard ormal distributio. The CUSUM cotrol chart give i (3.4) ad (3.5) is based o the perfect RSS, we eed to modify it to the case of imperfect rakig by substitutig for the variace of Xirssj give i (3.7) to get k (3.8) 1 0 Xirssj 1 X irssj i1 x where zi:. We use the average ru legth (ARL) to compare the RSS CUSUM cotrol charts to the usual CUSUM cotrol charts based o SRS data. The ARL assumes that the process is i cotrol with mea 0 ad stadard deviatio 0, ad at some poit i time the process may start to get out of cotrol i.e. the mea is shifted from 0 to 0 0 / with mea 0. We are assumig that the process is followig the ormal distributio ad variace 0 if the process is i cotrol, ad the shift o the process mea is 0 0. If 0 the process is i cotrol ad i this case if the poit is outside the cotrol limits it is a false alarm. 3.3 Cumulative sum (CUSUM) cotrol chart usig MRSS Let X i: m j deote the i th media of the i th set of size i the j th cycle if the set size is odd or eve. The MRSS mea Xmrssj of the j th cycle defied i the previous will be used to costruct the CUSUM for the MRSS data if the rakig is

6 370 Cumulative sum statistical cotrol charts usig raked set samplig data perfect. With the assumptio that the populatio mea µ is kow to be 0, the CUSUM cotrol chart for the populatio mea µ 0 usig MRSS data is proposed to be C max 0, X mrssj mrssj 0 k C mrssj1 (3.9) C max 0, 0 X mrssj k mrssj C mrssj1 (3.10) mrss mrss where the startig values are C 0 C 0 0. The value of k will be chose i the same way that we used i previous sectio to be k (3.11) 1 0 Xmrrsj where 1 X mrss j i: m i1 If cotrol. Cmrssj or ad i: m E X : i m E X i: m Crssj exceed the decisio iterval h, the process is cosidered to be out of As the case for RSS we will cosider the case of imperfect media raked set samplig (IMRSS) with differet values of ρ. Equatio (3.7) ca be writte as where x imrss j zi: m var X 1 x i1 is the variace of the variable of iterest X ad z ( i : m ) i th media i a sample of size from the stadard ormal distributio. (3.1) is the variace of the The CUSUM cotrol chart give i (3.9) ad (3.10) ca be modified for the case of imperfect rakig by substitutig for the variace of Ximrssj give i (3.1) to get k (3.13) 1 0 Ximrssj x where 1 z( i: m) i1 Ximrss. 4. COMPUTER SIMULATION We used computer simulatio to compare the CUSUM usig the usual simple radom samplig (SRS) data to the ewly developed CUSUM cotrol charts usig RSS ad MRSS data. I these comparisos we will use the average ru legth to compare the ewly developed CUSUM with the usual CUSUM usig SRS data.

7 Walid S. Al-Sabah Computer simulatio for CUSUM usig RSS To be able to compare the values of the average ru legth (ARL) usig RSS with existig ARL usig SRS, we used i our simulatio the same values of, k ad h that were used by Hawkis ad Olwell (1998), we simulate 400,000 replicatios. We calculate the values of the limits i (1) ad (13) usig the results of the order statistics for the stadard ormal distributio, see for example Harter ad Balakrisha (1996). The computer simulatios are ru for k = 0.00 ad 0.5; h =, 1.5, 1.50,, 3.00; ρ = 0.00, 0.5, 0.50, 0.75, 0.90, ; = 3 ad for δ = , 0.5, 0.375, 0.50, 0.75,, 1.50, We oly reported i this paper the case of = 3 simply because icreasig the set size will ot decrease the ARL. Results are show i Tables 1 ad. Note that whe ρ = 0 the correspodig values of ARL are equal to the SRS, i.e. the usual ARL usig simple radom samplig data. Cosiderig the results i Tables 1 ad the followig coclusios ca be made: 1. If the process is i cotrol i.e. = 0, RSS ad SRS have approximately the same ARL. But if the process starts to get out of cotrol i.e. > 0, RSS reduces the ARL substatially, for example if = 0.5, k = 0.5, h = 1.5 ad 1 the ARL is oly 5. as compared to 7.1 for the SRS.. Errors i rakig i.e. imperfect rakig will decrease the efficiecy ad the ARL will be larger. This icreasig i ARL will deped o the correlatio betwee the variable of iterest ad the cocomitat variable that we use to estimate the rak of the variable of iterest. 3. If the value of k icreases, the reductio i the ARL values usig RSS will be larger tha SRS. For example if 1, k =0.0, = 0.5, ad h = 1.5 the ARL is.89 for RSS as compared to 3.50 for SRS i.e. with 18% reductio i the ARL. But if k = 0.5 ad everythig else is the same the ARL is 5. for RSS as compared to 7.10 for SRS, i.e. with 37% reductio i the ARL. 4. If the h values icrease, the reductio i the ARL values usig RSS will be substatial. For example if 1, k = 0.5, = 0.5, ad h = 1 the ARL is 3.69 for RSS as compared to 4.74 for SRS i.e. with 8% reductio i the ARL. But if h = 3 ad everythig else remai the same the ARL is for RSS as compare to for SRS, i.e. with 59% reductio i the ARL. 4. Computer simulatio for CUSUM usig MRSS We use the average ru legth (ARL) to compare the MRSS CUSUM cotrol charts to the CUSUM cotrol charts based o RSS ad SRS data. The ARL assumes that the process is i cotrol with mea µ 0 ad stadard deviatio 0 ad at some poit i time the process may start to get out of cotrol. We are assumig that the process is followig the ormal distributio with mea µ 0 ad variace shift o the process mea is if the process is i cotrol, ad the Simulatio was used to compare the values of ARL usig MRSS with the ARL usig RSS ad SRS. We calculate the values of the limits i (3.9) ad (3.11) usig the results of the order statistics for the stadard ormal distributio, see for example Harter ad Balakrisha (1998).

8 37 Cumulative sum statistical cotrol charts usig raked set samplig data The computer simulatios are ru for the same values of δ, k, h,, ad umber of replicatios used i the previous sectio. Oce agai we are oly reportig the cases of k = 0.0 ad k = 0.5 with = 3. Results are show i Tables 3 ad 4. Cosiderig the results i Tables 3 ad 4 the followig coclusios ca be made: 1. If the process i cotrol i.e. = 0, MRSS, RSS ad SRS have approximately the same ARL. But if the process starts to get out of cotrol i.e. > 0, MRSS is reduces the ARL substatially, for example if = 0.5, k = 0.5, h = 1.5 ad 1 the ARL is 4.81 as compared to 5. ad 7.1 for RSS ad SRS respectively.. As i the case of RSS the errors i rakig i.e. imperfect rakig will decrease the efficiecy ad the ARL will be larger for usig MRSS. This icrease i ARL will deped o the correlatio betwee the variable of iterest ad the cocomitat variable that we use to estimate the rak of the variable of iterest. 3. If the value of k icreases, the reductio i the ARL values usig MRSS will be larger tha RSS. For example if 1, k = 0.0, = 0.5, ad h = 1.5 the ARL is for MRSS.74 as compared to 3.5 for SRS i.e. with 8% reductio i the ARL. But if k = 0.5 ad every thig else is the same the ARL is 4.81 for MRSS as compared to 7.07 for SRS, i.e. with 47% reductio i the ARL. 4. If the h values icrease, the reductio i the ARL values usig MRSS will be larger tha RSS. For example if 1, k = 0.5, = 0.5, ad h = 1 the ARL is 3.45 for MRSS as compared to 4.74 for SRS i.e. with 37% reductio i the ARL. But if h = 3 ad every thig else remai the same the ARL is 9.74 for MRSS as compared to for SRS, i.e. with 78% reductio i the ARL. 5. EXAMPLE I this example we use the sets of data from Muttlak ad Al-Sabah (003) collected from a soft drik bottle-fillig productio lie of the Pepsi Cola productio compay i Al-Khobar, Saudi Arabia. The data were collected by measurig the ufilled part of the bottle usig RSS ad MRSS samplig techiques with perfect rakig for sample sizes = 3. There are 55 radom samples of set size = 3 where collected usig RSS ad MRSS. To costruct CUSUM cotrol charts for the RSS ad MRSS data collected i this study, we first check the ormality assumptio for the populatio of iterest usig the data collected i this study it seem that this assumptio is reasoable. As for the values of mea ad stadard deviatio for each chart, we used the umber or replicatios (cycle) which are 55 cycles (samples) of size 3 each to come up with the over all meas ad the stadard deviatios for RSS ad MRSS data. To compare our ewly suggested CUSUM cotrol charts to the classic charts based o SRS, we used a data set collected usig SRS from 55 samples of size = 3 each, the cotrol chart is show i Figure 1. As for the RSS ad MRSS methods we replicated the cycle 55 times, the set (sample) size = 3 for each cycle, Figures ad 3 represet cotrol charts for RSS ad MRSS respectively. 6. CONCLUSIONS AND RECOMMENDATIONS Raked set samplig has bee demostrated to be a efficiet samplig method. The RSS method proved to be more efficiet whe uits are difficult ad costly to measure, but are easy ad cheap to rak with respect to a variable of iterest without actual

9 Walid S. Al-Sabah 373 measuremet. I this study we used the RSS ad MRSS to develop several CUSUM quality cotrol charts for the variables of iterest usig the sample mea. These charts are compared with the classical cotrol charts usig simple radom samplig data. It is clear that all the ewly developed charts are more efficiet tha the classical cotrol chart, but some of them are better tha others. The followig are some specific coclusios. 1. All ewly developed cotrol charts domiate the classical charts. If the process starts to get out of cotrol most of these ew charts reduced the average ru legth (ARL) substatially.. Errors i rakig will reduce the ARL for all the cases cosidered. The amout of reductio i the ARL will deped o the amout of errors committed i rakig the uits of the variable of iterest. For example if we are usig a cocomitat variable to rak our variable, the the ARL will deped o the correlatio betwee the two variables. 3. To over-come the problem of errors i rakig ad/or to icrease the efficiecy of estimatig the populatio mea, we suggest usig MRSS istead of RSS. We ca see that both methods will reduce the errors i rakig ad are easy to be used i real life. The MRSS domiates all other methods i terms of reducig the ARL if the process starts to get out of cotrol. Fially we recommed usig MRSS with odd sample size to build the CUSUM quality cotrol charts. Sice they reduce the ARL compared to RSS ad SRS method, ad they are easily implemeted i the field of study, i.e. we ca fid the smallest ad the largest i a set of five uits much easier tha rak all five uits i the set. ACKNOWLEDGEMENT The author gratefully ackowledges the help of Professor Hasse A. Muttlak, as well as the support of Kig Fahd Uiversity of Petroleum ad Mierals (KFUPM), Dhahra, Saudi Arabia. REFERENCES 1. Dell, T.R. ad Clutter, J.L. (197). Raked set samplig theory with order statistics backgroud. Biometrics, 8, Harter, H.L. ad Balakrisha, N. (1996). CRC Hadbook of Tables for the use of Order Statistics i Estimatio. CRC Press Boca Rato. 3. Hawkis, D.M. ad Olwell, D.H. (1998).Cumulative Sum Charts ad Chartig Improvemet. Spriger, New York, 4. McItyre, G.A. (195). A method for ubiased selective samplig usig raked sets. Aust. J. Agri. Res., 3, Motgomery, D.C. (1997). Statistical Quality Cotrol. 5th Editio, Wiley, New York. 6. Muttlak, H.A. (1997). Media raked set samplig. J. App. Statist. Scie., 6(4), Muttlak, H.A. ad Al-Sabah, W.S. (003). Statistical quality cotrol usig raked set samplig. J. App. Statist., 30, Stokes, S.L. (1977). Raked set samplig with cocomitat variables. Commuicatios i Statistics - Theory ad Methods A6, Takahasi, K. ad Wakimoto, K. (1968). O the ubiased estimates of the populatio mea based o the sample stratified by meas of orderig. A. Ist. Statist. Math., 0, 1-31.

10 374 Cumulative sum statistical cotrol charts usig raked set samplig data Table 1: The ARL values whe = 3 & k = 0 with differet values for ρ, & h usig RSS CUSUM ρ h

11 Walid S. Al-Sabah Table : The ARL values whe = 3 & k = 0.5 with differet values for ρ, & h usig RSS CUSUM ρ h

12 376 Cumulative sum statistical cotrol charts usig raked set samplig data Table 3: The ARL values whe = 3 & k = 0.0 with differet values for ρ, & h usig MRSS CUSUM ρ h

13 Walid S. Al-Sabah Table 4: The ARL values whe = 3 & k = 0.5 with differet values for ρ, & h Usig MRSS CUSUM ρ h

14 378 Cumulative sum statistical cotrol charts usig raked set samplig data CUSUM Figure 1: CUSUM Cotrol Chart usig SRS with = Figure. CUSUM Cotrol Chart usig RSS with = 3. CUSUM Figure 3: CUSUM Cotrol Chart usig MRSS with = 3. CUSUM