CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION

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1 CONTROL CHARTS FOR THE LOGNORMAL DISTRIBUTION Petros Maravelakis, Joh Paaretos ad Stelios Psarakis Departmet of Statistics Athes Uiversity of Ecoomics ad Busiess 76 Patisio St., 4 34, Athes, GREECE. Itroductio Cotrol charts are the tools of statistical process cotrol for detectig a chage i a process. The most kow are the Shewhart cotrol charts that where developed for cotrollig the mea ad variace of the distributio our observatios follow. Shewhart charts perform satisfactorily for detectig large shifts i the process. However, for small shifts they had poor results. This fact led Page i 954 to costruct CUSUM cotrol chart for detectig small ad moderate shifts i the process. Cotrol charts have foud great applicatios i idustry ad especially the oes for the ormal distributio. Nevertheless, a process may ot follow the ormal distributio-it may be positive or right skewed. Morriso (958), Joffe ad Sichel (968) ad Kotz ad Lovelace (998) provided examples of processes that follow the logormal distributio. Morriso (958), Kotz ad Lovelace (998) have developed Shewhart cotrol charts for the logormal distributio. Joffe ad Sichel (968) costructed a cotrol chart for testig sequetially arithmetic meas from a logormal populatio. I this paper, CUSUM procedures are developed for the logormal for testig the mea ad variace. I additio, a chart based o the sequetial probability ratio test (SPRT) is illustrated.

2 . Logormal distributio A positive radom variable X is said to be logormally distributed with two parameters µ ad if YlX is ormally distributed with mea µ ad variace. The probability desity fuctio is (x) x π ( l xµ ) exp, f, x > x There are four forms of the logormal distributio as Crow ad Shimizu (988) preset i their book: (a) The two-parameter distributio Λ(µ, a lower threshold of zero. (b) The three-parameter distributio Λ(τ, µ, ) that describe positive skew data with ) that describe positive skew data with a lower threshold of τ. This form ca be reduced to the two-parameter case by the traspositio YX-τ. (c) The three-parameter distributio Λ(θ, µ, ) that describe egative skew data with a upper threshold of θ. This form ca be reduced to the two-parameter case by the traspositio Yθ-X. (d) The four-parameter distributio Λ(τ, θ, µ, ) that describe skew data with upper ad lower thresholds of θ ad τ. I the followig, we will deal with the two-parameter distributio. The threeparameter may occur i practice but with the suitable trasformatios as it is show i (b), (c) it reduces to the two-parameter oe. The four-parameter is ot likely to have applicatios i process cotrol.

3 3. CUSUM Chart 3.. The geeral expoetial family. The probability desity for ay member of the expoetial family with a sigle parameter θ ca be writte as: { (y)b(θ) c(y) d(θ) } f (y θ) exp where θ is the parameter of the distributio ad Y is the radom variable. The joit desity for a radom sample of Y, where Y is a member of the expoetial family, is give by: f(y θ) exp (yi )b(θ) c(yi ) d(θ). Suppose we wat to test whether the process has goe from a i-cotrol parameter value θ to a out-of-cotrol value θ. We will use Wald s sequetial probability ratio test (SPRT) because CUSUM is a sequece of Wald sequetial tests [Page (954)]. The the variables Z i are: f(y i ) Zi l (Y i ) f (Y i ) The CUSUM scheme is: ad it gives a out-of-cotrol sigal if Therefore the CUSUM scheme is: ad it sigals whe { b(θ ) b(θ )} { d(θ ) d(θ )} (,D Z ) D max D > A. (,D (Y ){ b(θ ) b(θ )} d(θ ) d(θ )) D max i D > A. Let X (Y ) ad d(θ) d(θ ) k b(θ ) b(θ ). If b(θ) b(θ ) > the we rescale CUSUM by dividig it with this quatity ad 3

4 C max (,C X k) where D /( b(θ ) b(θ )) C ( b(θ ) b(θ )) h A / with this quatity ad ad the CUSUM sigals if > h C where. If (θ ) b(θ ) < the we rescale CUSUM by dividig it b C ( X k) mi,c where D /( b(θ ) b(θ )) C ( b(θ ) b(θ )) h A /. ad the CUSUM sigals if C < h where 3.. CUSUM schemes for the logormal distributio 3... kow Let be fixed ad kow. The above desity fuctio is writte i the form of the expoetial family of distributios as follows: f (x µ) exp l ( l x) ( π ) l x µ µ l x µ µ The (x)lx, b(µ), d(µ). Therefore d(θ) d(θ ) µ µ µ µ k b(θ) b(θ ) (µ µ ) The CUSUM scheme for the mea whe µ > µ will be: C C max(,c l X k) µ µ k 4

5 The scheme sigals whe > h C where h is chose to give a specified ARL. The CUSUM scheme for the mea whe µ < µ will be: C C mi(,c l X k) µ µ k The scheme sigals whe C < h µ kow Let µ be fixed ad kow. The desity fuctio of the logormal is writte i the form of the expoetial family of distributios as follows: The (x) ( ) f (x ) exp l x l x µ, b() ( l xµ ) ( π ) l, d()-l. Therefore d(θ) d(θ ) ll k b(θ) b(θ ) The CUSUM scheme for the variace whe > will be: C C max(,c ( l X µ ) k) k l l The CUSUM scheme for the variace whe < will be: C C mi(,c k l ( l X µ ) k) l 5

6 If oe looks at the CUSUM schemes for the variace, he will otice that it is affected by chages i µ as well as by chages i. However, we moitor µ separately, therefore we are able to distiguish chages i the CUSUM chart for the variace caused by µ A geeral result I the logormal CUSUM for moitorig, I have show that the logormal is a member of the expoetial family with parameters (x) ( x µ ) d()-l. Let Z ( ) l x µ. The the distributio of Z is: l, b(), [ z] P ( l X µ ) [ z] P[ l X µ z] P[ z l X µ ] FZ (z) P Z z P[ µ z l X µ z] F ( µ z ) F ( µ z ) l X l X. Therefore d µ fz(z) dz ( z ) d ( µ z ) exp z π exp z πz But this is a ( ), fl X(µ z) ( µ z µ ) ( µ z µ ) dz fl X(µ z) exp π Γ distributio with desity fuctio Cosequetly, (x) ~ f(x;,β) ( ), Γ. β x exp Γ() ( x / β) 6

7 3.3.. Moitorig i the logormal CUSUM It is kow that if Y~Γ(,β) ad ZaY the Z~Γ(,aβ). Therefore (x)~ Γ(, ). If we wat to moitor we just have to moitor for a scale chage of the gamma distributio from β to costructed CUSUM schemes for the gamma distributio. β. Hawkis ad Olwell (998) have Moitorig i the logormal CUSUM From the defiitio of the logormal distributio, it is kow that l xµ lx~n(µ, ) ad as a result ~ Ν(, ). Hece moitorig is just the case of moitorig for i the stadard ormal distributio. Hawkis ad Olwell (998) have also costructed CUSUM schemes for the ormal distributio Wald s sequetial probability ratio test (SPRT) CUSUM cotrol charts, as Johso (96) poited out, are roughly equivalet to the sequetial probability ratio test (SPRT). SPRT, as it will be show i the sequel, leads to a acceptace pla. This acceptace pla has bee used for determiig the i ad out-of-cotrol limits i CUSUM procedures. Suppose that we take a sample of m values x, x,..., x m, successively, from a populatio f(x, θ ). Cosider two hypotheses about θ, Η : θ θ ad Η : θ θ. The ratio of the probabilities of the sample is: L m m f(xi, θ) m f(x, θ ) i 7

8 We select two umbers A ad B, which are related to the desired ad β errors i a way we will explai later. The sequetial test is set up as follows: as log as B< L m <A we cotiue samplig at the first i that L m A we accept Η at the first i that L m B we accept Η A equivalet way for computatio is to use the logarithm of L m. The, the iequality becomes: m m logb < logf(x i, θ ) - log f(x i, θ ) < loga This family of tests is referred to as sequetial probability-ratio tests. If zlog{ i f (x i, θ )/ f(x i, θ )} the samplig termiates whe or zi zi logα logβ The z i s are idepedet radom variables with variace, say, >. Obviously m z i has a variace m. As m icreases the dispersio becomes greater ad the probability that a value of z i will remai withi the limits logb ad loga teds to zero. The mea z teds to a ormal distributio with mea /m ad therefore the probability that it falls betwee (logb)/m ad (loga)/m teds to zero. Cosider a sample for which L m lies betwee A ad B for the first - trials ad the L m A at the th trial, so we accept Η (ad reject Η ). By defiitio, the probability of gettig such a sample is at least A times as large uder Η as 8

9 uder Η. The probability of acceptig Η whe Η is true is ad that of acceptig Η whe Η is true is -β. Therefore: -β A or β A. (3.4.) Similarly, we see that whe we accept Η β B( ) or β B. (3.4.) Wald (947) showed that for all practical purposes the above iequalities hold as equalities. Thus: β A ad β B. β β Suppose that a ad b ad that the true errors of first ad secod kid for the limits a ad b are ' ad β '. The, from (3.4.): ad from (3.4.) Therefore ' β' a β β' β b. ' ( β') ' (3.4.3) β β 9

10 Furthermore or β( ') β β' (3.4.4) '( β) β'( ) ( β' ) β( ' ) ' β' β. (3.4.5) I practice ad β are small. From (3.4.3) ad (3.4.4) we see that the amout that ca exceed or β exceed β is egligible. I additio, from relatio (3.4.5) we see that either or β β. Therefore the use of ad β i place of Α ad Β ca oly icrease oe of the errors ad oly by a very small amout Applicatio of the SPRT to the CUSUM for the logormal distributio Based o the theory of Wald we will derive a CUSUM scheme for the parameter µ of the logormal distributio. Defie L ad L to be the likelihood fuctios of the radom sample uder Η ad Η, respectively, ad let the likelihood ratio L L be deoted by Λ. That is: L ( ) Xi π exp ( l Xi µ ) ad Λ is L Xi i µ ( π ) exp ( l X ) Λ exp ι ι ( l Xi µ ) ( l Xi µ ). We assumed that is kow which is reasoable if cotrol charts have bee used to moitor the process for a period. Defie A ad B as follows:

11 β A ad β B. We accept the lot if Λ Β. We reject the lot if Λ A. If B<Λ<A we cotiue samplig. After some calculatios, we coclude that (a) Accept the lot if Y Y S h (b) Reject the lot if Y Y S h (c) Cotiue to sample if Y <Y< Y where Y l X i, S µ µ, β l h ad µ µ h β l µ µ. I the above three cases h ad h are parallel lies referred as acceptace ad rejectio lies respectively. The results were developed for the case of µ> µ meaig that µ is a upper specificatio limit. I the same way, we ca derive results for the case µ< µ. 4. ARL performace Cotrol charts are usually evaluated usig the average ru legth (ARL). ARL is the average umber of samples util sigal. A computer program was writte for the computatio of the ARL based o the theory of Brook ad Evas (97). I this program the values of k ad h are provided ad we receive the value of the ARL. I the followig several results of the program are preseted ad also the effect of the decisio iterval h is illustrated. The symbols µ ad are the parameters of the logormal distributio.

12 µ,6 K h ARL µ K h ARL Coclusio I this paper, a ew CUSUM cotrol chart for the logormal distributio is preseted. The close relatioship of the CUSUM ad the SPRT is also show i detail. Various versios of the ew chart for moitorig are illustrated. Fially, a computer program for comparig differet cotrol charts usig average ru legth (ARL) is provided. Refereces Brook, D. ad Evas, D.A. (97). A Approach to the Probability Distributio of CUSUM Ru Legth, Biometrika, 59, Crow, E.L. ad Shimizu, K. (988). Logormal distributio, Theory ad Applicatios, Marcel Dekker, Ic., New York ad Bassel. Hawkis, D.M. ad Olwell, D.H. (998). Cumulative sum charts ad chartig for quality improvemet (Statistics for egieerig ad physical sciece), Spriger- Verlag.

13 Joffe, A.D. ad Sichel, H.S. (968). A chart for sequetially testig observed arithmetic meas from logormal populatios agaist a give stadard, Techometrics,, Johso, N.L. (96). A Simple Theoretical Approach to Cumulative Sum Cotrol Charts, Joural of the America Statistical Associatio, Kotz, S. ad Lovelace, C.R. (998). Process capability idices i theory ad practice, Edward Arold. Morriso, J. (958). The logormal distributio i quality cotrol, Applied Statistics, 7, 6-7. Page, E.S. (954). Cotiuous Ispectio Schemes, Biometrika, 4, -5. Wald, A. (947). Sequetial Aalysis, Joh Wiley, New York. 3

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