Construction of Control Charts for Some Circular Distributions

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1 IJIRST Iteratoal Joural for Iovatve Research Scece & Techology Volume 5 Issue 1 Jue 2018 ISSN (ole): Costructo of Cotrol Charts for Some Crcular Dstrbutos P. Srvasa Subrahmayam A. V. Dattatreya Rao Jot Drector Professor of Statstcs (Retred) Treasures & Accouts Departmet, Govt. of Adhra Pradesh, Vjayawada, Ida. Acharya Nagarjua Uversty, Gutur Dst. Adhra Pradesh, Ida Abstract Qualty s oe of the most essetal characterstcs of ay product. Motorg ad cotrollg qualty s a cotuous process. Statstcal Process Cotrol (SPC) s a qualty cotrol techque usg statstcal methods for motorg ad cotrollg the qualty. Cotrol charts are oe of the most mportat tools of SPC. Laha ad Gupta (2011) appled the cocept of cotrol charts to crcular dstrbutos such as Vo mses, Wrapped Cauchy, Wrapped Normal ad Cardod dstrbutos. Takg a cue from ths work, a attempt has bee made ths paper to exted ths cocept of cotrol charts to two crcular dstrbuto developed by Srvasa Subrahmayam et al (2017), amely the Wrapped Expoetal Iverted Webull dstrbuto (WEIWD) ad Wrapped New Webull Pareto Dstrbuto(WNWPD). For these two dstrbutos, the cotrol charts at dfferet sets of values of parameters are costructed, the theoretcal values for Cetral Ray (CR), Atclockwse Cotrol Ray (ACR) ad Clockwse Cotrol Ray (CCR) agles are obtaed, the acceptace rego ad rejecto rego at the theoretcal values are detfed ad the expaso ad cotracto of the acceptace rego wth respect to chage parameter values are studed. The estmates for CR, ACR ad CCR agles ad ther respectve crcular varaces are computed ad the mpact of smulato sze ad sample sze o the ACR ad CCR agles ad ther crcular varaces are studed. Also, a effort has bee made to fgure out a reasoable smulato sze as well as sample sze for fdg estmate values for CR, ACR ad CCR agles respect of these two crcular dstrbutos. Keywords: Crcular Statstcs, Wrappg, Expoetated Iverted Webull, New Webull Pareto, Qualty cotrol, Cotrol Charts I. INTRODUCTION Qualty s oe of the most essetal characterstcs of ay product. Motorg ad cotrollg qualty s a cotuous process. Statstcal Process Cotrol (SPC) s a qualty cotrol techque usg statstcal methods for motorg ad cotrollg the qualty. Cotrol charts are oe of the most mportat tools of SPC. The cotrol charts are costructed usg the sample data o qualty characterstc of a product, obtaed from the producto process. A cotrol chart cossts of a Cetre Le (CL) represetg the average of the qualty characterstc. Aother two les horzotal to the CL, the Upper Cotrol lmt (UCL) above the CL ad the other Lower Cotrol Lmt (LCL) below the CL are draw o the cotrol chart. These cotrol les are costructed such a way that f a process s uder cotrol, the most of the observatos the sample fall wth the rego covered betwee UCL ad LCL. If ay pot the sample falls outsde ths rego the t s a dcato that the process s out of cotrol. A example of a cotrol chart the lear case s show hereuder. Fg. 1.1: Lear Cotrol Chart II. CONTROL CHARTS TO THE CIRCULAR DISTRIBUTIONS Laha ad Gupta (2011) appled the cocept of cotrol charts to crcular dstrbutos such as Vo mses, Wrapped Cauchy, Wrapped Normal ad Cardod dstrbutos ad costructed Cetral Ray (CR), Atclockwse Cotrol Ray (ACR) ad Clockwse Cotrol All rghts reserved by 49

2 Ray (CCR) to the above metoed crcular dstrbutos ad studed the mpact of smulato sze o the crcular varaces of ACR ad CCR agles of these dstrbutos, the effect of varato of parameters of crcular dstrbutos o the varaces of ACR ad CCR agles ad also studed the Average Ru Legth (ARL) ad Meda of Ru Legth (MRL) of crcular cotrol charts. The cotrol chart respect of Crcular dstrbutos as llustrated by Laha ad Gupta (2011) s furshed below. Fg. 2.1: Crcular Cotrol Chart for Mea Drecto The above fgure represets the collecto of cocetrc crcles1, 2,..., wth th crcle havg radus of uts. Cetral Ray represets the mea agle. The rego bouded by CCR- CR-ACR the atclockwse drecto s the acceptace rego ad the rego bouded by ACR-CCR the atclockwse drecto s the rejecto rego for the process for whch the cotrols charts are costructed. Takg a cue from ths work, a attempt has bee made ths paper to exted ths cocept of cotrol charts to two crcular dstrbutos developed by Srvasa Subrahmayam et al (2017), amely the Wrapped Expoetal Iverted Webull dstrbuto (WEIWD) ad Wrapped New Webull Pareto Dstrbuto(WNWPD). I ths Paper, for these two selected crcular dstrbutos, the cotrol charts at dfferet sets of values of parameters are costructed, the theoretcal values for CR, ACR ad CCR agles are obtaed, the acceptace rego ad rejecto rego at the theoretcal values are detfed ad the expaso ad cotracto of the acceptace rego wth respect to chage parameter values are studed. The estmates for CR, ACR ad CCR agles ad ther respectve crcular varaces are computed, the mpact of smulato sze ad sample sze o the ACR ad CCR agles ad ther crcular varaces are studed. Also, a effort has bee made to fgure out a reasoable smulato sze as well as sample sze for fdg the estmates for CR, ACR ad CCR agles respect of these two crcular dstrbutos. Here the reasoable smulato sze or sample sze s cosdered as the sze at whch the correspodg estmated agles of CR, ACR ad CCR are close to ther respectve populato values wth mmum varace. Also, the cotrol charts costructed for the selected dstrbutos, all agles are show degrees ad for the graphcal represetato of CR, ACR ad CCR agles, east as zero drecto ad atclockwse as sese of rotato was cosdered. Crcular Dstrbutos: A crcular radom varable a cotuous crcular dstrbuto g : 0, 2 s sad to be followg a crcular probablty desty fucto of g (θ) f ad oly f g has the followg basc propertes 1) g ( ) 0, 2) 2 0 g( ) d 1 3) g ( ) g ( 2 k ) g s perodc, for ay teger k (Marda,2000) (3) Crcular Mea ad Varace: The crcular mea ad crcular varace for a gve set of agles ca be computed as below. Crcular Mea: Let p be a pot o the crcumferece of the ut crcle correspodg to the agle, 1(1) the the crcular mea or the mea drecto of,,..., s defed to be the drecto of the resultat of the ut vectors OP, OP,..., OP.The 1 2 Cartesa coordates of P are (cos, s ) so that the cetre of gravty of these pots s (C, S) where (1) (2) All rghts reserved by 50

3 The value 2 2 C 1 cos ad S 1 Costructo of Cotrol Charts for Some Crcular Dstrbutos s R C S gves the legth of the resultat vector. The drecto of the resultat vector s the mea 0 drecto. The drecto s gve by 0 Crcular Varace: 1 S C ta ( / ) f S 0, C 0 1 ta ( / ) f C 0 0 S C 1 ta ( S / C) 2 f S 0, C 0 A useful measure of the dstace betwee two agles ad s 1 cos( ) Hece the dsperso of agles ,..., about ts mea drecto,.e. crcular varace V, s gve by 1 V D( ) [1 cos( )] 1 III. METHODOLOGY To carry out the teded study, radom samples have bee geerated through smulato from the selected dstrbutos. As the Cumulatve Dstrbuto Fuctos of Wrapped Expoetal Iverted Webull dstrbuto ad Wrapped New Webull Pareto dstrbuto are ot the closed form, the smulato was mplemeted usg the Acceptace ad Rejecto method. Fdg CR, ACR AND CCR Agles for Dfferet Smulato Szes: The Step wse procedure for fdg CR, ACR ad CCR for dfferet Smulato szes s detaled below. 1) For a selected smulato sze say, N ad for a fxed sample sze k, radom samples of k agles each betwee 0 to 2 degrees are geerated. 2) The crcular mea of the th sub sample cosstg of k observatos s computed. 3) Sce N s the smulato sze, steps 1 ad 2 are repeated for N tmes. Rug N smulatos results a array of crcular meas,,...,. 1 2 N 4) Arrage ths array of crcular meas ascedg order. The crcular mea of all N crcular meas s computed ad s the Cetral Ray (CR). 5) Takg the level of sgfcace as 5% ad by elmatg 0.025% crcular meas from both the eds of the sorted array of crcular meas, at the ed of the array ad u at the begg of the array ca be obtaed. Drawg a le at gves L u the Atclockwse Cotrol Ray (ACR) ad drawg a le at gves the the Clockwse Cotrol Ray(CCR). Fdg Estmates ad Varaces of CR, ACR AND CCR Agles for a Gve Smulato Sze: After computg the CR, ACR ad CCR agles for a selected smulato sze, for fdg estmates ad crcular varaces for CR, ACR ad CCR agles, the procedure detaled secto A above wll be terated for ffty tmes. After the completo of ffty teratos, arrays of CR, ACR ad CCR agles, each cosstg of ffty agles wll be obtaed, for a gve smulato sze. For obtag a estmated value of ACR agle for a gve smulato sze, the crcular mea of the array of ACR agles s computed. Smlarly, crcular mea of the array of CCR agles gves a estmated value of CCR agle ad mea of the array of CR agles gves a estmated value of CR agle for a gve smulato sze. The crcular varace of the arrays of CR ACR ad CCR agles ca also be computed. Chagg the smulato sze N from 250, 500, 1000, 2000 ad up to 10,000, the correspodg estmated values for CR, ACR ad CCR agles ad ther crcular varaces are computed. Applyg ths procedure o both the selected dstrbutos WEIWD ad WNWPD at dfferet sets of values to the parameters the estmates ad varaces of CR, ACR ad CCR agles are obtaed for each smulato sze. Fdg CR, ACR AND CCR Agles for Dfferet Sample Szes: For fdg CR, ACR ad CCR agles for a gve sample sze, the smulato sze wll be fxed at 1,000 ad the sample sze s vared from 5, 10 ad up to 100 steps of 10 the procedure detaled subsecto A above ths secto. L All rghts reserved by 51

4 Fdg Estmates ad Varaces of CR, ACRS AND CCR Agles for A Gve Sample Sze: Costructo of Cotrol Charts for Some Crcular Dstrbutos After computg the CR, ACR ad CCR agles for a selected sample sze as detaled secto C above, for fdg estmates ad crcular varaces of CR, ACR ad CCR agles, ths procedure s terated for ffty tmes. After completo of ffty teratos, arrays of CR, ACR ad CCR agles, each cosstg of ffty agles for a gve sample sze are obtaed. For obtag a estmated value of ACR agle for a gve sample sze, the crcular mea of the array of ACR agles s computed. Smlarly, crcular mea of the array of CCR agles gves a estmated value of CCR agle ad mea of the array of CR agles gves a estmated value of CR agle for a gve sample sze. The crcular varace of the arrays of CR ACR ad CCR agles ca also be computed. Chagg the sample sze k from 5, 10 ad up to 100 steps of 10, the correspodg estmated values for CR, ACR ad CCR agles ad ther crcular varaces are computed. Applyg ths procedure o both the selected dstrbutos WEIWD ad WNWPD at dfferet sets of values to the parameters the estmates ad varaces of CR, ACR ad CCR agles are obtaed for each sample sze. Fdg Theoretcal Values of CR, ACR AND CCR Agles: The Populato (or) theoretcal value of the CR, ACR ad CCR agles for the selected two dstrbutos are computed by takg suffcetly large value for N, the smulato sze. Theoretcal values were computed for CR, ACR ad CCR agles by cosderg smulato sze as oe mllo. IV. CONSTRUCTION OF CONTROL CHARTS FOR WEIW DISTRIBUTION The probablty desty fucto of Wrapped Expoetated Iverted Webull Dstrbuto s ( c1) ( 2 k) c g ( ) c ( 2 k ) e k 0 Fve cases, each wth dfferet set of values to the parameters c where 0,2, c 0 ad 0., Table - 1 Cases cosdered for WEIWD Parameter Case 1 Case 2 Case 3 Case 4 Case 5 c Theoretcal Values of CR, ACR ad CCR Agles for Dfferet Cases: ad as detaled below are cosdered for the study. Usg the procedure explaed subsecto E of Secto III, the theoretcal values of parameters for WEIWD for the cases metoed above are computed ad tabulated. The tables alog wth correspodg graphs for case 1 s preseted below. Case 1: C 2 ad 2 Table 2 Theoretcal Values for Case 1 c CR ACR CCR Fg. 3: Graph of CR, ACR ad CCR for Case 1 All rghts reserved by 52

5 The acceptace rego the fgure 3 s the rego bouded by CCR CR ACR.e. from to the atclockwse drecto takg east as zero drecto. The Theoretcal Values obtaed for the rest of the cases are tabulated hereuder. Table - 3 Theoretcal values Obtaed for other Cases Aalyss of Theoretcal Values: Case c CR ACR CCR Theoretcal values for CR, ACR ad CCR agles for the Wrapped Expoetated Iverted Webull dstrbuto at dfferet values for parameters c ad are computed to study the expaso ad cotracto of the acceptace rego respose to the chages the value of parameters. The results are furshed below. Table 4 Theoretcal Values of CR, ACR ad CCR of WEIWD Sl. No c CR ACR CCR ACR -CCR Chage (1) (2) (3) (4) (5) (6) (7) = (5)-(6) (8) I the table 4, the colum (7) dcates the sze of the acceptace rego whereas colum (8) dcates the chage occurred the sze of the acceptace rego, whe the value of oe of the two parameters s creased by a oe ut. A postve value colum (8) dcates the expaso of acceptace rego ad a egatve value dcates the cotracto of acceptace rego. It ca be observed that wheever the value for the parameter c s creasg, the acceptace rego s decreasg. O the other had, wth the crease value for the parameter the acceptace rego s expadg. But the magtude of chage happeg both drectos gradually decreasg wth creasg values for the parameters c ad. Fdg CR, ACR AND CCR Agles ad Varaces for Dfferet Smulato Szes: For WEIW Dstrbuto, the CR, ACR, ad CCR agles ad ther respectve varace for Case 1 at parameter values c 2 ad 2, for dfferet smulato szes ragg from 250 to 10,000 are obtaed as tabulated below. Table 5 CR, ACR ad CCR Agles wth Crcular Varace Smulato Sze CR CVCR 1 ACR CVACR 2 CCR CVCCR Crcular Varace of Cetral Ray, 2. Crcular Varace of Atclockwse Cotrol Ray, 3. Crcular Varace of Clockwse Cotrol Ray The graph of crcular varaces for ACR agles for dfferet smulato szes s obtaed as below. All rghts reserved by 53

6 Fg. 4: Crcular Varace of ACR Agles The graph of crcular varaces for CCR agles for dfferet smulato szes s obtaed as below. Fg. 5: Crcular Varace of CCR Agles Comparg wth the theoretcal values of CR, ACR ad CCR agles for WEIWD, at c 2 ad 2 tabulated at table 2, wth values obtaed for CR, ACR ad CCR agles for dfferet smulato szes at table 5, It ca be observed that wth the creasg smulato sze, the CR, ACR ad CCR agles are close to ther respectve theoretcal values. Also, from fgures 4 ad 5, the crcular varaces of ACR ad CCR are moderate whe the smulato sze s aroud 250 to 500 but wth the creasg smulato sze, especally after 1000, a sharp reducto the crcular varaces ca be otced fally reachg to zero for hgher smulato szes. Also, the results for Cases 2, 3, 4 ad 5 are obtaed ad all of these outcomes cofrm the tedeces of CR, ACR ad CCR agles observed the frst case. Fdg CR, ACR AND CCR Agles ad Varaces for Dfferet Sample Szes: For WEIW Dstrbuto, the CR, ACR, ad CCR agles ad ther respectve varace for Case 1 at parameter values c 2 ad 2, for dfferet sample szes ragg from 5 to 100 are computed ad are tabulated below. Table 6 CR, ACR ad CCR Agles wth Crcular Varaces Sample Sze CR CVCR 1 ACR CVACR 2 CCR CVCCR Crcular Varace of Cetral Ray, 2. Crcular Varace of Atclockwse Cotrol Ray, 3. Crcular Varace of Clockwse Cotrol Ray The graph of CR, ACR ad CCR agle for dfferet sample szes s furshed below. All rghts reserved by 54

7 Fg. 6: CR, ACR ad CCR agles of WEIWD for dfferet Sample Szes The graph of crcular varaces for ACR agles for dfferet sample szes s obtaed as below. Fg. 7: Crcular Varace of ACR Agles The graph of crcular varaces for CCR agles for dfferet sample szes s obtaed as below. Fg. 8: Crcular Varace of CCR Agles From the fgure 6, t s clear that wth creasg sample sze the ACR ad CCR agles are comg close to CR agle. Also, from fgures 7 ad 8 the crcular varaces of ACR ad CCR are reachg zero wth creasg sample sze after a sharp reducto crcular varace at sample sze 10. Also, the results for Cases 2, 3, 4 ad 5 are also obtaed ad all of these outcomes cofrm the tedeces of CR, ACR ad CCR agles observed the frst case. V. CONSTRUCTION OF CONTROL CHARTS FOR WNWP DISTRIBUTION The probablty desty fucto of Wrapped New Webull Pareto Dstrbuto (WNWPD) s c 1 c ( 2 k ) g ( ) k 0 e c ( 2 k ), where 0, 2, c 0, 0 ad 0. Fve cases, each wth dfferet set of values to the parameters c, ad are cosdered for the study. All rghts reserved by 55

8 Table 7 Cases cosdered for WNWPD Parameters Case 1 Case2 Case 3 Case 4 Case 5 c Theoretcal Values of CR, ACR AND CCR Agles for Dfferet Cases: Costructo of Cotrol Charts for Some Crcular Dstrbutos Usg the procedure explaed subsecto E of Secto III, the theoretcal values of parameters for WNWPD for the cases metoed above are computed ad tabulated. The tables alog wth correspodg graphs for case 1 s preseted below. Case 1: C 2, 2 ad 2 Table 8 Theoretcal Values for Case 1 c CR ACR CCR Fg. 9: Graph of CR, ACR ad CCR for Case 1 From the fgure 9, the acceptace rego s from to the atclockwse drecto ad east as zero drecto. The rego excludg ths the crcle s the rejecto rego. The Theoretcal Values obtaed for the rest of the cases are tabulated hereuder. Table 9 Theoretcal Values obtaed for other cases Aalyss of Theoretcal Values: Case c CR ACR CCR Theoretcal values for CR, ACR ad CCR agles for the Wrapped New Webull Pareto dstrbuto at dfferet values of parameters c, ad are computed to study the expaso ad cotracto of the acceptace rego respose to the chages the value of parameters. The results are furshed below. Table 10 Theoretcal Values of CR, ACR ad CCR of WNWPD Sl. No c CR ACR CCR ACR -CCR Chage (1) (2) (3) (4) (5) (6) (7) (8) = (6) - (7) (9) All rghts reserved by 56

9 From the table 10 above, t ca be observed that wheever the value for the parameter c ad are creasg the acceptace rego s shrkg. Also, relatvely more cotracto of acceptace rego s happeg whe the parameter c s creased tha. O the other had, wheever the value for the parameter s creased the acceptace rego s expadg. Fdg CR, ACR AND CCR Agles ad Varaces for Dfferet Smulato Szes: Case 1: WNWPD at C 2, 2 ad 2 For WNWP Dstrbuto, the CR, ACR, ad CCR agles ad ther respectve varace for Case 1,.e. at parameter values C 2, 2 ad 2, for dfferet smulato szes ragg from 250 to 10,000 are obtaed as tabulated below. Table 11 CR, ACR ad CCR Agles wth Crcular Varace for Case 1 Smulato Sze CR CVCR 1 ACR CVACR 2 CCR CVCCR Crcular Varace of Cetral Ray, 2. Crcular Varace of Atclockwse Cotrol Ray, 3. Crcular Varace of Clockwse Cotrol Ray The graph of crcular varaces for ACR agles for dfferet smulato szes s obtaed as below. Fg. 10: Crcular Varace of ACR Agles for Case 1 The graph of crcular varaces for CCR agles for dfferet smulato szes s obtaed as below. Fg. 11: Crcular Varace of CCR Agles for Case 1 All rghts reserved by 57

10 Comparg wth the theoretcal values tabulated at table 8 for CR, ACR ad CCR for WNWPD at C 2, 2 ad 2 wth values obtaed for CR, ACR ad CCR for dfferet smulato szes furshed above at table 11, It ca be otced that wth the creasg smulato sze, the CR, ACR ad CCR agles are approachg to ther respectve theoretcal values. Also, from fgures 10 ad 11 the crcular varaces of ACR ad CCR are more whe the smulato sze s aroud 250 ad 500 but wth the creasg smulato sze, especally after 1000, t s clearly evdet that after a sharp reducto the crcular varaces are attag zero for both ACR ad CCR agles. Also, the results obtaed for Cases 2, 3, 4 ad 5 cofrms the tedeces of CR, ACR ad CCR agles, observed the case 1. Fdg CR, ACR ad CCR Agles ad Varaces for Dfferet Sample Szes: Case 1: WNWPD at C 2, 2 ad 2 For WNWP Dstrbuto, the CR, ACR, ad CCR agles ad ther respectve varace for Case 1,.e. at parameter values of C 2, 2 ad 2 for dfferet sample szes ragg from 5 to 100 are computed ad are tabulated below. Table 12 CR, ACR ad CCR Agles wth Crcular Varace for Case 1 Sample Sze CR CVCR 1 ACR CVACR 2 CCR CVCCR Crcular Varace of Cetral Ray, 2. Crcular Varace of Atclockwse Cotrol Ray, 3. Crcular Varace of Clockwse Cotrol Ray The graph of CR, ACR ad CCR agle for dfferet sample szes s furshed below. Fg. 12: CR, ACR ad CCR agles for Case 1 The graph of crcular varaces for ACR agles for dfferet sample szes s obtaed as below. Fg. 13: Crcular Varace of ACR Agles for Case 1 The graph of crcular varaces for CCR agles for dfferet sample szes s obtaed as below. All rghts reserved by 58

11 Fg. 14: Crcular Varace of CCR Agles for Case 1 From the fgure 12, t s clear that wth creasg sample sze the ACR ad CCR agles are comg close to CR agle. Also, from fgures 13 ad 14 the crcular varaces of ACR ad CCR are more whe the sample sze s aroud 5 ad 10 but wth the creasg sample sze, especally after 10, t ca be otced that after a sudde fall the crcular varaces reachg zero for both ACR ad CCR agles. The results obtaed for Cases 2, 3, 4 ad 5 also cofrms the tedeces of CR, ACR ad CCR agles, observed the Case 1. Effects of Parameters o Acceptace Rego: VI. CONCLUSIONS For both the crcular dstrbuto, the acceptace rego the cotrol charts s shrkg wth the creasg value of the shape parameter c. O the other had wth the creasg value of the acceptace rego s expadg. I the Wrapped New Webull Pareto dstrbuto, whe s creasg, the acceptace rego s decreasg. Effects of Smulato Sze o CR, ACR ad CCR agles: I both the crcular dstrbutos, t ca be otced that wth the creasg smulato sze, the CR, ACR ad CCR agles are close to ther respectve theoretcal values ad ther respectve crcular varaces are approachg to zero. Also, t s otced that the smulato sze 1000 would be a reasoable smulato sze for both the crcular dstrbutos. Effects of Sample sze o CR, ACR ad CCR agles: I both the crcular dstrbutos, t ca be otced that wth the creasg sample sze, the ACR ad CCR agles are comg close to ther respectve CR agles ad also ther respectve crcular varaces are approachg zero. Also, t s otced that the sample sze 20 would be a reasoable sample sze for both the crcular dstrbutos. REFERENCES [1] Dattatreya Rao, A.V., Ramachadra Sarma, I., Grja S.V.S., (2007). O wrapped verso of some lfe testg models. Commucato Statstcs - Theory ad Methods, 36, [2] Flah A.H. Elsalloukh, E. Med ad M. Mlaova (2012), The Expoetated Iverted Webull Dstrbuto, Appl. Math. If. Sc. 6, No. 2, [3] Jammalamadaka S. Rao, Se Gupta, A., Topcs Crcular Statstcs, World Scetfc Press, Sgapore. [4] Laha, A.K, Gupta A., (2011), Statstcal Qualty Cotrol of Drectoal Data,2d IIMA Iteratoal Coferece o Advaced Data Aalyss, Busess aalytcs ad Itellgece, Ahmedabad, Dated 8-9 Ja [5] Laha, A.K. (2002). Cotrol Charts for agular observatos statstcs for qualty mprovemet, Edtos: Ach, A.B.; Chatterjee, A., ad Chattopadhyay, A.K., Ida assocato for productvty, qualty ad relablty, Kolkata. [6] Marda, K.V. ad Jupp, P.E. (2000), Drectoal Statstcs, Joh Wley, Chchester. [7] Srvasa Subrahmayam P, Dattatreya Rao A.V ad Grja S.V.S (2017), O Wrappg of Expoetated Iverted Webull Dstrbuto, IJIRST Iteratoal Joural for Iovatve Research Scece & Techology, ISSN (ole): , Volume 3, Issue 11, Aprl 2017, pg. o [8] Srvasa Subrahmayam P, Dattatreya Rao A.V ad Grja S.V.S (2017), O Wrappg of New Webull Pareto Dstrbuto, IJARR Iteratoal Joural of Advaced Research ad Revew, ISSN (ole): , Volume 2(4), Aprl 2017, pg. o [9] Tahr M. H, Gauss M. Cordero, Ayma Alzaatreh, M. Masoor, & M. Zubar, A New Webull-Pareto Dstrbuto: Propertes ad Applcatos, Commucatos Statstcs - Smulato ad Computato Vol. 45, Issue. 10,2016. All rghts reserved by 59