GINI MEAN DIFFERENCE AND EWMA CHARTS. Muhammad Riaz, Department of Statistics, Quaid-e-Azam University Islamabad,

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1 GINI MEAN DIFFEENCE AND EWMA CHATS Muhammad iaz, Deparmen of Saisics, Quaid-e-Azam Universiy Islamabad, Pakisan. Saddam Akbar Abbasi, Deparmen of Saisics, Quaid-e-Azam Universiy Islamabad, Pakisan. Absrac: For an improved monioring of process parameers, an EWMA conrol charing scheme is proposed. The proposal inroduces Gini mean difference based EWMA conrol scheme for locaion and scale parameers using asympoic limis. The comparison of he proposed schemes are made wih he EWMA conrol schemes based on sample range, sample sandard deviaion S and Downon s esimaor used by Khoo (2004). Key Words: Conrol Chars; Downon s esimaor; EWMA; Gini mean difference; locaion and scale parameers; Normaliy; Ouliers. 1. Inroducion In saisical Process Conrol ool ki, conrol char is he mos commonly used and he mos powerful ool for process monioring. Process parameers (e.g. locaion and scale) need a imely deecion of unfavorable variaions which may cause deerioraion in qualiy of he producs. EWMA and CUSUM conrol srucures address smaller shifs whereas Shewhar conrol schemes address larger shifs in process parameers. In EWMA, CUSUM and Shewhar seup, robus conrol schemes are always desirable. 1

2 There are many robus conrol chars in qualiy conrol lieraure e.g. ocke (1992) proposed robus EWMA based on sample range and sample mean, Khoo (2004) proposed proposed downon s based EWMA schemes, Maravelakis e al. (2005) examines robusness o non normaliy of he EWMA Conrol Chars for he spread parameer where as Soumbos and Sullivan (2002) examines robusness o non normaliy of he mulivariae EWMA conrol char ec. Ng and Case (1989) proposed wo chars EWMAS and EWMASM for monioring spread and locaion parameers respecively based on sample range.. EWMAS char is proposed by ploing he saisic following asympoic limis. r$ ( 1 ) $ = λ + λ r 1 agains he d3 λ UCL = 1+ 3 d2 2 λ CL = d3 λ LCL = 1 3 d 2 λ 2 (1.1) where λ (lying beween 0 and 1) is he weigh assigned o each observaion, d and d 2 3 are well known conrol char coefficiens available in qualiy conrol lieraure (e.g. see Mongomery (2001)) as a funcion of sample size (n) and is he average of sample s esimaed from iniially seleced samples assumed o come from an in-conrol process. EWMASM char was proposed by ploing he saisic Z λx ( 1 λ) 1 following conrol limis: ˆ = + Z ˆ agains he 2

3 3 λ UCL = X + d 2 2 n λ CL Zˆ = o = X 3 λ LCL = X d n 2 λ 2 (1.2) where X is he grand mean of average sample X s esimaed from iniially seleced samples assumed o come from an in-conrol process. Khoo (2004) proposed wo chars namely EWMAVD and EWMAMD chars for monioring spread and locaion parameers respecively based on Downon s esimaor. EWMAVD char was proposed by ploing he saisic ˆ λ ( λ) he following conrol limis: Y 1 ˆ = S + Y agains -1 ˆ 2 λ UCL = Yo + 3ˆ σˆ = c4 3 ( 1 c Y + 4) σ 2 λ CL Yˆ c = o = 4σ ˆ 2 λ 3ˆ LCL = Yo + σ ˆ = max 0, c4 3 ( 1 c Y 4) σ 2 λ (1.3) where c 4 is a well known conrol char coefficien available in qualiy conrol lieraure * (e.g. see Mongomery (2001)) as a funcion of sample size (n), σ is he Downon s Esimaor defined as: n * 2 π 1 σ = n n 1 + i= 1 2 ( ) i ( n 1) X() (1.4) i and σ * is he average of sample σ s esimaed from iniially seleced samples assumed o come from an in-conrol process. 3

4 ˆ = + 1 Z agains he EWMAMD char was proposed by ploing he saisic Z λx ( λ) $ 1 following conrol limis ˆ 3 λ UCL = Zo + 3 ˆ σzˆ = X + σ n 2 λ CL Zˆ = o = X ˆ 3 λ LCL = Zo 3 ˆ σzˆ = X σ n 2 λ (1.5) The similar EWMA conrol srucures based on sample saisic S for monioring process spread and locaion parameers are defined respecively in erms of SdV and SdM chars as: SdV char is defined by ploing he saisic ˆ λ ( λ) following conrol limis: Y 1 ˆ = S + Y agains he -1 UCL = c + 3 λ 2 ( 1 c4 ) S 4 2 λ c4 CL = S 2 λ ( 1 c4 ) S LCL = c4 3 2 λ c4 (1.6) where S is he average of sample S s esimaed from iniially seleced samples assumed o come from an in-conrol process. SdM char is defined by ploing he saisic Z λx ( λ) Z $ 1 conrol limis ˆ = + 1 agains he following 4

5 3 λ S UCL = X + n 2 λ c4 CL Zˆ = o = X 3 λ S LCL = X n 2 λ c 4 (1.7) The following secion inroduces EWMA conrol srucures for monioring locaion and scale parameers based on Gini mean difference using asympoic limis. 2. The Proposed Chars The proposals are based on Gini mean difference (G), which was used by Gini (1912), as a measure of variabiliy. Le x1, x2, x3,..., x n be an i.i.d random sample, hen G is defined as: G = n n j= 1 i= 1 j i x n( n 1) / 2 i x j (2.1) Gini (1912) had shown superioriy of sample saisic G over he sample saisic as an esimae ofσ, in erms of efficiency and robusness agains non normaliy. David (1968) showed ha π 2 G serves as an unbiased esimae ofσ. iaz and Saghir (2007) proposed a Shewhar ype conrol char namely G char for robus monioring of process scale parameerσ. They defined he wo random variables K and M as: π K = G, (2.2) 2 M K =, (2.3) σ 5

6 and developed he design srucure of G-char in erms of coefficiens b 3 and he quanile poins of M. These quaniies (i.e. Tables 1 and 2 respecively. b 3 and quanile poins of M) are given in Appendix We propose here he use of K in EWMA ype paern for improved monioring of variabiliy and locaion parameers. The GVA char for monioring process variabiliy is proposed as: ˆ 2 λ UCL = Yo + 3ˆ σ ˆ = c4 3 ( 1 c Y + 4) K 2 λ CL Yˆ c K, = o = 4 ˆ 2 λ 3ˆ LCL = Yo σ ˆ = max 0, c4 3 ( 1 c Y 4) K 2 λ (2.4) where K he average of sample K s esimaed from iniially seleced samples assumed o come from an in-conrol process. Afer consrucing he conrol srucure of GVA char we plo he EWMA saisic ( λ) Yˆ = λs + 1 Yˆ following Khoo (2004). Any -1 Ŷ falling ouside he conrol limis is an indicaion of ou of conrol siuaion wih respec o scale parameer. Similarly an asympoic limi EWMA scheme namely GMA char is proposed for monioring he locaion parameer as: ˆ 3 λ UCL = Zo + 3 ˆ σ ˆ = X + Z K n 2 λ CL Zˆ = o = X, ˆ 3 λ LCL = Zo 3 ˆ σ Zˆ = X K n 2 λ (2.5) 6

7 Afer consrucing he conrol srucure for GMA char, we plo he EWMA saisic ( λ) Z $ 1 Zˆ = λx + 1 following Khoo (2004). Any Z ˆ falling ouside he conrol limis is an indicaion of ou of conrol siuaion wih respec o locaion parameer. 3. Comparisons In his secion, comparison of he proposed EWMA charing schemes is made wih ha of Khoo (2004) proposed scheme and he well known and S based EWMA schemes for locaion and spread parameers. For comparison purposes, he abiliy o deec ouliers is used as a performance measure. Following Khoo (2004), we have chosen m=10 subgroups of sizes n = 5 and 10 by considering he following wo siuaions: a) Un-conaminaed disribuion where all he samples are generaed from he sandard normal disribuion i.e. N (0, 1). b) Conaminaed disribuions where ouliers are inroduced in he daa in four differen combinaions namely C05N3 (i.e. a siuaion where 95% observaion come from N (0, 1) and 5% from N (0, 9) ), C10N3 (i.e. 90% from N (0, 1) & 10% N (0, 9)), C05N5 (i.e. 95% from N (0, 1) & 5% from N (0, 25)) and C10N5 (i.e. 90% from N (0, 1) & 10% from N (0, 25)). The subgroups (m=10) of sizes n=5 and 10 are generaed for boh he above menioned siuaions (a) and (b), asympoic limis are compued for all he EWMA conrol srucures based on, S, G and * σ for spread and locaion parameers. Afer he compuaion of he conrol limis, he number of subgroups for which he EWMA saisics fall ouside heir respecive conrol limis, are couned. The whole procedure is simulaed 10,000 imes and he proporion of subgroups ha fall ouside he limis is 7

8 calculaed for all he EWMA conrol srucures. The resuling proporions are repored in Appendix Tables 3 and 4 for spread and locaion parameers respecively. These simulaion resuls are in close agreemen wih hose of Khoo (2004) provided for EWMAVD, EWMAS, EWMAMD and EWMASM, which shows he validiy of our rouines used o obain he simulaion resuls. For ease in comparison among differen conrol schemes under discussion, graphs of proporion of ouliers agains values of λ are ploed. The graphs of columns iled N (0, 1) for n=5 and 10, CO5N3 for n=5 and C10N3 for n=10 (from Appendix Table 3) are provided here in he following Figures for scale parameer. The graphs of columns iled N (0, 1) for n=5, C1ON3 for n=5, C0N5 for n=10 and C10N3 for n=10 (from Appendix Table 4) are provided here in he following Figures for locaion parameer. Fig 3.1: Proporion of ouliers deeced by EWMAS, GVA, SdV and EWMAVD for monioring process scale parameer for N(0,1) and n=5 Proporion of ouliers G S D Lambda 8

9 Fig 3.2: Proporion of ouliers deeced by EWMAS, GVA, SdV and EWMAVD for monioring process scale parameer for N (0, 1) and n=10 Proporion of ouliers G S D Lambda Fig 3.3: Proporion of ouliers deeced by EWMAS, GVA, SdV and EWMAVD for monioring process scale parameer for C05N3 and n=5 Proporion of ouliers G S D Lambda 9

10 Fig 3.4: Proporion of ouliers deeced by EWMAS, GVA, SdV and EWMAVD for monioring process scale parameer for C10N3 and n=10 Proporion of ouliers G S D Lambda Fig 3.5: Proporion of ouliers deeced by EWMASM, GMA, SdM and EWMAMD chars for monioring process locaion parameer for N(0,1) and n=5 Proporion of ouliers G S D Lambda 10

11 Fig 3.6: Proporion of ouliers deeced by EWMASM, GMA, SdM and EWMAMD chars for monioring process locaion parameer for C10N3 and n=5 Proporion of ouliers Lambda G S D Fig 3.7: Proporion of ouliers deeced by EWMASM, GMA, SdM and EWMAMD chars for monioring process locaion parameer for C05N5 and n= Proporion of ouliers G S D Lambda 11

12 Fig 3.8: Proporion of ouliers deeced by EWMASM, GMA, SdM and EWMAMD chars for monioring process locaion parameer for C10N3 and n= Proporion of ouliers G S D Lambda The bars in he above chars represen he proporion of ouliers deeced using differen EWMA conrol schemes. The bars referred o as, G, S and D respecively represen he proporion of ouliers deeced by EWMAS, GVA, SdV and EWMAVD for scale parameer (in Figures ) and EWMASM, GMA, SdM and EWMAMD for locaion parameer ( in Figures ). From he above graphs i is clear ha: i) for in conrol siuaions (i.e. unconaminaed disribuions), he proporion of ouliers deeced (i.e. false alarm rae) by he proposed GVA do no exceed he respecive false alarm raes of EWMAS, EWMAVD and SdV chars. The same is he case wih GMA char as compared o EWMASM, EWMAMD and SdM chars; ii) for ou of conrol siuaions (i.e. conaminaed disribuions), he proporion of ouliers deeced by he proposed GVA char is higher han hose of EWMAS and SdV chars and almos equal o ha of 12

13 EWMAVD char. Similarly he proporion of ouliers deeced by he proposed GMA char is higher han hose of EWMASM and SdM chars, and almos equal o ha of EWMAMD char. All his is due o he fac ha he limis of he proposed chars based on G are less influenced by he presence of ouliers, which makes he proposed srucures of GVA and GMA chars more capable of deecing ouliers. Hence he proposed GVA char is superior o EWMAS and SdV chars and almos equally efficien o EWMAVD char for scale parameer, in erms of ouliers deecion. Also he proposed GMA char is superior o EWMASM and SdM chars and almos equally efficien o EWMAMD char for locaion parameer, in erms of ouliers deecion. 4 Conclusion The proposed GVA and GMA chars are more efficien, in erms of oulier deecion, han and S based EWMA design srucures and almos equally efficien o he Khoo (2004) Downon s esimaor based EWMA design srucures. 13

14 eferences David, H.A. (1968). Gini s mean difference rediscovered. Biomerika, 55, Gini, C. (1912). Variabilia e Muabilia, Bologna. Khoo, M.B.C. (2004). Some Conrol Chars for he Process Mean and Variance based on Downon s Esimaor, Inernaional Engineering Managemen Conference Maravelakis, P.E., Panareos, J. and Psarakis, S. (2005). An Examinaion of he obusness o Non Normaliy of he EWMA Conrol Chars for he Dispersion, Communicaions in Saisics - Simulaion and Compuaion, Volume 34, Issue 4 Ocober 2005, pages Mongomery, D. C. (2001). Inroducion o Saisical Qualiy Conrol, 4h ediion. Ng, C. H. and Case, K. E. (1989). Developmen and evaluaion of conrol char using exponenially weighed moving averages, Journal of Qualiy Technology, vol. 21, pp iaz, M. and Saghir, A. (2007). Monioring process variabiliy using Gini s mean difference, Quaniaive Managemen 4 (4), ocke, D. M. (1992). 41, pp X Q and Q chars: robus conrol chars, The Saisician, vol. Soumbos,Z.G., Sullivan, J. H. (2002). obusness o non normaliy of he mulivariae EWMA conrol char, Journal of Qualiy Technology. 34:

15 Appendix Table 1: Conrol Char Consans of G -Chars n b

16 Table 2: Facors for G- Chars Probabiliy Limi n M M M 0.01 M 0.05 M 0.10 M 0.20 M 0.25 M 0.50 M 0.75 M 0.80 M 0.90 M 0.95 M 0.99 M M

17 Table 3: Proporion of ou-of-conrol for EWMAS, GVT, SdV and EWMAVD Chars for monioring process scale parameer Sae of Conrol EWMAS n In Conrol Siuaion ou of Conrol Siuaion Disribuion N(0,1) N(0,1) C05N3 C10N3 C05N5 C10N5 C05N3 C10N3 C05N5 C10N5 λ GVT SdV EWMAVD λ λ λ

18 Table 4: Proporion of ou-of-conrol for EWMASM, GMT, SdM and EWMAMD Chars for monioring process locaion parameer Sae of Conrol EWMASM n In Conrol Siuaion ou of Conrol Siuaion Disribuion N(0,1) N(0,1) C05N3 C10N3 C05N5 C10N5 C05N3 C10N3 C05N5 C10N5 λ GMT SdM EWMAMD λ λ λ