Control Charts for Mean under Shrinkage Technique

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1 Helderma Verlag Ecoomic Quality Cotrol ISSN Vol 24 (2009), No. 2, Cotrol Charts for Mea uder Shrikage Techique J. R. Sigh ad Mujahida Sayyed Abstract: I this paper a attempt is made to study cotrol charts for the mea usig the shrikage techique. The operatig characteristic (OC) fuctio ad probability of the error of first kid of the cotrol chart with symmetrical limits are calculated. The OC curves for the shrikage techique are also studied i the case of kow value σ for the mea. Keywords: Operatig characteristic, type 1 error, miimum variace ubiased liear estimator. 1 Itroductio Cotrol charts are used to distiguish betwee variatio i a process that ca ot be ecoomically idetified ad corrected (chace causes) ad those that ca be (assigable causes). It is assumed that the patter of chace causes follows some stable probability distributio. Shewhart selected the ormal distributio relyig o (a) (b) (c) the cetral limit theorem, a series of experimets with three differet distributios, ad idustrial experiece. The cotrol limits of these traditioal charts are exact oly if the process characteristic uder cosideratio is ormally distributed. If the true process distributio is ukow ad if the cost of Type I ad Type II errors are high, the cotrol charts that are based o less restrictive distributioal assumptios are useful. I may settigs, while the process is i cotrol, the process readigs have a costat mea ad variace. I such a settig, the X chart is used to moitor the mea. For moitorig ormally distributed data, Hillier [5] pioeered the research o the effect of estimatig the cotrol limits for X ad R charts. Queseberry [8] studied the effect of the sample size o the estimated limits for X ad X charts. Other related studies ca be foud i Motgomery [7], Castillo [2], Yag et al. [15], Kag et al. [6] ad Beeya [1]. I this cotext, the shrikage estimator is very powerful estimator that has bee studied by may authors like Cui et al. [4], Chiou ad Miao [3], Xu [14], Sigh ad Saxea [10], Sigh ad Vakim [11] ad Sigh et al. [9]. I the light of the above work we ivestigate the effect of the shrikage techique o the performace of the cotrol chart for the mea. I this paper we develop the ecessary techical uderpiigs for cotrol charts for the mea uder shrikage techiques ad illustrate its performace.

2 256 J. R. Sigh ad Mujahida Sayyed 2 OC ad Type I Error of the Cotrol Charts uder Shrikage Techique Thompso [12, 13] has cosidered the advisability of shrikig a Miimum Variace Ubiased Liear Estimator (MVULE) towards the atural origi µ 0 by multiplyig it by a shrikig factor c. The atural origis may arise for ay oe of a umber of reasos; e.g., we are estimatig a parameter µ ad, 1. we believe µ 0 is close to the true value of µ, or 2. we fear that µ 0 may be ear the true value of µ; i.e., some bad thig may happe if i fact µ µ 0 ad we do ot kow about it. Thompso [12, 13] cosidered the populatio mea ad proposed the estimator: ˆµ s = cˆµ + (1 c)µ 0 = c(ˆµ µ 0 ) + µ 0 (1) where µ 0 is believed to be close to the true value µ ad ˆµ is a usual estimator such as the Maximum Likelihood Estimator (MLE). The value of c is obtaied by miimizig: [ E[(ˆµ s ˆµ) 2 ] = E (c(ˆµ µ 0 ) + (µ 0 µ)) 2] (2) Optimizatio yields: Such that ĉ = (ˆµ µ 0 ) 2 (ˆµ µ 0 ) 2 + ˆV [ˆµ] (3) Thus the shrikage estimator ˆµ s for µ is: ˆµ s = (ˆµ µ 0 ) 2 (ˆµ µ 0 ) 2 + ˆV [ˆµ] (ˆµ µ 0) + µ 0 (4) The above illustrated techique modifyig a estimator is called shrikage. Let X be a radom variable with probability desity fuctio such that both E[X] = µ ad V [X] = σ 2 exist. The sample mea X is MVULE for µ. We shall cosider the estimator c X, where 0 < c 1. Such shrikage is reasoable i the case where the prior value of the populatio mea is zero. Usig the Mea Square Error (MSE) as risk fuctio, the value of c is determied by miimizig: MSE[c X] = E[(c X µ) 2 ] (5) We obtai: c = µ2 µ 2 + σ2 (6) ad

3 Cotrol Charts for Mea uder Shrikage Techique 257 MSE[c X] = = = ( µ 2 µ 2 + σ2 µ 2 µ 2 + σ2 ) 2 ( ) ( µ 2 + σ2 µ 2 2 µ 2 + σ2 σ2 ) µ 2 + µ 2 σ 2 + ν 2 (7) where ν = σ µ is the coefficiet of variatio (CV). The cotrol chart for the mea is set up by drawig the cetral lie at the process average µ ad the cotrol limits at µ±k σ, where σ is the process stadard deviatio, is the sample size ad k is a fixed umber. The OC fuctio gives the probability that the cotrol chart idicates the process average as beig µ whe it is actually ot µ but µ σ = µ + γ (8) + ν 2 It is derived by itegratig the distributio of mea with µ as the process average betwee the limits of the cotrol chart. The distributio of the sample mea X is give by ( ) ν 2 x µ f X(x) = φ σ (9) σ +ν 2 where φ is the desity fuctio of the stadardized ormal distributio. The OC fuctio is obtaied, after replacig µ i (9) by µ ad itegratig the desity fuctio betwee the limits of the cotrol chart. After some simplificatio the OC as fuctio of γ is as follows: L(γ) = Φ(kT + γ) + Φ(kT γ) 1 (10) with ( + ν 2 ) 1 2 T = (11) ad γ ragig betwee ±5.0. The probability of a error of the first kid gives the probability of searchig for assigable causes whe i fact, there are o such causes, or i other words, it is the probability that the sample value lies outside the cotrol limits whe the process average ad variatio remai uchaged. This false alarm probability deoted α is give by: α = 1 µ+k σ µ k σ f X (x)dx = 2Φ( KT ) (12)

4 258 J. R. Sigh ad Mujahida Sayyed where Φ is the distributio fuctio of the stadardized ormal distributio. The OC fuctio ad the probability of a error of the first kid for a X chart for Shewhart with kow σ are give by: L(γ) = Φ(k + γ) + Φ(k γ) 1 (13) α = 2Φ( k) (14) The values of the error of the first kid are calculated ad preseted i Table 1 for k = 2,3, = 5,10,15 ad ν = 2,4,6 for usig the shrikage estimator. From (14) the use of 2σ ad 3σ cotrol limits for the Shewhart chart whe the stadard deviatio σ is kow will produce a error of the first kid as α = ad α = respectively. The values displayed i Table 1 clearly idicate that for k = 2,3, the error of the first kid decreases as the coefficiet of variatio icreases. It is also see that whe ν 4 for k = 2,3 ad = 5,10,15 the probability that the sample values lie outside the cotrol limits will be egligible. Table 1: Values of error of first kid uder Shrikage Estimator. Sample size k σ ν = 2 ν = 4 ν = The calculated values of the OC fuctio as fuctio of γ for k = 3, = 5,10,15 are cotaied i Table 2. From Table 2 it is obvious that the OC icreases as the coefficiet of variatio icreases. For v = 2,4 ad γ = ±2.0 the OC fuctios adopts values of ad , respectively. While from (13) the Shewhart OC fuctio for = 5, k = 3 ad γ = ±2.0 adopts the , which is a good support for selectig the cotrol chart for the mea usig the shrikage estimator. 3 Coclusio The values show i Table 2 clearly idicate that the use of the shrikage estimator may be very beeficial i particular if there are some prior iformatio available about the value of the process mea. Actually, this is the case for most idustrial productio processes. There is aother very ofte give situatio i which the shrikage techique may be used. This is whe the cetral lie ad the cotrol limits are based o a target value.

5 Cotrol Charts for Mea uder Shrikage Techique 259 I both cases the i-cotrol state of the process ca be very reliably judged usig the shrikage estimator yieldig a much better protectio agaist costly false alarms which may cosiderably reduce the process profit. Note that for implemetig the shrikage techique o complicated chages must be made withi the already available SPC. Table 2: Values of OC uder Shrikage Estimator. Sample size γ σ ν = 1 ν = 2 ν = 3 ν = 4 ν = 5 ν = 6 5 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

6 260 J. R. Sigh ad Mujahida Sayyed Ackowledgemet: The authors are highly thakful to the head i statistics Prof. H.O. Sigh for his costat ecouragemet. Refereces [1] Beeya, J.C. (2001): Performace of Number-betwee g type Statistical Quality Cotrol Charts for Moitorig Adverse Evets. Health Care Maagemet Sciece 4, [2] Castillo, D.E. (2002): Statistical Process Adjustmet for Quality Cotrol. Wiley & So, New York, NY. [3] Chiou, P. ad Miao, W. (2007): Shrikage Estimatio for the Differece betwee a Cotrol ad Treatmet mea. Joural of Statistical Computatio ad Simulatio 77, [4] Cui, X., Hwag, G.J.T., Qiu, J., Blades, N.J. ad Churchill, G.A. (2005): Improved Statistical Test for Differetial Gee Expressio by Shrikig Variace Compoets Estimates. Biostatistics 6, [5] Hillier, F.S.(1969): X ad R Chart Cotrol Limits Based o a Small Number of Subgroups. Joural of Quality Techology 1, [6] Kag, C. W., Lee, M.S., Seog, Y.J. ad Hawkis, D.M. (2007): A Cotrol Chart for the Coefficiet of Variatio. Joural of Quality Techology 39, [7] Motgomery, D.C. (2001): Itroductio to Statistical Quality Cotrol. 4th Ed. Joh Wiley & Sos, New York, NY. [8] Queseberry, C. P. (1993): The Effect of Sample Size o Estimated Cotrol Limits for X ad X Cotrol Charts. Joural of Quality Techology 25, [9] Sigh, H.P., Saxea, S., Alle, J., Sigh, S. ad Samaradche F. (2002): Estimatio of Weibull Shape Parameter by Shrikage towards a Iterval uder Failure Cesored Samplig, Radomess ad Optimal Estimatio i Data Samplig. America Research Press, Rehoboth 1, [10] Sigh, H.P. ad Saxea, S. (2003): A Class of Shrikage Estimators for Variace of a Normal Populatio. Brazilia Joural of Probability ad Statistics 17, [11] Sigh, H.P. ad Vakim, C. (2008): Some Classes of Shrikage Estimators for Estimatig the Stadard Deviatio towards a Iterval of Normal Distributio. Model Assisted Statistics ad Applicatios 3, [12] Thompso, J.R. (1968): Some Shrikage Techiques for Estimatig the Mea. Joural of America Statistical Associatio 63,

7 Cotrol Charts for Mea uder Shrikage Techique 261 [13] Thompso, J.R. (1968): Accuracy Borrowig i the Estimatio of the Mea by Shrikage to a Iterval. Joural of America Statistical Associatio 63, [14] Xu, S. (2007): Derivatio of the Shrikage Estimator of Quatitative Trait Locus Effects. Geetics 177, [15] Yag, Z., Xie, M., Kuralmai, V. ad Tsui, K.L. (2002): O the Performace of Geometric Charts with Estimated Cotrol Limits. Joural of Quality Techology 34, J. R. Sigh ad Mujahida Sayyed School of Studies i Statistics Vikram Uiversity Ujjai , M.P. Idia mujahida.sayyed@rediffmail.com

point estimator a random variable (like P or X) whose values are used to estimate a population parameter

point estimator a random variable (like P or X) whose values are used to estimate a population parameter Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity