A Simulation Study of the Relative Efficiency of the Minimized Integrated Square Error Estimator (L2E) For Phase I Control Charting

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1 Joural of Moder Applied Statistical Methods Volume 10 Issue 1 Article A Simulatio Study of the Relative Efficiecy of the Miimized Itegrated Square Error Estimator (L2E) For Phase I Cotrol Chartig Joh N. Dyer Georgia Souther Uiversity, jdyer@georgiasouther.edu Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical Theory Commos Recommeded Citatio Dyer, Joh N. (2011) "A Simulatio Study of the Relative Efficiecy of the Miimized Itegrated Square Error Estimator (L2E) For Phase I Cotrol Chartig," Joural of Moder Applied Statistical Methods: Vol. 10 : Iss. 1, Article 27. DOI: /jmasm/ Available at: This Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.

2 Joural of Moder Applied Statistical Methods Copyright 2011 JMASM, Ic. May 2011, Vol. 10, No. 1, /11/$95.00 A Simulatio Study of the Relative Efficiecy of the Miimized Itegrated Square Error Estimator (L2E) For Phase I Cotrol Chartig Joh N. Dyer Georgia Souther Uiversity, Statesboro, GA, USA Parameter estimates used i cotrol chartig, the sample mea ad variace, are based o maximum likelihood estimatio (MLE). Ufortuately, MLEs are ot robust to cotamiated data ad ca lead to improper coclusios regardig parameter values. This article proposes a more robust estimatio techique; the miimized itegrated square error estimator (L2E). Key words: Phase I cotrol chartig, SPC, L2E, MLE, parameter estimatio. Itroductio Process moitorig usig cotrol charts is the most commo method used i statistical process cotrol (SPC). I the literature two phases of cotrol chartig are distiguished: Phase I ad Phase II cotrol chartig. Phase I cotrol chartig cosists of two stages: Stage 1, the retrospective stage, ad Stage 2, the prospective stage (Koig & Does, 2000). Durig Phase I, the appropriate cotrol chartig methods must be determied, ad the appropriate process parameters estimated (Joes, 2002). The techiques associated with Phase I iclude aalyzig sample data usig gauge repeatability ad reliability (GR&R) studies to ivestigate measurig system accuracy ad variability, usig capability idices to determie if a process is capable of producig withi specificatio, usig histograms ad probability plots to verify distributioal assumptios, usig outlier detectio tools (Ramsey & Ramsey, 2007) to detect ad remedy special causes of variatio i the process, ad obtaiig reliable estimates of the process parameters (Motgomery, 1997). Thus, part of Phase I ca Joh N. Dyer is a Associate Professor of Iformatio Systems i the College of Busiess Admiistratio, with research streams i statistics, iformatio systems, ad fiace. him at: jdyer@georgiasouther.edu. be cosidered a data editig process wherei outlyig or cotamiated data are removed from the sample to eable estimatio of the appropriate process parameters. Phase II cotrol chartig is the actual use of the desired cotrol chart to moitor ad cotrol a process i regards to chages i the process parameters (Woodall, 2000), distributioal chages, ad the radomess of the process. The costructio of a Phase II cotrol chart is based o the parameter estimates obtaied i Phase I. Commo Phase II cotrol charts iclude the followig (applied to either idividual process observatios or subgroups): the Shewhart-type, the expoetial weighted movig average (EWMA), ad the cumulative sum (CUSUM), amog others (Dyer, Adams & Coerly, 2003). It is crucial that the data collected i Phase I are good data, meaig, free from outliers (cotamiated data) ad represetative of typical process data with o special causes of variability. Cotamiated data ca lead to ureliable parameter estimates which, i tur, lead to improper coclusios regardig distributio assumptios, process capability ad cotrol chart desig. The use of most cotrol charts requires the estimatio of the mea, µ, ad stadard deviatio, σ (or a fuctio thereof), of the i-cotrol (IC) process. A process is said to be IC whe oly commo cause variatio is preset, otherwise it is cosidered out-of-cotrol (OC). 300

3 JOHN N. DYER The estimates used for the true process mea, µ, ad stadard deviatio, σ, are typically sample statistics, specifically, the sample mea, x, ad the sample stadard deviatio, s, obtaied from the good data. The sample statistics used i Phase I cotrol chartig are based o the priciple of maximum likelihood estimatio, that is, the sample mea ad sample variace are maximum likelihood estimates (MLEs) of µ ad σ 2, respectively. Some of the practical deficiecies of MLEs are their lack of resistace to outliers ad their geeral o-robustess with respect to model misspecificatio (Rudemo, 1982). For example, cosider the followig 5 data values: 4, 5, 6, 7 ad 100, ad estimates based o MLEs (the sample mea ad stadard deviatio). The sample mea ad variace of all five data values are 24.4 ad 1,781, respectively. If the data value of 100 is idetified as a outlier ad removed, the the ew MLEs for the mea ad variace are 5.5 ad 1.69, respectively. Although the magitude of the outlier is absurdly large, it is obvious that the MLEs caot resist the ifluece of the large value. The values of the ew MLEs are dramatically differet, but they are more represetative of the true ature of the data values. Recall, oe emphasis of Phase I cotrol chartig is to idetity ad remove outliers, hece providig reliable estimates of the true process parameters. It should also be oted that, although MLEs are oresistat to outliers, they are typically preferred because of their costructive ature as well as their asymptotic optimality properties. To overcome the deficiecies of MLEs ad better eable the practitioer to obtai reliable parameter estimates, this article proposes the use of a specific oparametric desity estimatio techique usig a form of the itegrated square error (ISE) estimator, also called L2E. Scott (2001) provides the theoretical costruct of the L2E ad the iterested reader is ecouraged to review the article. I this study, the L2E techique is show to provide parameter estimates that are robust to cotamiated data ad to be costructive i ature. For example, cosiderig the full data set previously discussed, the L2E estimates of the mea ad variace (obtaied through a simply executed Excel spreadsheet algorithm) are 5.5 ad 2.25, respectively. Notice how the L2E estimates are robust to the iclusio of the outlier. Although Scott (2001) itroduced the L2E as a estimator of process parameters, evideces the estimator s robustess to outliers i large data sets, ad shows its costructive ature, this research explores the properties of the L2E as a alterative estimator to MLE across a broad rage of sample sizes ad a broad rage of data cotamiatio affectig the mea aloe, the variace aloe, ad the mea ad variace together. This study also compares the absolute differece betwee MLE ad L2E over the rage of sample sizes ad cotamiatios (mea, variace, ad mea-variace), ad shows that the L2E estimates are as good as MLE estimates i almost all cases. Additioally, the relative efficiecy of MLE versus L2E estimates is compared across all cases ad it is show that the L2E estimates are more robust i most cases tha MLE estimates. The literature related to Phase I cotrol chartig for uivariate processes is limited. Readers are referred to Chou & Champ (1995), Koig & Does (2000), Newto & Champ (1997), Sulliva & Woodall (1996), ad Woodall (2000). Surprisigly, the focus of the majority of the literature is devoted to methods for multivariate Phase I SPC (Alt & Smith, 1988; Sulliva & Woodall, 1994; Sulliva, Barrett & Woodall, 1995; Woodall, 2000). Overview of the Phase I Eviromet Durig Phase I, process data are collected ad aalyzed to eable Phase II cotrol chartig. After the data are collected, the SPC method ca be cosidered as the combiatio of Phase I ad Phase II applicatios. The geeral SPC method ca be thought of i terms of four desig steps. The first three steps occur i the Phase I eviromet ad step 4 occurs i the Phase II eviromet. Step 1: Idetify the desired cotrol chart (for moitorig idividual observatios or subgroup data), the required parameters, ad the desired IC average ru legth (ARL). The IC ARL is the average umber of samples take util a IC 301

4 EFFICIENCY OF MINIMIZED INTEGRATED SQUARE ERROR ESTIMATOR (L2E) process idicates a statistic outside of the cotrol limits. Step 2: Determie the subgroup size,, ad the umber of subgroups, m, which will be used to estimate the parameters of the IC process. Obtai a referece sample of m subgroups of size 1 observatios. Step 3: Esure that the referece sample is represetative of the IC process, simultaeously estimatig the required cotrol chart parameters usig a robust techique, such as, L2E (recommeded herei) or a iteratively robust techique like MLE. Step 4: Apply the desired cotrol chart to a ogoig process, moitorig, cotrollig ad adjustig the process as it evolves. I Step 1, the typical choice of cotrol chart is related to the desire for quick detectio of extreme chages i process parameters versus evetual detectio of mior chages i process parameters (Dyer, Adams & Coerly, 2003; Li & Adams, 1996). The Shewhart-type cotrol charts are commoly used for the former, ad the EWMA ad CUSUM cotrol charts are used for the latter. The choice of the IC ARL i Step 1 ivolves practical ad ecoomic cosideratios, depedig largely o the costs associated with false alarms versus cocealmet of true process chages (Dyer, Adams & Coerly, 2003). I Step 2, the subgroup size () is a fuctio of the samplig frequecy, the process output rate, ad practical cosideratios ad limitatios regardig time ad costs. Marsaglie, Maclare & Bray (1964) provide a discussio of the selectio of a appropriate subgroup size () ad samplig frequecy to desig cotrol charts. The choice of the umber of subgroups (m) is most likely a ecoomic cosideratio (Joes, 2002). If cotamiated data exist i the referece sample, the parameter estimates obtaied ca be adversely affected if MLEs are used to obtai parameter estimates (L2E to a lesser degree). Small referece samples ted to magify the adverse effects of estimatio. A widely accepted heuristic is that m = 30 subgroups from a process will provide reasoable estimates (Joes, 2002); Queseberry (1993) suggests at least m = 100 subgroups of size = 5 to estimate the parameters for the Shewhart-type cotrol chart. Joes, Champ & Rigdo (2001) showed that a m much greater tha 100, up to m = 400, is ofte required whe desigig a EWMA cotrol chart. I Step 3, the referece sample obtaied i Step 2 is aalyzed i order to estimate the ukow parameters ad to determie the state of the process (IC versus OC). This is also the stage whe distributioal ad radomess assumptios are verified, as well as whe GR&R ad capability studies are coducted. Cocerig parameter estimatio, if MLEs are used, the resultig values are the estimates used to costruct a iitial cotrol chart with limits set accordig to the desired IC ARL i Step 1. I Stage 2, the cotrol charts are used for prospective moitorig of the referece sample to determie departures from the estimated parameters. The cotrol charts are primarily used to detect cotamiated data or oradom process output, that is, data resultig from special cause variatio. Step 3 is ofte a iterative process, wherei cotamiated data are idetified (to the degree possible) ad removed usig a cotrol chart based o the iitial parameter estimates (MLEs). Ay cotamiated data idetified are ivestigated ad removed, ew MLEs are obtaied, a ew cotrol chart is costructed usig the MLE values ad more cotamiated data are removed. The process of parameter estimatio ad cotrol chart removal of cotamiated data cotiues util sufficiet experiece has bee accumulated so that the IC parameters are effectively cosidered to be kow through estimatio. It should also be oted that if a large degree of cotamiated data exist i the referece sample (as a percet of the sample size), or the magitude of cotamiated data is large (measured i terms of shifts i the process mea or variability), the the iitial cotrol limits may be iflated to a poit where the cotamiated data are hidde ad uidetifiable. If this is the case, the Phase II parameter 302

5 JOHN N. DYER estimates will be ureliable. If L2E estimates are used istead, it will be show that the iterative process i Step 3 might be miimized by providig a more robust set of parameter estimates i the first iteratio, which will lead to a more robust set of cotrol limits, thus eablig more efficiet detectio ad removal of cotamiated data. Methodology The L2E Estimatio Techique The L2E estimatio criterio for the two-parameter ormal desity techique requires the miimizatio of the L2E fuctio with respect to the parameters µ ad σ. (See Scott (2001) for the derivatio of the geeral L2E criterio ad specificatio of the twoparameter ormal desity.) Suppose a sample of size 1 is draw from a ormal distributio with mea, µ, ad stadard deviatio, σ. Let the sample data be represeted by x 1, x 2,, x, ad let the uivariate ormal desity be deoted by φ(x µ, σ). The miimizatio of the ormal L2E fuctio (equatio 1) with respect to µ ad σ produces the L2E estimates, μ, σ, that is, the estimatio criterio is show as: 1 2 L2E μ,σ =arg mi - φ ( xi μ,σ ). μ,σ 2σ π i=1 (1) Observe that the L2E miimizes a fuctio of the sum of the desities; however, the MLE ca be show to maximize a fuctio of the product of the desities. For values of x extremely distat from µ, the desity value approaches zero. As a result, the L2E utilizes oly the largest portio of the data that matches the model (good data), that is, x values located withi a reasoable distace of µ ± 3σ. I effect, the L2E criterio igores cotamiated data, hece geerally providig more robust parameter estimates. Because MLE must accout for all the data, the fits ofte blur the distictio betwee good data ad cotamiated data (Scott, 2001). I cases wherei there are o cotamiated data, the L2E ad MLE estimates are early equal. It ca be show through cosistecy theory that, for a large sample of ucotamiated data, MLE is a very good estimator (Mood, Graybill & Boes, 1970); other estimators, such as the L2E may be just as good, but ot better. I this study the L2E is show to be just as good whe the referece sample is ucotamiated ad better i almost all simulated cases whe cotamiatio exists. Results Compariso with MLEs Ufortuately there are few example data sets that cover the rage of samples sizes ad cotamiatio types ad levels described herei. Motgomery (1997) provides some of the most refereced data sets i SPC research, but ufortuately oe of these have sufficiet examples required to cover the 96 cases of sample sizes ad cotamiatio types ad levels described i this article. Simulatio results are therefore used to ivestigate the behavior of the L2E estimates across a broad rage of sample sizes as well as types ad levels of data cotamiatio. I lieu of borrowig a example data set, the simulatio results are used to reveal the behavior of the L2E estimates over a broad rage of cases ad a example applicatio is provided to assist the user i applyig the L2E techique. Regardig the simulatio results, Tables 1a ad 1b reveal average L2E ad MLE estimates for µ ad σ 2 (σ 2 reported as σ) based o averagig 10,000 simulatios of = 100 ormal pseudo-radom variables represetig differig levels ad degrees of good versus cotamiated data. (A complete descriptio of the simulatio desig is provided i the Appedix.) The good data (IC process) are radom variables represetig a ormal (µ = 0, σ = 1) process, N(0, 1). The cotamiated data are draw from a ormal process with parameters that vary from the IC process. Levels of cotamiatio refer to the umber of cotamiated data values (c) i a sample of size = 100 ad degrees of cotamiatio refer to whether the cotamiated data has experieced a mea shift aloe, a shift i the stadard deviatio aloe, or a shift i both the mea ad stadard deviatio. Cotamiatio levels i Tables 1a ad 1b correspod to = 5, 15, 25 ad 45. Degrees of cotamiatio 303

6 EFFICIENCY OF MINIMIZED INTEGRATED SQUARE ERROR ESTIMATOR (L2E) correspod to the followig shifts (for c = 5, 15, 25 ad 45): Mea shifts (aloe) of µ = 0.5, 1.0, 2.0, ad 3.0 (16 cases) Stadard deviatio shifts (aloe) of σ = 1.5, 2.0, 2.5, 3.0 (16 cases) Simultaeous mea ad stadard deviatio shifts represetig combiatios of all mea ad stadard deviatio shifts aloe (64 cases). Tables 1a ad 1b display simulatio results providig 96 comparisos for average L2E versus MLE estimates of µ ad σ. For the IC process data ( = 100 radom variables geerated from a N(0, 1) process, the resultig simulatio based estimates are μ (L2E) = , = , σ (L2E) = , ad σ (MLE) = I Tables 1a ad 1b the estimates of µ ad µ are show as μ (L2E),, σ (L2E) ad σ (MLE). For all mea shifts ad stadard deviatio shifts aloe, the mathematical expectatio ad stadard deviatio (based o the levels ad degrees of cotamiatio) match the simulated MLE results. I derivig the expected value, let X be the mixture of two ormally distributed samples of size, where X -c is the ucotamiated distributio with E(X -c ) = µ -c, ad X c is the cotamiated distributio with E(X c ) = µ c (recall, c is the umber of cotamiated data values i the combied sample of size ). I this case, the E(X ) is the weighted average expectatio of each distributio of data, where the weights are the sample sizes from each distributio relative to the total sample size. Thus, c c E( X ) = E( X c) + E( X c). I the case where the ucotamiated data distributio has E(X -c ) = 0, c the E( X ) = E( X c). For example, for X c ~N(3, 1) where = 100 ad c = 45, the 45 E( X ) = (3) = This value matches 100 the simulated value give by μ (MLE) i Table 1b. All simulated values for μ (MLE) (for both mea ad stadard deviatio shifts aloe) match the mathematical expectatios. This is expected give that is locatio ivariat to distributioal chages due to shifts i either the mea or stadard deviatio. The same ca be observed for the stadard deviatio estimates, σ (MLE), where σ c c ( X ) = Var ( X c) + Var ( X c) E X = ad ( c) whe ( ) 0 Var X = 1. All simulated values for σ (MLE) (for stadard σ. This is deviatio shifts aloe) match ( X ) expected because σ (MLE) is scale ivariat to distributioal chages due to shifts i the mea aloe or the stadard deviatio aloe. For cases where the mixed distributio has experieced both a mea shift ad a stadard deviatio shift, σ (MLE) is ot scale ivariat; hece, the variace is ot the weighted average of mixed variace compoets. Simulatio Result Compariso with MLEs The simulatio results reveal that i all cases {abs( μ (L2E) µ) abs( µ)}, ad i 95% of cases {abs(σ (L2E) σ) abs(σ (MLE) σ)}. That is, the L2E estimates i almost all cases are as good (ad ofte much better) as the MLE 304

7 JOHN N. DYER estimates. This attests to the cotetio that the L2E estimators are as robust, or more robust, tha MLE estimators. Observe i Tables 1a ad 1b that μ (L2E) is robust for most shifts i µ c, for all c.45, ad more robust tha i all cases. The relative efficiecy measures i Tables 2a ad 2b idicate that the worst cases are those with large c, for µ c 2. Whe µ -c = 0, the relative efficiecy for either mea estimator is defied as RE µ = 1- abs( μ ) where μ is the estimate of µ -c ad is the mea of the IC process. Table 3 displays the percet frequecy distributio of relative efficiecy measures for all cases simulated. Notice that, for mea shifts aloe, 57% of μ (L2E) have RE µ > 0.80 versus 44% of. For shifts i the mea ad stadard deviatio (simultaeously), the frequecy of RE µ > 0.80 is 81% for μ (L2E) ad oly 43% for. It appears that μ (L2E) is most robust whe both a mea ad stadard deviatio shift has occurred. The relative efficiecy for a stadard deviatio estimate is defied as abs σ RE σ = 1 σ ( ) σ c c, where σ is the estimate of σ -c, the stadard deviatio of the IC process. Because σ -c = 1 i all simulatio cases, RE σ =1-abs(σ -1). Agai, observe i Tables 1a ad 1b that σ (L2E) is c robust for most shifts i σ c, for all.45, ad particularly whe µ -c < 1. Notice also that σ (L2E) is more robust tha σ (MLE) i 95% of all cases. It appears that σ (MLE) is less robust whe all of µ -c, σ -c, ad c are large. The relative efficiecy measures i Tables 2a ad 2b also idicates that these are the worst cases for σ (L2E). Note i Table 3 that, for stadard deviatio shifts aloe, 87% of σ (L2E) have RE σ > 0.80 versus 50% of σ (MLE). For shifts i both the mea ad stadard deviatio (simultaeously), the frequecy of RE σ > 0.80 is 69% for σ (L2E) ad oly 31% forσ (MLE). It appears that σ (L2E) is more robust whe oly a shift i the stadard deviatio has occurred. L2E Applicatio Example As oted, oe advatage of usig MLE is its costructive ature. I other words, it is simple to average a collectio of data values or calculate the stadard deviatio. The L2E estimates are also costructive i ature, but require optimizatio techiques. Specifically, the L2E fuctio give by equatio 1 must be formulated ad miimized subject to costraits. This ca be readily accomplished i a spreadsheet eviromet with little or o kowledge of programmig or miimizatio techiques. The authors suggest usig Microsoft Excel ad the spreadsheet add-i Solver. The data ca be displayed i the spreadsheet, the L2E fuctio ca be formulated usig the data ad fuctios of the data as iput, ad the Solver fuctio ca be ivoked to provide the L2E estimates via Solver s built-i optimizatio algorithm. The data ca represet idividual observatios or subgroup averages. If idividual observatios are used, the the resultig L2E estimates are those for process µ ad σ. If subgroup averages are used, the resultig L2E estimates are those for µ ad σ (stadard error of the mea, SE). I the latter case, multiplyig the estimate of SE by yields the estimate for σ. For practitioers familiar with optimizatio, the L2E estimatio problem ca be 305

8 EFFICIENCY OF MINIMIZED INTEGRATED SQUARE ERROR ESTIMATOR (L2E) CSS Table 1a: L2E ad MLE Estimates of µ ad σ µ σ μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) viewed i the istructioal form give by objective: miimize 1 2 L2E = - φ xi μ,σ 2σ π i=1 by chagig the valuesμ, σ subject costraits: σ > 0. Figure 1 displays the author s spreadsheet i fuctioal form, before usig to 306

9 JOHN N. DYER Solver to miimize the L2E fuctio. The data values 4, 5, 6, 7, 100 are iput ito colum B, cells B11 to B15. The MLE sample mea ad stadard deviatio, from the MLE variace, (24.4, 42.7) are calculated ad displayed i colum A, cells A5 ad A6, respectively, usig the built-i Excel fuctio formulas show i Figure 2. Figure 2 displays the same spreadsheet i formula/fuctio view, allowig replicatio of cell formulas by the practitioer. Figure 1, colum A, cells A11 to A15, cotai the calculated ormal probability desity fuctio (Npdf) values resultig from the built-i Excel fuctio show i Figure 2. Because the Npdf fuctio requires iput values for the mea ad stadard deviatio, the MLE estimates are iitially used, ad these values are temporarily iput ito the L2E estimate cells, colum B, cells B5 ad B6. Cells B5 ad B6 will evetually be overwritte ad cotai the L2E estimates, as provided by Solver. Figure 1, cell A2, displays the L2E fuctio value that is to be miimized, ad Figure 2 displays the formula give by equatio 1 as a fuctio of both the CSS Table 1b: L2E ad MLE Estimates of µ ad σ µ σ μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE)

10 EFFICIENCY OF MINIMIZED INTEGRATED SQUARE ERROR ESTIMATOR (L2E) Table 2a: Relative Efficiecy of L2E ad MLE Estimates of µ ad σ CSS µ σ μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) sample size () i cell B8 ad the summed Npdf values. Prior to ivokig the Solver fuctio, the L2E fuctio value (show i Figure 1) is calculated usig the MLE mea ad stadard deviatio, but referecig the cells for the L2E mea ad stadard deviatio. Figure 3 displays the Solver dialogue box referecig (1) the miimized L2E value cell (A2) as the target cell to miimize, (2) the cells to be chaged to produce the miimum L2E value (B5 ad B6), ad (3) the costrait requirig the stadard deviatio to be o-egative. Selectig the Solve butto ivokes Solver to produce the L2E estimates of µ ad σ whose values will overwrite the MLE values temporarily stored i cells B5 ad B6. After solvig for the L2E estimates, the actual value of the miimized L2E fuctio is of o practical use ad ca be discarded. The L2E estimates of the mea ad stadard deviatio (based o this example) are 5.5 ad 1.5, respectively. 308

11 JOHN N. DYER CSS Table 2b: Relative Efficiecy of L2E ad MLE Estimates of µ ad σ µ σ μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) Table 3: Percet Frequecy of L2E ad MLE Estimates of µ ad σ withi a Rage of Relative Efficiecy Rage of Relative Efficiecy µ Shifts Aloe σ Shifts Aloe µ ad σ Shifts μ (L2E) σ (L2E) σ (MLE) μ (L2E) σ (L2E) σ (MLE) % 19% 56% 19% 67% 19% 47% 17% % 25% 31% 31% 14% 25% 23% 14% % 13% 6% 13% 6% 13% 9% 20% % 6% 6% 19% 5% 6% 13% 19% % 13% 0% 13% 3% 13% 5% 19% < % 18% 0% 6% 5% 24% 3% 11% 309

12 EFFICIENCY OF MINIMIZED INTEGRATED SQUARE ERROR ESTIMATOR (L2E) Figure 1: Fuctioal Form Excel Spreadsheet (Prior to usig Solver fuctio) Figure 2: Formula/Fuctio View Excel Spreadsheet Figure 3: Solver Dialogue Box Referecig the Miimized L2E Value Cell Coclusio The importace of Phase I cotrol chartig was discussed, particularly the estimatio of appropriate parameters to eable Phase II cotrol chartig. The geeral SPC method was show to be a collectio of steps that iclude both Phase I ad Phase II cotrol chartig. For the Phase I eviromet, the miimized itegrated square error estimator, L2E, was itroduced as a robust parameter estimatio techique ad suggested as a alterative to MLEs. Regardig maagerial implicatios, the L2E estimatio techique was described ad show to be easily costructed ad applied i a spreadsheet eviromet. It was also show to be a robust alterative to MLE estimatio ad just as simple to apply. The study also provided isights to the importace of clea data whe costructig cotrol charts based off of the Phase I processes ad how the L2E estimator ca facilitate robust parameter estimatio required i SPC applicatios. A simulatio study revealed that the L2E estimates of µ ad σ for a ormal distributio are as good, ad i most cases better, tha MLE estimates whe the referece sample is cotamiated by shifts i the mea, the variace, or both the mea ad variace. Tables based o the simulatio results compare the absolute ad relative performace of both the L2E ad MLE estimators. Fially, a example was provided to eable a SPC practitioer, with little or o kowledge of programmig or optimizatio, to readily apply the L2E techique. Although this article discussed the applicatio of L2E estimators i the SPC eviromet (assumig a uivariate ormal distributio), the techique ca also be adapted to eable robust parameter estimatio whe discrete (Poisso) or multivariate processes are to be moitored ad cotrolled. Additioally, the L2E is oly oe of several oparametric desity estimators that ca be cosidered i the Phase I eviromet. Other estimators that might be of research iterest iclude M- Estimators ad estimators based o Helliger s distace criterio. 310

13 JOHN N. DYER Refereces Alt, F. B., & Smith, N. D. (1988). Multivariate process cotrol. I Hadbook of Statistics 7, P. R. Krishaiah & C. R. Rao (Eds.), New York: Elsevier. Bowma, A. W. (1984). A alterative method of cross-validatio for the smoothig of desity estimates. Biometrika, 71, Chou, S., & Champ, C. W. (1995). A compariso of two phase I cotrol charts. Proceedig of the Quality ad Productivity Sectio of the America Statistical Associatio, Dyer, J. N., Adams, B. M., & Coerly, M. D. (2003). The reverse movig average cotrol chart ad comparisos of forecast based moitorig schemes. Joural of Quality Techology, 35(2), Joes, L. A. (2002). The statistical desig of EWMA cotrol charts with estimated parameters. Joural of Quality Techology, 34, Joes, L. A., Champ, C. W., & Rigdo, S. E. (2001). The performace of expoetially weighted movig average cotrol charts with estimated parameters. Techometrics, 43, Kiderma, A. J., & Ramage, J. G. (1976). Computer geeratio of ormal radom variables. Joural of the America Statistical Associatio, 71, Koig, A. J., & Does, R. J. M. M. (2000). CUSUM charts for prelimiary aalysis of idividual observatios. Joural of Quality Techology, 32, Li, W. S. W., & Adams B. M. (1996). Combied cotrol charts for forecast-based moitorig schemes. Joural of Quality Techology, 28, Marsaglia, G. (1964). Geeratig a variable from the tail of a ormal distributio. Techometrics, 6, Marsaglia, G., & T. A. Bray (1964). A coveiet method for geeratig ormal variables. SIAM Review, 6, Marsaglia, G., MacLare, M. D., & Bray, T. A. (1964). A fast procedure for geeratig ormal radom variables. Commuicatios of the ACM, 7, Motgomery, D. C. (1997). Itroductio to statistical quality cotrol, 3 rd Ed. New York, NY: Joh Wiley. Mood, M. M., Graybill, A. G., & Boes, D. C. (1970). Itroductio to the theory of statistics, 3 rd Ed. New York, NY: McGraw-Hill. Newto, P. B., & Champ, C. W. (1997). Probability limits for Shewhart Phase I xbarcharts. Proceedig of the Southeast Decisio Scieces Istitute Aual Coferece, Atlata, Georgia, Queseberry, C. P. (1993). The effect of sample size o estimated limits for X ad X cotrol charts. Joural of Quality Techology, 25, Ramsey, P. H., & Ramsey, P. P. (2007). Optimal trimmig ad outlier elimiatio. Joural of Moder Applied Statistical Methods, 6(2), Rudemo, M. (1982). Empirical choice of histogram ad kerel desity estimators. Scadiavia Joural of Statistics, 9, Scott, W. S. (2001). Parametric statistical modelig by miimum itegrated square error. Techometrics, 43, Sulliva, J. H., & Woodall, W. H. (1996). A cotrol chart for prelimiary aalysis of idividual observatios. Joural of Quality Techology, 28, Sulliva, J. H., & Woodall, W. H. (1994). A compariso of multivariate cotrol charts for idividual observatios. Techical Report, Applied Statistics Program, Uiversity of Alabama, Sulliva, J. H., Barrett, J. D., & Woodall, W. H. (1995). Iterpretig the retrospective T2 cotrol chart based o idividual observatios. Idia Associatio for Productivity, Quality, ad Reliability Trasactios, 20, Woodall, W. H. (2000). Cotroversies ad cotradictios i statistical process cotrol, Joural of Quality Techology, 32, Appedix: Simulatio Descriptio The simulatio program was desiged ad compiled usig Microsoft Visual Basic 6.0, executed i Microsoft Excel 2000 usig ormal radom variates geerated ad imported from 311

14 EFFICIENCY OF MINIMIZED INTEGRATED SQUARE ERROR ESTIMATOR (L2E) Microsoft FORTRAN PowerStatio for Widows, Versio 4.0, FORTRAN 90. Each simulatio was coducted accordig to steps provided below. A series of 100 N(0, 1) radom variates was geerated by FORTRAN MSIMSL subroutie RNNOA. Routie RNNOA geerates pseudoradom umbers from a stadard ormal (Gaussia) distributio usig a acceptace/rejectio techique due to (Kiderma & Ramage, 1976). I this method, the ormal desity is represeted as a mixture of desities over which a variety of acceptace/rejectio methods due to (Marsaglia, 1964), (Marsaglia & Bray, 1964), ad (Marsaglia, Maclare & Bray, 1964) are applied. The fial parameter estimates for each of the 96 cases were based o 10,000 simulatios, which provided a maximum margi of error of 0.02 i estimatio of the MLE meas, with 95% cofidece. These variates were the simulated observatios, X i s, for each of the cases ivestigated. c. For estimatio of the mea ad stadard deviatio (the 64 cases of both a mea ad stadard deviatio shift), a shift i each parameter was iduced i the simulated observatios affectig c of the = 100 variates. Agai, the values of c = 5, 15, 25 ad 45, ad the magitudes of shifts were µ c = 1.50, 2.00, 2.50, 3.00 ad σ c = 1.50, 1.00, 2.00 ad Every combiatio of c, µ c, ad σ c produced the 64 cases. Step 2: The idividual L2E ad MLE estimates of µ ad σ (10,000 for each estimate, per case) were calculated usig the procedures described i the article. Step 3: The average L2E ad MLE estimates of µ ad σ for each case was obtaied by averagig over the 10,000 idividual estimates for each estimator. Step 1: a. For estimatio of the mea (the 16 cases of a mea shift oly), a shift i the mea was iduced i the simulated observatios affectig c of the = 100 variates. The values of c = 5, 15, 25 ad 45 (levels of cotamiatio), ad the magitudes of shifts were µ c = 1.50, 2.00, 2.50 ad 3.00 (degrees of cotamiatio). Every combiatio of c ad σ c produced the 16 cases. b. For estimatio of the stadard deviatio (the 16 cases of a stadard deviatio shift oly), a shift i the stadard deviatio was iduced i the simulated observatios affectig c of the = 100 variates. Agai, the values of c = 5, 15, 25 ad 45, ad the magitudes of shifts were σ c = 1.50, 1.00, 2.00 ad Every combiatio of c ad µ c produced the 16 cases. 312