Two-Sided Cumulative Sum (cusum) Control Chart for Monitoring Shift in the Shape Parameter of the Pareto Distribution

Transcription

1 J. Stat. Appl. Pro. 7, No. 1, Journal of Statistics Applications & Probability An International Journal Two-Sided Cuulative Su cusu Control Chart for Monitoring Shift in the Shape Paraeter of the Pareto Distribution Shei Sayibu Baba Albert Luguterah Departent of Statistics, Faculty of Matheatical Sciences, University for Developent Studies, Ghana Received: 18 Mar. 2017, Revised: 25 Oct. 2017, Accepted: 11 Nov Published online: 1 Mar Abstract: In this study, we propose a Cuulative Su control chart for onitoring shift in the shape paraeter of the Pareto distribution when the scale paraeter is known. The V-ask ethod of constructing Cuulative Su control chart was eployed together with the Sequential Probability Ratio Test. We investigated the features of the V-ask, observed that the lead distance, the ask angle the Average Run Length of the Cuulative Su control chart changed considerably given a sall shift in the shape paraeter of the Pareto distribution. Keywords: Cuulative Su, paraeter, V-ask, two-sided sequential 1 Introduction An organization will attract aintain a large nuber of custoers, if it is able to iprove control the quality of its products. This desired quality cannot be achieved without eploying efficient effective quality control easures. Statistical Process Control SPC charts are used widely by quality engineers for onitoring the stability of different process over tie. A control chart helps to distinguish between noral process variation unusual variation due to special causes. In order to onitor a process quality, practitioners frequently use the Shewhart s control charts like x chart, p-chart R-charts aong others. These charts are effective in detecting large shift larger than 1.5 σ in the process paraeter. Industry players therefore cannot liit theselves to the use of the Shewhart control charts since it is not always a preferred choice when it coes to detecting sall to oderate shift in the process paraeter. Faced with this challenge, a ore robust control chart with greater sensitivity is needed than what the Shewhart s charts can provide. One such ethod is the use of Cuulative Su CUSUM control chart. The CUSUM control charts have the ability to detect sall shifts that is less than 1.5 σ by charting a statistic that incorporates current previous data values fro the process. It is also able to deterine when a process shift has occurred. The CUSUM control chart is a natural fit for situations where process average is expected to shift fro a specified target there is the need to ake tiely process adjustents to bring the process back to the target. The sensitivity of a CUSUM control chart largely depends on the type of distribution used. This akes the Pareto distribution a suitable cidate in the construction of a CUSUM control chart because of its skewness heavy tail. Any sall change in the process affects the ean variance of the distribution. The CUSUM control chart was first proposed by [1] to address the shortfalls of the Shewhart control charts. [2] stated that CUSUM control charts are ost sensitive SPC charts to signal a persistent sall step change in a process paraeter. A great deal of research has been done in the area of CUSUM control charts: [3] developed alpha-cut control chart for attributes data: This approach provided the ability of the CUSUM control chart in detecting the tightness of the inspection by selecting a suitable alpha level. [4] applied CUSUM control chart using sequential probability likelihood ratio test. [5], applied the CUSUM control chart in agriculture to check the quality of production bulk tank soatic cell counts in ilk. Also, [6] developed unified CUSUM control chart for onitoring siultaneous shifts in the paraeters of the Elang-Truncated Exponential Distribution therefore derived the paraeters of the CUSUM chart proposed. Again, [7] applied the CUSUM the Shewhart charts to onitor oestrus detection efficiency in dairy cows. In their analysis, Corresponding author e-ail: b.shei@yahoo.co

2 30 S. S. Baba, A. Luguterah: Two-sided cuulative su cusu control chart... the SPC charts detected changes in oestrus detection efficiency in tie to be potentially useful in diary anageent. It was pointed out that the correct design of the used control chart is iportant to receive eaningful classification results. Recently, [8] developed a one-sided CUSUM control chart for onitoring the shape paraeter of the Pareto distribution. And lately [11], developed a one sided CUSUM control chart for onitoring shift in the scale paraeter δ of the new Weibull-Pareto distribution. In this study, we extend the study of [8] therefore develop a two-sided CUSUM control chart for onitoring shifts in the shape paraeter of the Pareto distribution. 2 Pareto Distribution The Pareto distribution is naed after an Italian civil engineer econoist Vilfredo Pareto. It is a skewed distribution with heavy or slowly decaying tail. It is a powerful probability distribution that is applied in econoics, sociology, statistics. It was used by Pareto in odelling national incoe wealth distribution in any huan endeavour. The Pareto distribution is used in odelling any situations which are not evenly distributed. If the ro variable has a Pareto distribution, then the density function is given by; f x x;γ,c= γcγ x γ+1 1 where x c the paraeters γ > 0 c > 0 are the shape scale paraeters respectively. The corresponding cuulative distribution function is given by; c γ F x x;γ,c=1 2 x The ean variance of the Pareto distribution are given by; σ 2 = µ = γc,γ > 1 3 γ 1 γc 2 γ 2γ 1 2,γ > Sequential Probability Ratio Test The Sequential Probability Ratio Test SPRT plays a ajor role in the developent of an acceptance sapling plan. [9] SPRT is a joint, subject-by-subject, Likelihood Ratio Test LRT, where each subject constitutes a stage. According to[10], the CUSUM control charts are roughly equivalent to the SPRT. The SPRT has been used extensively in the developent of an acceptance sapling plan this acceptance sapling plan has been used in deterining the in out of control liits in CUSUM procedures. Suppose that we take a saple of values x 1,x 2,...,x successively, fro a population fx,θ. Consider two hypotheses about θ, H 0 : θ = θ 0 H 1 : θ = θ 1. The ratio of the probabilities of the saple is; L = Π fx i,θ 1 Π fx i,θ 0 We select two nubers A B, which are related to the desired β errors. The sequential test is set up as follows; 1.As long as B<L < A we continue sapling 2.At the first i that L A we accept H 1 3.At the first i that L B we accept H 0 An equivalent way of coputation is to use the logarith of L. Then, the inequality becoes: B<Σ fx i,θ 1 Σ fx i,θ 0 <A 6 5 The faily of test is referred to as SPRT. if: { } x i,θ 1 Z i = x i,θ 0 7

3 J. Stat. Appl. Pro. 7, No. 1, / 31 then the sapling terinates when or z i A 8 z i B 9 The z i s are independent ro variables with variance, say σ 2 > 0: Obviously, Z i has a variance σ 2. As increases the dispersion becoes greater the probability that a value of Z i will reain within the liits B A tends to zero. The ean, Z, tends to a noral distribution with variance σ 2, therefore the probability that it will fall between B tends to zero. If we take a saple for which L lies between A B for the first n 1 trials, then L A at the n th trial, so that we accept H 1 reject H 0. Then by definition, the probability of getting such a saple is at least A ties as large under H 1 as under H 0 : The probability of accepting H 1 when H 0 is true is,, that of accepting H 1 when H 1 is true is 1 β. This is expressed by: A 1 β 10 siilarly, if we accept H 0 B β 1 Wald1947 showed that for all practical purposes10 11 hold as equalities. Thus 11 A= 1 β 12 B= β 1 In practice β are sall the use of A B can increase one of the errors by a very sall aount CUSUM control chart for onitoring the shape paraeter The SPRT ethod is used to construct a two-sided CUSUM control chart for onitoring the shape paraeter of the Pareto distribution, when the scale paraeter is known. The Likelihood Ratio Test for testing the null hypothesis of no shift in the shape paraeter as against the alternative hypothesis of a shift in the shape paraeter is given by: L 1 L 0 = Π fx i,,c Π fx i,,c where L 1 is the likelihood of the new value of the shape paraeter, γ L 0 is the likelihood of the original value of the shape paraeter,γ. The continuation region of the SPRT discriinating between the two hypotheses is given by: 14 β 1 < Π c x +1 c γ < 0 x +1 Π 1 β where β are types I II errors. taking logarith of 15, the continuation region becoes β 1 < Π c x +1 Π c x +1 < siplifying 16 we obtain: [ ] β < 1 + c + 1 β x< 1 β

4 32 S. S. Baba, A. Luguterah: Two-sided cuulative su cusu control chart... β 1 [ ] < + c + x< 1 β 18 A shift in the shape paraeter affects the ean variance of the Pareto distribution. Assuing is the target value > is the changed values due to a shift in the value of the paraeter, the SPRT will be terinated by rejecting or accepting the null hypothesis or by continuing the sapling. The sapling process is stopped by rejecting the null hypothesis if L 1 L 0 A; which gives a rejection line >. On the other h, if L 1 L 0 A is obtained for the case of <, then another rejection line is obtained. These two rejection lines give a syetrical asking: The observations in the saple are used with the ask in a sequential pattern. Consideration 18 setting β = 0 the inequality in 18 reduces to: [ ] + c + + x< 19 Re-arranging 19 keeping the ter containing x at the left-h side of the inequality, we obtain: x [ + c ] + 20 which iplies x [ + c ] + 21 now equation 21 can be wriiten in the for of a straight line as: x i f + g 22 where f = γ1 + c 23 g = 24 Let X 1,X 2,...,X be a saple fro a Pareto distribution. If the points, x iare plotted with a suitable scale, then the ordinates of the points represent the cuulative su of the data. Equations are the effects of a shift in the population paraeter,γ. Figure 1 shows a sizeable shift in the paraeters if x i falls outside the lines G 1 H 1 G 1 H 1. The chart is interpreted by placing the ask over the last plotted point as shown in figure 1 If any of the points lies below G 1 H 1, then it indicates a decrease in γ if any of the points falls above, G 1 H 1 then it shows an increase in γ. If all the points plotted are within the two lines G 1 H 1 G 1 H 1 of the V-ask, then the process is in control: If there is a point below G 1 H 1 then it is concluded that the process is out of control has a positive shift of the process ean if there is a point found above G 1 H 1 then the process is out of control has a negative shift of the process ean.

5 J. Stat. Appl. Pro. 7, No. 1, / 33 Fig. 1: : v- ask for two-sided cusu The paraeters of the CUSUM chart, known as the angle of the ask the lead distance, are obtained. Fro Figure 1, tanθ 1 = slope of the line G 1 H 1 = f tanθ 1 = slope of the line G 1 H 1 = f, hence: [ γ1 ] θ 1 = tan 1 + c 25 where > [ γ1 ] θ 1 = tan 1 + c 26 where where < Various values of the ask angle, theta, are shown in 1 for various values of the scale paraeter, c, shifts in the shape paraeter: The results show that the values of the angle, theta, decreases with an increase in the value of for any given value of c. Also, as the value of increases the value of theta increases with increasing value of c. Again increasing values of, decreases the value of theta Table 1: Values of the angles θ for controlling the paraeters c

6 34 S. S. Baba, A. Luguterah: Two-sided cuulative su cusu control chart... Table 2: Value of the lead distance for the paraeters of the Pareto distribution 5 Average Run Length of the two-sided CUSUM control chart The Average Run Length for the two-sided CUSUM control chart is given by: ARL= + 27 Proof by using: where Then: ARL= E[Z] γ=γ1 28 Z = fx,,c fx,,c 29 Z = c x +1 c γ0 x γ1 Z = c x 31

7 J. Stat. Appl. Pro. 7, No. 1, / 35 Taking natural logarith of 31: hence: where Then: substituting 31 into 29 we obtain: substituting 36 into 26, the ARL is obtained as: Z = γ1 c+ x 32 E[Z]= +γ1 c+ E[x] 33 E[x]= x fx,,cdx 34 c c E[x]= c x γ xdx [ E[x]= c c 0 γ1 2 cc ] e + 1c [ γ1 + γ1 2 E[x]= c ] γ γ1 E[Z]= + 38 ARL= + 39 Various values of the ARL are given in 3. Fro the results, for a given value of, an increase in the value of, the value of the ARL decreases. Also, an increase in the ratio,, with a given, decreases the value of the ARL a decrease in the value of a fixed value of, increases the ARL. Table 3: Average Run Length ARL for controlling the paraeter γ Practical Deonstration of the use of the Two-sided CUSUM control chart In this section, the application of the proposed CUSUM control chart is illustrated using hypothetical data siulated fro the Pareto distribution. The first ten observations were siulated with = 2.5 c=1.5; the last five observations were siulated with =5. Table 4 displays the siulated data their cuulative su. The paraeters of the V-ask were calculated using = 2.5 c=1.5; = 5 =0.01. The lead distance the angle of the ask were respectively. The saple nuber was plotted against the cuulative su of the data. The V-ask was then placed at the last plotted point to onitor whether the process is in control or out of control as shown in figure 2. It can be

8 36 S. S. Baba, A. Luguterah: Two-sided cuulative su cusu control chart... Fig. 2: two-sided cusu plot for the siulated hypothetical data established fro figure 2 that the process was out of control as observations 1 to 5 fell above line G 1 H 1 indicating an increase in γ. This eans an action ust be taken in order to bring the process back to control. Table 4: Siulated data for the two-sided CUSUM control chart saple Datax x CUSUM Conclusion Control charts play a key role as long as the quality of a product is concerned in the industrial establishent. Therefore proposing control charts for onitoring the quality of products is not trivial. This CUSUM will particularly be helpful in deterining when the assignable cause has occurred in situations in which two directions of a syste are to be onitored. It also offers considerable perforance iproveent relative to the Shewhart charts. It can be applied in the cheical process industries. The liitation of this CUSUM is that when the ask angle the lead distance are not well deterined, can lead to inappropriate syste adjustents. Further studies can be done on this distribution by looking at siultaneous shift in both the shape scale paraeters of the distribution. The results of the study showed that the size of the ask angle Θ decreases as the value of increases with any given value of the scale paraeter c. However, given any fixed value of, the angle Θ increases for an increase in the value of c. With regards to the lead distance, it can be observed that an increase in the value of given the values of c, the value of d decreases. Again the value of d increases with fixed value of c. However, d decreases as c increases in value, if values are aintained. On the ARL, it can be established that as the value of increases, the ARL tends to decrease with any given value of. Also, as the value of decreases, the ARL also increases for fixed values of.

9 J. Stat. Appl. Pro. 7, No. 1, / 37 Acknowledgeent The authors wish to thank the Editor the reviewers for their suggestions coents which helped in iproving the final version of the paper.the authors also wish to thank Mr Suleana Abdul-Sala for his support on the Latex write-up. References [1] E. S. Page, Continuous Inspection Schees, Bioetrics, 41 1, [2] D. M. Hawkins D. H. Olwell, Cuulative su charts charting for quality iproveent, Springer-Verlag, New York, [3], M. Gulbay, C. Kahraa, D. Ruan, Agir;-Cut., Fuzzy Control Charts for linguistic data, International journal of Intelligent Systes, 1912, [4] G. A. Barnard, Control charts stochastic processes, J. Royal Stat. Soc. B Methodological.21, [5] I. Thysen, Monitoring bulk tank soatic cell counts by a ulti-process Kalan filter, Acta Agriculturae Scinavica, Sect. A, Anial Science, 43, [6] A. Luguterah, Unified cuulative su chart for onitoring shifts in the paraeters of the Erlang-Truncated exponential distribution. Far East Journal of Theoretical Statistics501, [7] A. De Vries, B. J. Conlin, Design perforance of statistical process control charts applied to estrous detection efficiency, Journal Dairy Science,86, [8] S. Nasiru, One-sided cuulative su control chart for onitoring shifts in the shape paraeter of Pareto distribution. Int. J. Productivity Quality Manageent, 192, [9] A. Wald, Sequential Analysis, John Wiley, New York, NY 1947 [10] N. L. Johnson, A siple theoretical approach to cuulative su control charts. Journal of the Aerican Statistics Association, Nature 564, [11] S. B. Shei, J. Haruna, A. Suleana, One-Sided Cuulative Su Control Chart for Monitoring Shift in the Scale Paraeter Delta δ, of the New Weibull-Pareto Distribution: International Journal of Probability Statistics,Vol. 6 No. 4, pp doi: /j.ijps Shei Sayibu Baba received his Higher National Diploa in Statistics fro Taale Polytechnic, Bsc in Matheatical Sciences Statistics option an Msc degree in Applied Statistics both at the University for Developent Studies, Navrongo Ghana. His research interests are in the areas of control charts Tie Series. He has published in International journal of Statistics Applications International Journal of Probability Statistics. And also has a vast expericience in teaching at the various levels of education in Ghana. Albert Luguterah is a Full tie Senior Lecturer Dean at the Faculty of Matheatical Sciences at the University for Developent studies, Navrongo, Ghana. Dr Luguterah Holds a BSc Statistics with Matheatics fro the University of Ghana, Msc in Statistics fro the University of Cape Coast, a PhD in Applied Statistics fro the University for Developent Studies All in Ghana. Dr Luguterah is a eber of several statistical organizations in Ghana abroad. He is widely researched has participated also presented in several local International conferences.