Robustness of DEWMA versus EWMA Control Charts to Non-Normal Processes

Transcription

1 Journal of Modern Appled Sascal Mehods Volume Issue Arcle Robusness of D versus Conrol Chars o Non- Processes Saad Saeed Alkahan Performance Measuremen Cener of Governmen Agences, Insue of Publc Admnsraon, Ryadh, Saud Araba Follow hs and addonal works a: hp://dgalcommons.wayne.edu/masm Par of he Appled Sascs Commons, Socal and Behavoral Scences Commons, and he Sascal Theory Commons Recommended Caon Alkahan, Saad Saeed (3) "Robusness of D versus Conrol Chars o Non- Processes," Journal of Modern Appled Sascal Mehods: Vol. : Iss., Arcle 8. DOI:.37/masm/ Avalable a: hp://dgalcommons.wayne.edu/masm/vol/ss/8 Ths Regular Arcle s brough o you for free and open access by he Open Access Journals a DgalCommons@WayneSae. I has been acceped for ncluson n Journal of Modern Appled Sascal Mehods by an auhorzed edor of DgalCommons@WayneSae.

2 Journal of Modern Appled Sascal Mehods Copyrgh 3 JMASM, Inc. May 3, Vol., No., /3/$95. Robusness of D versus Conrol Chars o Non- Processes Saad Saeed Alkahan Performance Measuremen Cener of Governmen Agences, Insue of Publc Admnsraon Ryadh, Saud Araba Exponenally weghed movng average () and double (D) conrol chars were desgned under he normaly assumpon. Ths sudy consders varous skewed (Gamma) and symmerc non-normal () dsrbuons o examne he effec of non-normaly on he average run lengh (ARL) performance of and D. ARL performances were nvesgaed and compared usng Mone Carlo smulaons. Resuls show ha D chars can be desgned o be robus o non-normaly, ha he ARL performances of and D chars were more robus o dsrbuons and D was more robus o non-normaly for larger values of he smoohng parameer. Key words: Average run lengh, conrol chars,, D, robusness, non-normaly. Inroducon A popular conrol char used o deec small shfs n a process mean s he (Robers, 959). In an effor o ncrease he sensvy of conrol chars o deec small shfs and drfs n a process, a double (D) conrol char was developed by Shamma and Shamma (99). Zhang () has conduced exensve sudes on D conrol chars for he mean. Lke mos commonly used conrol chars, he radonal and D conrol chars for monorng process means were developed under he assumpon of normaly. The behavor of he conrol char performance for non-normal dsrbuons has been nvesgaed. Borror, e al. (999) used he Markov chan mehod and smulaons o sudy he average run lengh (ARL) performance of he conrol chars for he mean of Saad Saeed Alkahan s an Asssan Professor of Appled Sascs. Dr. Alkahan eaches courses n sascs such as SPC, regresson, mulvarae sascs, expermenal desgn, SAS and SPSS. He also provdes consulaons n sascs and has publshed several arcles. Emal hm a: alkahansas@yahoo.com. skewed (gamma) and heavy-aled () symmerc non-normal dsrbuons. They concluded ha he ARL performance of a well-desgned conrol char was robus o volaons of he normaly assumpon. As a par of an exensve sudy of he effec of non-normaly and auo-correlaon on he performance of conrol chars, Soumbos and Reynolds () concluded ha some combnaons of conrol chars for deecng small shfs n a process mean and/or varance can be desgned o be robus o he volaon of normaly assumpon. Mongomery (5) found ha an appropraely desgned char wll perform well even for nonnormal daa. Smulaon sudes on he robusness of an conrol char for process mean monorng have been conduced by Borror, e al. (999) and by Kosh and Kalgonda (). In addon, Human, e al. () ran an exensve smulaon o sudy he robusness of an conrol char for ndvdual observaons. They nvesgaed he n-conrol robusness of he desgns suded by Borror, e al. (999) and found ha, wh some ypes of non-normal daa, cauon should be aken no o overuse chars. Sudes relaed o he concep of robusness o non-normaly of he conrol sasc have also been conduced. For 8

3 SAAD SAEED ALKAHTANI example, Ln and Chou () nvesgaed he robusness of and - conrol chars wh varable samplng nervals o nonnormaly; Shau and Hsu (5) suded he robusness of he conrol char o nonnormaly for auo-correlaed process; and Calzada and Scarano (3) nvesgaed he robusness of he Max conrol chars o he volaon of normaly. The robusness o non-normaly of D conrol chars for deecng shfs n a process mean has no been nvesgaed; hus, hs sudy consders he robusness of D and compares o usng Mone Carlo smulaons. Background If X, where =,,..., m, s a sequence of random varables aken from a normal dsrbuon wh mean and varance σ, hen he conrol sasc s defned as: Z = λx + ( λ) Z (.) where < λ s a smoohng parameer and Z = (.e., he n-conrol or arge process mean). Typcally, s unknown and s esmaed from he prelmnary sample by he sample mean X. The conrol lms for he conrol char are: λ UC + Lσ ( λ) λ C λ LC Lσ ( λ) λ (.) UCL C LCL = + L λ σ λ λ σ λ = L The D conrol sasc ( λ) ( λ) Y = λx + Y Z = λy + Z (.3) Z s defned as: (.) such ha < λ< and Y = Z =. I can be shown ha (see Appendx A) and Z = λ ( + )( λ) X = + λ( λ) Y + ( λ) Z ( ) E Z = (.5) (.6) + ( λ) ( + + )( λ) ( )( λ) ( λ) 3 ( λ ) Z = σ λ σ ( ) (.7) The conrol lms for he D conrol char are: where L s he dsance beween he conrol lms and he cener lne (CL) measured n σ uns. For large values of, he conrol lms become: 9

4 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES UC + ( λ) ( + + )( λ) ( )( λ) ( λ) ( λ ) L 3 σ λ C LC ( ) + ( λ) ( + + )( λ) ( )( λ) ( λ) ( λ ) L 3 σ λ ( ) (.8) where L s as defned. For large values of, he conrol lms become: UC + Lσ C LC Lσ ( + ) λ λ λ ( λ ) 3 ( + ) λ λ λ ( λ ) 3 () For boh and D conrol chars, he conrol sascs n (.) and (.), respecvely, are ploed on he conrol char and he process s consdered o be ou of conrol f he ploed pon les ousde he LCL and UCL. Borror, e al. (999) used varous symmerc and skewed non-normal dsrbuons o sudy he robusness of he conrol chars for process mean. They consdered he dsrbuons wh dfferen numbers of degrees of freedom (.e., df =, 6, 8,, 5,, 3,, 5). The mean and he varance of he df dsrbuon are: = and σ = such df ha df >, respecvely. For he skewed dsrbuon, he auhors consdered Gamma dsrbuons denoed Gam ( α, β ) wh α =, 3,,,.5 and β =. The mean and varance of α he gamma dsrbuon are: = and β α σ =, respecvely. β Average Run Lengh Performances of vs D Ths sudy consders dsrbuons wh df =,,,, and 5 and Gam ( α, β ) wh α =, 3,,,.5 and β = o compare he effec of non-normaly on he RL performance of boh and D conrol chars. For λ =.,.,.3,.,.5, he values of L for boh conrol chars were chosen such ha he n-conrol ARL 37. (as used by Borror, e al. 999) when he process followed a normal dsrbuon. As shown heren, he robusness sudy of chars reproduces Borror s fndngs and hey were consdered here for he ease of comparson. Mehodology All RL calculaons were compleed based on, Mone Carlo smulaons for each scenaro, usng SAS V. 9. RANNOR and RANGAM funcons. The smulaons were conduced as follows:. Pseudo random numbers from normal, gamma, and dsrbuons were generaed by SAS V. 9. RANNOR and RANGAM funcons.. The conrol sascs Y for and for D were compued. Z 3. The conrol sasc was compared wh an expermenal LCL and UCL and a run lengh was obaned and recorded.. Afer, smulaon runs, he mean of he, derved RL (ARL) and he sandard error of ARL (SEARL) values were obaned. 5

5 SAAD SAEED ALKAHTANI 5. The values of L were chosen such ha he compued n-conrol ALR s almos equal o 37.. Seps from o were run for each scenaro of he combnaon of he prevously assgned dfferen values of he parameers λ, α, β, df and shfs. Tables and summarze he resuls of he calculaed L values and he n-conrol ARL along wh her SEARL n parenhess for and D conrol char for and Varous Gamma and dsrbuons. The followng are noed:. For Gamma and dsrbuons, he nconrol ARL for boh and D were reasonably close o 37. for small values of λ (.e., λ <.) especally wh larger α and df values of gamma and dsrbuons, respecvely (.e., when boh dsrbuons approach normaly).. The degree of n-conrol ARL deeroraon for boh and D was less for dsrbuon han for gamma dsrbuon. In general, for gamma and dsrbuons wh larger parameers α, β and df respecvely, he n-conrol ARL values for were beer (larger) han hose for D for small values of λ (λ <.). Conversely, he n-conrol ARL values for D were beer (.e., larger) han hose for for larger values of λ (λ >.) parcularly wh gamma dsrbuon. Tables 3 and show he ou-of-conrol ARLs for he and D Conrol Chars for Varous Gamma Dsrbuons and shfs n he mean measured n sandard devaon uns. For small shfs n he process mean (shf =.5) and λ >., he ou-ofconrol ARLs for boh and D are sgnfcanly less han he value ha would be obaned f he process was normal; oher han Table : In-conrol ARL for and D Conrol Char for Varous and Varous Gamma Dsrbuons (SEARL) D (SEARL) λ L () 37.3 () 37. (7) 37. () () 37. (5) 37.6 (5) 37.9 (8) 36 () (6) Gam(,) 33 () 6.5 (.56).7 (.) 59.9 (.57) 35.8 (.37) 33. () 59. (.59). (.) 6 (.6) 3 (.) Gam(3,) 3 (3.) 38.9 (.38) 8.6 (.8) (.).8 (.) 36.5 (3.) 39. (.39) 76. (.7). (.38).3 (.8) Gam(,) 36. (3.6) 6.3 (.) (.5) 3. (.3). (.) 97.3 (.9) 5.7 (.) 5. (.8). (.9) 3. (.) Gam(,) 7. (.69) 63. (.6) 7.6 (.7) 9 (.9) 77. (.78) 6. (.63) 6.7 (.63) 7.3 (.7) 9. (.9) 77.8 (.78) Gam(.5,) 8.9 (.) 33.9 (.3) 9.6 (.9) 75.8 (.75) 6.6 (.63) 8. (.3) 3 (.3) 9 (.95) 7.9 (.7) 6. (.6) 5

6 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES ha, he ARLs are comparable. Generally, he behavor of robusness o gamma dsrbuons of boh chars was smlar. Tables 5 and 6 show ou-of-conrol ARL s for he and D Conrol Chars for Varous Dsrbuons and shfs. The ARL performance of and D for boh and normal dsrbuons are comparable excep for he case ha shf =.5 and df < ; ha s, he dfference beween he ALRs of boh normal and dsrbuon s consderable. For he n-conrol case, he degree of ou-of-conrol ARL deeroraon for boh and D s less for dsrbuon han for gamma dsrbuon. Concluson The effec of non-normaly on he ARL performances for and D was nvesgaed usng Mone Carlo smulaons. SAS V. 9. RANNOR and RANGAM funcons were used o generae daa from varous normal, gamma, and dsrbuons and o perform he calculaons for all scenaros. Resuls show ha, n general, he nconrol ARL performances of boh and D conrol chars were more robus for he dsrbuon han for gamma. The degree of robusness of he and D conrol chars o non-normaly ncreased for smaller values of smoohng parameer and as he and gamma dsrbuons approach normaly. In addon, for gamma and dsrbuons, he nconrol ARL values for were more robus han hose for D for small values of λ (λ <.); however, he n-conrol ARL values for D were more robus han hose for for large values of λ ( λ >.) specfcally wh gamma dsrbuon wh larger parameers Smlarly o he n-conrol case, he ouof-conrol ARLs of and D were more robus for dsrbuon han for gamma. However, some deals needed o be consdered. I was noced ha he ou-of-conrol ARL for and D was sgnfcanly less for gamma comparng o normal-heory ARL for small shf (shf =.5) and large smoohng Table : In-conrol ARL for and D Conrol Char for Varous and Varous Dsrbuons (SEARL) D (SEARL) λ L () 37.3 () 37. (7) 37. () () 37. (5) 37.6 (5) 37.9 (8) 36 () (6) (8) 3.3 () 35.3 (6) 39.6 () 3.6 (3.6) 39. (6) 38.6 () 33 (3) 336. (6) 37. (3.6) (6) 3 () () 3.8 (3.9) 36. (3.9) 3.6 () 35.7 (8) 38.7 (3.) 3. (3.) 3 (3.) 35.8 (7) 3 (3.9) 3 (.97) 66.9 (.6) 5.8 (.58) 3.9 () 3 (3.9) 98.3 (.9) 68. (.65) (.58) 33. (3.) 8 (). (3).7 (.) 75. (.75) 37. (3.9) 76. (.7) 3 (.9).5 (.5) 8. (.75) 68.6 (.6) 87.5 (.8) 8. (.7) 7. (.6). (.) 6.3 (.6) 9.6 ().6 (.38) 8. (.9).5 (.) 5

7 SAAD SAEED ALKAHTANI value ( λ >.) and comparable oherwse. For he dsrbuon, he ou-of-conrol ARL for and D were comparable o he normal-heory ARL excep for shf =.5 and df <. In addon, for larger values of λ he ou-of-conrol ARLs for D were smlar or slghly beer han hose for. Shamma and Shamma (99) saed: Baxley (99) repored resuls for a smulaed ndusral process requrng a larger λ (λ=.35) bu he opmal char requres ha λ=.5. d chars wll be more sensve o cases whch can be bes modeled by d models wh larger values as compared o snce such values wll be non-opmal for chars (p. ). Based on boh he resuls of hs sudy and Shamma and Shamma s repor, D conrol chars should be consdered n pracce because he s non-opmal for larger values of λ. Also, he varably of he smulaed average run lengh for D s generally smaller han ha for. These properes should movae he use of D n ndusral process. References Baxley, R. V. (99). Dscusson. Technomercs, 3, 3-6. Borror, C. M., Mongomery, D. C., & Runger, G. C. (999). Robusness of he conrol char o non-normaly. Journal of Qualy Technology,, Calzada, M. E., & Scarano, S. M. (3). The robusness of he Max char o non-normaly. Communcaons n Sascs- Smulaons and Compuaons, 3(), Gradsheyn, I. S., & Ryzhk, I. M. (979). Table of negrals, seres, and producs. Walham, MA: Academc Press. Human, S. W., Krznger, P., & Chakrabor, S. (). Robusness of he conrol char for ndvdual observaons. Journal of Appled Sascs, 38(), Kosh, V. V., & Kalgonda, A. A. (). A sudy of robusness of he exponenally weghed movng average conrol char: a smulaon approach. Inernaonal Journal of Advanced Scenfc and echncal research, (), Ln, Y. C., & Chou, C. Y. (). Robusness of and combned X - conrol chars wh varable samplng nervals o non-normaly. Journal of Appled Sascs, 38(3), Mongomery, D. C. (5). Inroducon o sascal qualy conrol (5 h Ed.). New York, NY: John Wley & Sons, Inc. Robers, S. W. (959). Conrol chars ess based on geomerc movng average. Technmercs, (3), Shamma, S. E., & Shamma, A. K. (99). Developmen and evaluaon of conrol chars usng double exponenally weghed movng averages. Inernaonal Journal of Qualy & Relably Managemen, 9(6), 8-5. Shau, J. H., & Hsu, Y. C. (5). Robusness of he conrol char o nonnormaly for auocorrelaed processes. Qualy Technology & Quanave Managemen, (), 5-6. Soumbos, Z. G., & Reynolds, M. R. (). Robusness o non-normaly and auocorrelaon of ndvduals conrol chars. Journal of Sascal Compuaon and Smulaon, 66(), Zhang, L. Y. (). conrol chars and exended conrol chars. Unpublshed docoral dsseraon. Unversy of Regna, Saskachewan, Canada. 53

8 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES Table 3: Ou-of-conrol ARL s for he Conrol Chars for and Varous Gamma Dsrbuons λ =..698 λ =. 56 λ =.3.99 Dsrbuon Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Shf (Number of Sandard Devaons) (.8) 78.8 (.7) 76.6 (.69) 76. (.67) 7.5 (.68) 7 (.67) 9.3 (.7) 8. (.8) 7 (.77) 7.7 (.7) 67. (.65) 6.7 (.6) 9. (.6) 79. (.76) 7.9 (.7) 68.7 (.67) 6. (.59) 56.3 (.55) 7.9 (.9) 7.9 (.) 8.5 (.) 9. (.) 3. (.3) (.) 36. (.3) 33. (.9) 3 (.9) 3 (.9) 3. (.8) 3 (.9) 6.8 (.) 36. (.3) 3 (.3) 3 (.3) 33. (.3) 3 (.3) (.7).3 (.9).8 (.7) (.9).6 (.9) (.7)

9 SAAD SAEED ALKAHTANI Table 3 (connued): Ou-of-conrol ARL s for he Conrol Chars for and Varous Gamma Dsrbuons λ =..956 λ = Dsrbuon Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Shf (Number of Sandard Devaons) (.66) 75. (.7) 7.8 (.69) 6. (.6) 5.7 (.5) 7.8 (.6) 95. () 7.9 (.69) 66.7 (.66) 59. (.6) 9. (.8) (.) 59. (.56) 38.3 (.36) 36.5 (.35) 3 (.3) 3 (.3) 3. (.9) 7. (.68) 38.9 (.37) 36. (.35) 35.8 (.35) 3.3 (.3) 3. (.9).7.6 (.).7 (.) (.).9 (.).3 (.) (.). 5. (.) (.).8 (.) 55

10 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES D λ =. 96 D λ =..8 D λ =.3.69 Dsrbuon Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Table : Ou-of-conrol ARL s for he D Conrol Chars for and Varous Gamma Dsrbuons Shf (Number of Sandard Devaons) (.8) 78.3 (.69) 75.3 (.66) 75.3 (.69) 73.9 (.68) 7.6 (.67).7 (.6) 8. (.78) 79.9 (.77) 7.6 (.7) 67.9 (.6) 6 (.6) 9. (.) 78.8 (.76) 7. (.7) 68.5 (.65) 6. (.57) 5.7 (.5) 7.5 (.) 8. (.) 8.8 (.) 8.7 (.) 9.9 (.3) 3. (.) 36.6 (.3) 33. (.9) 33. (.9) (.9) 3.9 (.9) (.3) 6. (.) 36.7 (.33) 35. (.3) 33.9 (.3) 3 (.3) 3.9 (.3) (.7). (.7).6.9. (.9).3. (.9). (.9) 3. (.) (.3) (.)

11 SAAD SAEED ALKAHTANI D λ =..59 D λ = Table (connued): Ou-of-conrol ARL s for he D Conrol Chars for and Varous Gamma Dsrbuons Dsrbuon Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Gam(,) Gam(3,) Gam(,) Gam(,) Gam(.5,) Shf (Number of Sandard Devaons) (.69) 7 (.73) 6 (.68) 6. (.6) 5 (.5) 8.8 (.8) 9.6 (5) 7 (.7) 66.8 (.65) 58.5 (.59) 9.9 (.9).5 (.) 58.3 (.56) 37.8 (.36) 36. (.3) 3.8 (.33) 33. (.3) 3.6 (.3) 7.8 (.69) 38.8 (.37) 37.7 (.36) 35. (.3) 3.6 (.3) 3.9 (.3).9.7 (.) (.) (.) 3.9 (.) (.). (.) (.).8.8 (.) 57

12 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES λ =..698 λ =. 56 λ =.3.99 Table 5: Ou-of-conrol ARL s for he Conrol Chars for and Varous Dsrbuons Dsrbuon Shf (Number of Sandard Devaons) (.8) 87. (.78) 88.9 (.78) (.8) 9. (.8) 9.6 (.86) 9.3 (.7) 7. (.).8 (.6) 9.6 (.3) (.7). (.) 9. (.6) (.3) 3.9 (.).7 (.39) 6.5 (.) (.3) 7.9 (.9) 7.9 (.9) 8.5 (.) 8.5 (.) 8.3 (.) 3. (.) 36. (.3) 3 (.3) 36.6 (.3) 36.3 (.3) 37. (.3).3 (.35) 6.8 (.) 7. (.3) 6.5 (.3) 6.6 (.3) 7.6 (.5) 5.8 (.7) (.9) (.9)

13 SAAD SAEED ALKAHTANI λ =..956 λ = Dsrbuon 5 5 Table 5 (connued): Ou-of-conrol ARL s for he Conrol Chars for and Varous Dsrbuons Shf (Number of Sandard Devaons) (.66) 6 (.59) 6.9 (.6) 9.5 (.6) 8. (.8) 9 (.93) 95. () 8. (.77) 7. (.69) 57. (.5) 3. (.3) 87. (.88) 59. (.56) 58. (.55) 56.7 (.55) 56. (.5) 5. (.5) 56. (.53) 7. (.68) 69. (.67) 7.5 (.68) 66.7 (.65) 6.5 (.6) 58.8 (.57) (.) 6. (.) 8.6 (.7) (.) 6. (.) 8.6 (.7)

14 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES Table 6: Ou-of-conrol ARL s for he D Conrol Chars for and Varous Dsrbuons D λ =. 96 D λ =..8 D λ =.3.69 Dsrbuon Shf (Number of Sandard Devaons) (.8) 88. (.78) 87.8 (.78) 88. (.78) 89. (.79) 9 (.85).7 (.6). (.) 9.6 (.) 6.9 (.5).8 (.6) 7.3 (.) 9. (.).3 (.37). (.) 3.3 (.9) 5. (.).7 (.98) 7.5 (.) 7.9 (.) 8. (.) 7.8 (.) 8. (.) 3. (.) 36.6 (.3) 36.6 (.3) 36. (.3) 36.6 (.3) 37. (.3).9 (.36) 6. (.) (.) 6.6 (.) 5.3 (.) 7.6 (.5) 9.5 (.7) (.9)

15 SAAD SAEED ALKAHTANI D λ =..59 D λ = Table 6 (connued): Ou-of-conrol ARL s for he D Conrol Chars for and Varous Dsrbuons Dsrbuon 5 5 Shf (Number of Sandard Devaons) (.69) 6 (.6) 6.9 (.6) 6.9 (.5) 3. (.9) 93. (.9) 9.6 (5) 73. (.75) 6 (.67) 58.7 (.57) 5.8 (.) 85.7 (.86) 58.3 (.56) 56. (.5) 58.3 (.57) 58. (.55) 55. (.5) 5.6 (.53) 7.8 (.69) 68. (.65) 69.9 (.68) 66. (.65) 6.7 (.6) 57. (.56) (.) 8.3 (.7)

16 D VS. CONTROL CHART ROBUSTNESS TO NON-NORMAL PROCESSES Appendx A To oban he D sasc z defned n equaon (.5), repeaed subsuons were appled o equaons n (.) o oban y and z rewren as: y = λx + ( λ) y (A) = λ λ x + λ y and Subsung (A) no (A) resuls n: = ( ) ( ) ( λ) z = λy + z = λ( λ) y + ( λ) z = (A) k z = λ( λ) λ ( λ) x + ( λ) y + ( λ) z k = k= k ( ) ( ) ( ) ( ) = λ λ λ x + λ λ y + λ z = k= k l ( ) ( ) ( ) ( ) = λ λ λ x + λ λ y + λ z = l= l l λ ( l )( λ) xl λ( λ) y ( λ) z l= = (A3) Replacng l wh, (A3) s: z = λ ( + )( λ) x + λ( λ) y + ( λ) = z (A) where was assumed, whou loss of generaly, ha y = z =. The followng quanes were presened by Gradsheyn and Ryzhk (979). For a and ( ) a a na n n n+ k ka = (A5) k= a ( a) ( ) ( ) a+ a n+ a + n + n a n a n n+ n+ n+ 3 k ka = (A6) 3 k= are needed o prove he quaons (.6) and (.7). ( a) 6

17 SAAD SAEED ALKAHTANI Equaon (.6) Proof z Appendx A (connued) = E ( z ) = E λ ( + )( λ) x + λ ( λ) y + ( λ) z = λ λ λ λ λ ( )( ) E( x) ( ) E( y) ( ) E( z) = = λ = λ = + ( + )( λ) + λ( λ) + ( λ) Leng k = + n he frs erm λ k z = k( λ) + λ( λ) + ( λ) λ k= and applyng equaon (A.5) o he frs erm wh a = λ and n = resuls n Equaon (.7) Proof σ + λ ( λ) ( λ) ( λ) z = ( ) ( ) + λ λ + λ λ [ ( λ) ] ( λ ) ( λ) ( λ) = λ λ( λ) ( λ) + + λ λ = ( λ) λ( λ) + λ( λ) + ( λ) z = Var z Var x y z = [ ] = λ ( + )( λ) + λ( λ) + ( λ) = ( ) ( λ) ( ) = λ + + = λ + = ( ) ( λ) σ + = ( λ) Var x Applyng equaon (A.6) wh a = λ, k = + and n =, resuls n: σ λ = ( λ ) ( ) ( λ) + ( λ) ( + ) ( λ) + ( + )( λ) ( λ) ( λ ) ( λ) ( ) ( λ) ( )( λ) ( λ) 3 ( λ ) z = λ + + σ σ 63