Using the IML Procedure to Examine the Efficacy of a New Control Charting Technique

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1 Paper Usig the IML Procedure to Examie the Efficacy of a New Cotrol Chartig Techique Austi Brow, M.S., Uiversity of Norther Colorado; Bryce Whitehead, M.S., Uiversity of Norther Colorado ABSTRACT For may years, cotrol charts have bee utilized to moitor processes, improve quality, ad icrease profitability. However, the body of literature tilts overwhelmigly toward charts moitorig ormally distributed processes. I practice, the uderlyig distributio of a process may ot follow a ormal distributio, ad may of those techiques may ot be most effective. Mukherjee, McCracke, ad Chakraborti (2015) suggested three cotrol charts for simultaeous moitorig of the locatio ad scale parameters for processes followig the shifted expoetial distributio. This study examies their proposed, Shifted Expoetial Maximum Likelihood Estimator-Max Chart, (SEMLE-max) ad suggests utilizatio of pealized maximum likelihood estimators (MLE) istead of traditioal MLEs because of ubiasedess ad miimum variace amog ubiased estimators. The ew chart, the Pealized SEMLEmax chart is costructed usig similar methodology, ad simulated data is used to compare average ru legths of the proposed chart to those obtaied by the origial chart. INTRODUCTION I the work published by Mukherjee, McCracke, ad Chakraborti (2015), ample justificatio was give to the creatio ad practical implemetatio of o-ormally distributed statistical process cotrol charts. Specifically, the authors focus was upo moitorig time-to-evet processes, as frequetly modeled by the shifted expoetial distributio: f(x) = 1 θ e 1 θ (x η), x > η, < η < I the publicatio, the authors itroduced three cotrol charts explicitly desiged for such processes. The cotrol chart of iterest i this study is their proposed, Shifted Expoetial Maximum Likelihood Estimator Max Chart, or SEMLE-max. I this chartig scheme, the maximum likelihood estimators (MLE) for the scale ad locatio parameters are used to build two plottig statistics based upo the stadard ormal distributio. I order to avoid issues arisig from joit moitorig of two processes, they authors suggest usig a sigle plottig statistic for each respective sample take, which is the absolute value of the maximum of the two respective statistics. The drawback to usig the traditioal MLEs i this circumstace is that they are biased estimators of the parameters. Sarha (1954) derived two ew estimators for the scale ad locatio parameters of the shifted expoetial distributio, usig a least squares techique, which are ubiased ad uiformly miimum variace for ubiased estimators. Zheg (2013) showed these estimators ca be derived from a pealized likelihood fuctio. Thus, i this study we recommed utilizig Sarhas estimators as opposed to the traditioal MLEs for the desirable qualities aforemetioed. We will use Mote Carlo simulatio methods i PROC IML to compare the i-cotrol ad out-of-cotrol average ru legths (IC-ARL & OOC-ARL) betwee the origial chart, ad the proposed modified chart. ESTIMATORS Similar to the derivatio of the traditioal MLEs for the scale ad locatio paramters ( θ ML = X X 1:, η ML = X 1: ), Zheg (2013) itroduced a pealty to the likelihood fuctio of the radom sample. The purpose of the pealty is with specific respect to the overestimatio of η, as P[η ML = X 1: > η] = 1. Thus, 1

2 the differece betwee the miimum order statistic ad η is defied to be the likelihood fuctio s pealty, ad this differece scales the likelihood fuctio, as follows: (X 1: η)l(θ, η) = (X 1: η) 1 θ e 1 θ (x i η) Usig the traditioal MLE derivatio techique, the pealized MLEs for η ad θ are: Ad: l(l(θ, η)) θ l(l(θ, η)) η i=1 = θ + 1 θ 2 (x i η ) = 0 i=1 θ = 1 (x i η ) i=1 1 = X 1: η + θ = 0 η = X 1: θ Pluggig η ito the equatio for θ, the estimators the become: These estimators are ubiased: η = X 1: X 1 θ = 1 (X X 1: ) E[η ] = E [ X 1: X 1 ] = 1 1 (E[X 1: ] E[X ]) = 1 1 ( (θ + η) η θ) = η E[θ ] = E [ 1 (X X 1: )] = 1 (E[X ] E[X 1: ]) = ( 1) (θ + η θ ) = θ Sarha (1954) also proved these estimators to be uiformly miimum variace amog the ubiased estimators. It should also be oted that θ is distributed as a gamma radom variable, with parameters θ/( 1) ad κ = 1, early idetically to the origial MLE, which was show i Mukherjee et all (2015). A closed form for the distributio of η could ot be derived, ad thus, its desity was estimated usig simulatio. CONSTRUCTION OF THE CONTROL CHART As oted i Mukherjee et al (2015), there are a array of issues associated with joitly moitorig two or more ukow parameters. Oe solutio proposed i their work was calculatig two idetically distributed statistics, ad desigatig the plottig statistic for a give sample as the maximum betwee the two. These two statistics are stadardized (as they are ot idetically distributed i their origial form) usig the iverse cumulative desity fuctio (CDF) of the stadard ormal distributio, i.e.: B 1 = Φ 1 (F(x)) B 2 = Φ 1 (G(x)) 2

3 Where Φ 1 deotes the iverse CDF of the stadard ormal distributio, F(x) deotes the CDF of η, ad G(x) deotes the CDF of θ. For a give sample of the process beig moitored, both B 1 ad B 2 will be calculated ad the maximum of their absolute value, say M i, becomes the plottig statistic. Mathematically, M i = max( B 1, B 2 ) Sice oly positive values are yielded by the plottig statistic, oly a upper cotrol limit (UCL) is ecessary to specify. For a desired i-cotrol average ru legth (IC-ARL), called ARL 0, Mukherjee et al (2015) showed the UCL ca be determied by: UCL = Φ 1 (0.5 (1 + 1 (ARL 0 ) 1 )) Thus, for a desired IC-ARL of 500, the UCL to be used is Note, this IC-ARL will be used i comparig out-of-cotrol ARL (OOC-ARL) performace of the modified chart to the chart origially proposed i Mukherjee et al (2015). Therefore, for a give sample, the process is deemed OOC whe M i > UCL. Of ote, a OOC sigal ca assist i idicatig which of the two parameters has shifted away from its specified IC value. Whe M i = B 1 > UCL, this implies η may have shifted, ad coversely, whe M i = B 2 > UCL, this implies θ may have shifted. However, whe both B 1 ad B 2 are greater tha the UCL, it becomes difficult to determie which of the two values (or possibly both) is sigalig the OOC without physical ivestigatio. SIMULTATION METHODS & RESULTS Give the guidelies for the cotrol charts costructio as outlied i the previous sectio, the code for the simulatio was writte i similar steps. First, a radom sample from the OOC shifted expoetial distributio had to be simulated. This was performed usig the iverse probability itegral trasform. This trasformatio utilizes the kowledge of a radom variables cumulative desity fuctio (CDF) beig distributed uiformly betwee 0 & 1 (i.e., Y = F(x) UNIF(0,1)). Values from the uiform distributio are evaluated i a give radom variables iverse CDF. I this case, the shifted expoetial distributios CDF was set equal to Y ad solved for X as give by: Solvig for X: Y = 1 e 1 θ (x η) X = η θ l(1 Y) Thus, values radomly sampled from UNIF(0,1) are iput for Y, ad the resultig X values are distributed as the shifted expoetial for a give value of η ad θ. Usig two DO loops i the DATA step, 10,000 samples, each of sample size = 5, were simulated, ad the estimators for the respective parameters from the OOC distributio were obtaied usig PROC MEANS. At this poit, the plottig statistic for θ, B 2, was obtaied usig the PROBCHI ad QUANTILE fuctios i the DATA step. Here, the probability associated with a give observed value of θ (usig PROBCHI) was iputted ito the iverse CDF of the stadard ormal distributio (i.e., QUANTILE( NORMAL ) ), ad the absolute value of the resultig z-score is the value B 2 for a give sample. Seeig as the desity of η was estimated usig simulatio by ecessity, a hadful of extra procedures were implemeted i order to obtai its plottig statistic, B 1. Usig the same procedure described above, a radom sample from the IC shifted expoetial distributio was obtaied. This radom sample acts as the desity for η. Usig PROC IML, the OOC value of η from each respective sample was iteratively compared to the simulated IC distributio i order to obtai a pseudo-probability. Z-scores for this vector of pseudo-probabilities was obtaied i order to yield the plottig statistic B 1 i a early idetical maer as described for B 2. Now that values for B 1 ad B 2 for each of the respective 10,000 samples are obtaied, they ca be plotted agaist the specified UCL of 3.29 i order to evaluate the effectiveess of these ew estimators. This process was performed iteratively usig PROC IML. Whe a sample plotted above the UCL, thus sigalig a OOC process, the sample umber was 4 recorded, ad the mea differece betwee each 3

4 observed OOC poit was take to be the OOC-ARL for a give pair of OOC values of θ ad η. The etire aforemetioed process was performed 10 times for each pair of OOC values of θ ad η, ad mea of those results were take to be the OOC-ARL used for compariso agaist the Mukherjee et al (2015) chart s performace. The procedure described above was performed for 20 paris of OOC values of η ad θ, where the specified IC values were η = 0 ad θ = 1, respectively, ad the sample size take was = 5. The results of this study are give i the below table, deoted by ARL S, ad the results obtaied i the origial study are deoted ARL C. Table 1: Compariso of Simulated OOC-ARL η θ ARL S ARL C CONCLUSION As show i Table 1, the utilizatio of the pealized MLEs give improved performace over that of the stadard estimators for the dowward shifts i θ coupled with icreases i η. Both chartig schemes, as the locatio parameter shifts upward away from its omial IC value, are highly effective i quickly detectig these large shifts, which is a similar coclusio draw i Mukherjee et al (2015). However, cosiderig the small scale, oe may assume the aalyzed OOC shifts are small, but the magitude of the shifts is quite large. While the proposed modificatios to the estimators yield improvemets i some cases, such as the istace whe OOC value of θ = 0.75 ad η = 0.5, a ))C-ARL of , while a improvemet over , is still far too slow for a 25% shift i θ ad a icreased η. 4

5 A big cotributor to this relatively slow performace is the costructio of the chart itself. While creatively thought out ad mathematically elegat, this techique leds itself oly to detectio of very extreme values, as evideced by the quick detectio rate at large OOC η. However, cosider the case of θ. Sice it ca be trasformed ito a χ 8 2 radom variable i the case whe = 5, it ca easily be determied what the z-score would be for a particular quatile of the χ 8 2 distributio. Say θ is perfectly estimated to be 0.50, a 50% reductio i the IC value of θ. The probability associated with this estimate is about 0.24, which is associated with a z-score of about -0.70, which certaily would ot sigal a shift i θ. I fact, settig the UCL to 3.29 suggests that i order for a OOC sigal to be detected, the lower-tailed probability associated with our estimators would have to be approximately Thus, as show by the results i Table 1, uless η has shifted substatially from 0, it is highly improbable to detect a OOC shift. Cosequetly, it would ot be recommeded to implemet this scheme with such a high UCL uless specific circumstaces warrat it. While a smaller UCL (for example, 2.78) would sacrifice the IC performace of the chartig scheme i both istaces, it would make up the differece by beig far more sesitive to shifts i either parameter. 5

6 REFERENCES Mukherjee, A., McCracke, A., & Chakraborti, S. (2015). Cotrol charts for simultaeous moitorig of parameters of a shifted expoetial distributio. Joural of Quality Techology, 47(2), Sarha, A.E. (1954). Estimatio of the Mea ad Stadard Deviatio by Order Statistics. The Aals of Mathematical Statistics, Vol. 25, Zheg, M. (2013). Pealized Maximum Likelihood Estimatio of Two-parameter Expoetial Distributios (Doctoral dissertatio, Master Thesis, Uiversity of Miesota). 6

7 CONTACT INFORMATION Your commets ad questios are valued ad ecouraged. Cotact the authors at: Austi Brow & Bryce Whitehead Uiversity of Norther Colorado & SAS ad all other SAS Istitute Ic. product or service ames are registered trademarks or trademarks of SAS Istitute Ic. i the USA ad other coutries. idicates USA registratio. Other brad ad product ames are trademarks of their respective compaies. 7

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Topic 9: Sampling Distributions of Estimators

Topic 9: Sampling Distributions of Estimators Topic 9: Samplig Distributios of Estimators Course 003, 2018 Page 0 Samplig distributios of estimators Sice our estimators are statistics (particular fuctios of radom variables), their distributio ca be